Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Transcription

1 Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative Finance VU University Amsterdam March 2014 Abstract This thesis studies the hedging effectiveness of alternative hedging strategies for illiquid FX options. Generally seen the FX market is more liquid for spot transactions than for trades in FX options. We therefore focus on hedging the illiquid FX option against changes in the underlying volatility, since even for illiquid FX options; one can still easily invest in the underlying FX rate of the illiquid FX option and hedge the FX option price sensitivity against changes in the spot price of the underlying FX rate without making the hedged portfolio that much more illiquid. Trying to hedge an illiquid FX option against changes in the underlying volatility, we do not use hedging instruments depending on the FX rate of the illiquid FX option itself, as is preferably done, since it should be assumed that these hedging instruments suffer from similar illiquidity issues and would therefore make the hedged portfolio only more illiquid. Instead, we use liquid hedging instruments depending on other underlying FX rates option to construct our hedging strategies to hedge against volatility changes of the illiquid FX option. The backtest results show that our hedging strategies perform generally seen over time better than simply hedging against changes in the spot price of the underlying FX rate with the traditional BS-delta hedging strategy. Thus by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option itself, we are able to also reduce the volatility risk of an illiquid FX option instead of only reducing the spot price risk of the underlying FX rate, without making the hedged portfolio more illiquid. 1 Head of MRM Trading Quantitative Analytics FX, Credit, and CVA, ING Bank. 2 Quantitative Analyst, ING Bank. 3 Associate Professor at Faculty of Economics and Business Administration, VU University Amsterdam.

2 Thesis Summary Hedging illiquid FX options is a big issue for financial institutions holding large FX option portfolios in illiquid (emerging) markets. So far there is not yet that much literature found about hedging illiquid options, especially not when the underlying asset of the illiquid option is assumed to be an FX rate. This thesis studies the hedging effectiveness of alternative hedging strategies for illiquid FX options. The thesis mainly focus on hedging the illiquid FX option against changes in the underlying volatility, since even for illiquid FX options; one can still (easily) directly invest in the underlying FX rate of the illiquid FX option and hedge the illiquid FX option price sensitivity against changes in the spot price of the underlying FX rate without making the hedged portfolio that much more illiquid. However, trying to hedge the illiquid FX option against changes in the underlying volatility, the thesis does not use hedging instruments depending on the underlying FX rate of the illiquid FX option as is normally done, since it should be assumed that these hedging instruments suffer from similar illiquidity issues as the illiquid FX option suffers from (like high liquidation costs) and would therefore make the hedged portfolio only more illiquid. Instead, the thesis proposes to use liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option to construct the hedging strategies to hedge against underlying volatility changes of the illiquid FX option. The proposed alternative hedging strategies can be divided into empirical hedging strategies and into analytical model based hedging strategies. The empirical hedging strategies are based on a standard linear regression model where the hedging instrument weights are estimated with the ordinary least squares method, or they are based on a time-varying regression model where the hedging instrument weights are estimated with the Kalman Filter. The empirical hedging strategies are innovative since they are provided within a complete framework which fully explains which possible regressors to use while focusing entirely on hedging FX options, how to construct the regressors correctly based on historical data and how to estimate the unknown hedging instrument weights on a daily basis. The analytical model based hedging strategies are derived from the triangular relationship between currencies and from the stochastic intrinsic currency volatility framework of Doust. Both strategies are derived especially for options where the underlying asset is an FX rate, thus for FX options only. The analytical triangular hedging strategy uses the innovative insight of the triangular relationship between currencies, where one dependent FX rate can be written as the product of two independent FX rates with a common currency, to neutralize the illiquid FX option against changes in the independent volatilities instead of neutralizing the illiquid FX option against changes in its own underlying dependent volatility directly. The analytical intrinsic hedging strategy, derived from the sophisticated stochastic intrinsic currency volatility framework of Doust, is innovative compared to the other strategies in the sense that the analytical intrinsic hedging strategy focuses on hedging against changes in the intrinsic currency volatility instead of focusing on traditional hedging against changes in the traditional Black Scholes implied volatility (thus not intrinsic volatility) of the FX option. This framework provided in the thesis of intrinsically hedging against intrinsic currency volatility changes is as far as we know completely new. Generally seen, the main innovative contribution of this thesis is thus that it provides a complete empirical framework as well as a complete analytical model based framework to hedge illiquid FX option against underlying volatility changes without making the hedged portfolio more illiquid, based on the concept of using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. The thesis gives a backtest where the hedging performance of the different hedging strategies is extensively tested. From the backtest results we found out that all the hedging strategies perform generally seen over time better than our benchmark strategy. Thus besides of only reducing the spot price risk of the underlying FX rate, the hedging strategies are also able to reduce volatility risk of an 2

3 illiquid FX option without making the hedged portfolio more illiquid by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. Generally seen we can also state that the hedging strategies are quite robust in the sense that performing worse than benchmark strategy does not happen that often and when it happens it is usually not that much worse. 3

4 1 Introduction 6 2 Methodology Summary of used Existing Theory BS-model/Gahrman-Kohlhagen Valuation Model The Greeks The Volatility Smile Triangular Relationship between Currencies The Stochastic Intrinsic Currency Volatility Framework Description of our Approach Volatility Surface Interpolation General Introduction Hedging Strategies Empirical Hedging Strategies Analytical Triangular Hedging Strategy Analytical Intrinsic Hedging Strategy Hedge Effectiveness Measures The Backtest Data Dataset Dataset Results Estimated Volatility Smiles Main Backtest Results for (Dataset 1) Backtest Results based on the Cumulative at Expiration Measure for (Dataset 1) Main Backtest Results for (Dataset 1) Backtest Results based on the Cumulative at Expiration Measure for (Dataset 1) Main Backtest Results for (Dataset 2) Backtest Results based on the Cumulative at Expiration Measure for (Dataset 2) Conclusion and Recommendations Appendix A Estimated Volatility Smiles B vs. : Main Backtest Results for (Dataset 1) C vs. : Main Backtest Results for (Dataset 1) D vs. : Main Backtest Results for (Dataset 2) E vs. : Cumulative at expiration measure for (Dataset 1).256 4

5 F vs. : Cumulative at expiration measure for (Dataset 1). 263 G vs. : Cumulative at expiration measure for (Dataset 2). 270 H S.D.E. of Exchange Rate derived from Intrinsic Currency Values 276 I Derivation of Matrix

6 1 Introduction The Foreign Exchange (FX) market is a large decentralized market in which participants and financial institutions are able to trade currencies. Because of its enormous trading volume it is one of the largest asset class in the world with a high liquidity. The largest and most liquid part of the FX market can be assigned to spot transactions followed by trades in forward contracts, swap contracts and FX options. An FX option gives you the right to sell money in one currency and buy money in another currency at a fixed point in time and at a pre-determined FX rate. FX options are usually used by companies to reduce their currency risk on FX rates. A financial institution holding a large portfolio of FX options can reduce the exposure (hedge) to different risk factors, if they are unsure of which direction the market will go and they do not want to take the risk of losing large amounts of money. Hedging an FX option against changes in the spot price of the underlying FX rate (delta hedging) is done by increasing or decreasing your position in the underlying FX rate, while hedging against changes in the underlying volatility (vega hedging) is normally done by buying or selling other financial derivatives which are depending on the same underlying FX rate. It is convenient to (vega) hedge with liquid hedging instruments. The difference between liquid and illiquid FX options can be described as follows. Liquid FX options have a high open interest and are therefore frequently traded. These liquid FX options can be sold quickly for cash at the market price and contain a low bid-ask spread. Typically when an FX option is liquid, the underlying is the FX rate of a heavily traded currency pair. However it is possible to have an illiquid FX option, even when the underlying is the FX rate of a heavily traded currency pair. This is mostly seen when the option is far away from its expiration date and/or deep in/out-the money. For illiquid FX options, meaning that they are not so frequently traded because of a low open interest, it is harder to sell the FX option quickly for cash at the market price. Typically when an FX option is illiquid, the underlying is the FX rate of a not so heavily traded currency pair. An illiquid FX option contains a high bid-ask spread and often needs to be sold at discount. Furthermore it is hard to construct a (reliable) volatility smile for these illiquid FX options, since the needed data can be missing (not being updated) for many consecutive days. As a result of this low liquidity, it might be beneficial to (vega) hedge an illiquid FX option with hedging instruments depending on other underlying FX rates than the original underlying FX rate of the illiquid FX option itself. In this thesis we study the hedging effectiveness of alternative hedging strategies for illiquid FX options. The different hedging strategies which are introduced by us can be divided into empirical hedging strategies and into analytical model based hedging strategies. The empirical hedging strategies are based on a standard linear regression model or on a time-varying regression model, while the analytical model based hedging strategies are derived from the original Black-Scholes (BS) option pricing model [1], the triangular relationship between currencies and derived from the stochastic intrinsic currency volatility framework introduced by Doust [2]. In general the FX market tends to be more liquid for spot transactions than for trades in FX options. In this study we therefore mainly focus on hedging the illiquid FX option against changes in the underlying volatility, since even for illiquid FX options; one can still (easily) directly invest in the underlying FX rate of the illiquid FX option and hedge the illiquid FX option price sensitivity against changes in the spot price of the underlying FX rate without making the hedged portfolio that much more illiquid. However, when we try to hedge an illiquid FX option against changes in the underlying volatility, we do not use hedging instruments depending on the underlying FX rate of the illiquid FX option (as is normally done), since it should be assumed that these hedging instruments suffer from similar illiquidity as the illiquid FX option suffers from (like high liquidation costs) and it would make the hedged portfolio only more illiquid. Instead, we use liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option to construct our hedging strategies for illiquid FX options and we analyze the hedge effectiveness of the hedging strategies by performing a backtest. 6

7 The outcomes of this study could be especially of interest for big financial institutions holding large portfolios of illiquid FX options, for example in emerging markets, and who want to reduce the volatility risk of these portfolios but without creating a more illiquid portfolio, i.e. increasing the liquidity risk. The layout of this thesis is as follows. Section 2 is a summary of the relevant existing theory we used and gives furthermore a description of the approach we developed to hedge illiquid FX options. Section 3 describes the different datasets being used to study the performance of the hedging strategies. We use two datasets, one big dataset for all the hedging strategies except for the intrinsic hedging strategy and one smaller dataset for all the hedging strategies, that is including the intrinsic hedging strategy. Section 4 shows the main backtest results of the performance of the different hedging strategies. The first part of Section 4 shows the backtest results according to the bigger dataset (i.e. excluding the intrinsic hedging strategy) for two different illiquid FX options and the second part of Section 4 shows the backtest results according to the smaller dataset (i.e. including the intrinsic hedging strategy) for one specific illiquid FX option. Chapter 5 gives some conclusive remarks and suggestions for further research. The appendix contains mainly the backtest results of the standard linear regression model and the time-varying regression model and some mathematical derivations we used to derive the intrinsic hedging strategy. 2 Methodology 2.1 Summary of used Existing Theory BS-model/Gahrman-Kohlhagen Valuation Model The original BS option pricing model [1] is widely used to price stock options. Garman and Kohlhagen [3] extended the BS-model to deal with the existence of two interest rates (one interest rate for each currency of a currency pair) in order to be able to price FX options. Each FX rate is corresponding to a specific currency pair. The notation we use for an FX rate is the corresponding currency pair is noted by. With the notation we mean that is the foreign currency and is the domestic currency. An FX rate is valued in the domestic currency, which is also known as the numeraire or base currency. An FX rate buying/selling one unit of the foreign currency, one has to pay/receive and can be interpreted as follow: by units of the domestic currency. For example, if we consider currency pair and the FX rate has a spot value of, it means one has to pay to buy, equivalently by selling one will receive. If we assume that an FX rate follows a geometric Brownian Motion (GBM) under the risk-neutral measure: where is a standard Wiener process, is the risk free rate of domestic currency, is the risk free rate of foreign currency and is the volatility of FX rate spot price returns; one can show that by applying Itô calculus to the solution for is given by [( ) ] 7

8 This means that is normally distributed with mean and variance. The payoff at maturity of a plain vanilla FX option (European put/call option) is given by [ ( )] where is the strike price (denoted in the domestic currency ), is the spot value of the underlying FX rate at time (also denoted in the domestic currency ), is the expiration time of the option and is a binary variable, which is for a call option and for a put option. Notice that FX options are mostly European style options. So the holder can only exercise at time The value of an FX option at time is computed as the discounted expected payoff at maturity, ( ) [ ] where is all the information available till time Solving Equation results in the pricing formula for European style FX options: ( ) [ ] where : time to maturity, forward price of underlying, ( ) distribution. ( ), and is the cumulative standard normal Equation is known as the Gahrman-Kohlhagen pricing formula for FX options [3]. However, we will refer to it as the BS option pricing formula, since it is an extension of the original BS option pricing formula for stock options. More information about Equations and FX options can be found in the book of Wystup [4] The Greeks In the BS-model, the Greeks of an option measure the sensitivity of the option price with respect to changes in the input parameters. Thus the Greeks are an important tool when one wants to hedge against the risk due to possible changes in those input parameters. The delta of an option measures the sensitivity of an option value with respect to small changes in the underlying spot price. The BSdelta is defined as The delta of a call option is a positive number between the range [ ] and the delta of a put option is a negative number between the range [ ]. If the underlying spot price changes by a small amount, that is the option price changes by. The delta changes over time since the time to maturity gets smaller and the underlying spot price changes randomly; therefore (continuously) rebalancing is needed for a proper delta hedge. 8

9 Another important Greek is called the vega. It measures the sensitivity of an options value with respect to changes in the options volatility. The vega is defined as where is the probability density function of the standard normal distribution. The vega value is always positive. By multiplying with, it can be interpreted as the value by which the option price increases, if the volatility increases with One can price an FX vanilla option by using the BS option pricing formula. The input parameters and are known and the input parameters are well observable from the market. The volatility is harder to observe and has to be estimated. By having the market prices of FX options and all the input parameters except for solve the BS option pricing formula for, one can use these available market prices and to find the implied volatility which was used to come up with those market prices. If the BS-model holds, the implied volatility has to be constant throughout. In reality one finds that the implied volatility changes from day to day and even on a specific day, it differs across strikes and time to maturities. This implies that the BS model does not hold and that the market does not value the returns of the underlying as log normally distributed. Often the implied volatility is observed as a (skewed) smile curve The Volatility Smile We now look at some financial derivatives which can be used to decompose the implied volatility. The long/short straddle is a financial derivative containing a long/short call and a long/short put on the same underlying with the same strike and the same time to maturity. If we have that the strike is equal to the underlying forward price, the straddle is said to be at the money The butterfly is a financial derivative containing a long strangle and a short straddle. A strangle is similar to the straddle, except that the call and the put have different strikes, so they do not have the same strike. The delta is quoted in terms of volatility as: where is the volatility corresponding to strike, is the volatility corresponding to strike for which the delta of the put option is and is the volatility corresponding to strike for which the delta of the call option is. The risk reversal is a financial derivative containing a long out the money call option and a short out the money put option on the same underlying and with the same time to maturity. The delta is quoted in terms of volatility as: From this it follows that the volatility of a delta call option (call option with delta value of ) is given by: 9

10 [ ] [ ] The volatility of a delta put option (put option with delta value of ) is given by: [ ] [ ] With these three volatility points: and ; a smile can be constructed by interpolating between those points. The smile can be extended with the points and if we know the and volatility values. An example of a volatility smile in the delta space is shown in Figure. Figure 2.1: Examples of volatility smiles in the delta space for different maturities where the underlying is the exchange rate corresponding to currency pair. It is common to denote the implied volatility smile in the delta space, but by knowing the delta and the corresponding implied volatility, we can retrieve the strike as { ( ) } and denote the implied volatility smile in the strike space. 10

11 2.1.4 Triangular Relationship between Currencies Consider the following FX rates: and corresponding respectively to three different currency pairs and. Assume that each FX rate follows a GBM like stated in Equation and that the Wiener processes are correlated as. Note that the product of the two FX rates and with common currency is a new FX rate: Thus by assuming to be the dependent FX rate, it can be written as the product of the two independent FX rates and (triangular relationship). Using Itô s-lemma, we can show that the process ( ) follows the stochastic differential equation ( ) ( ) [ ] [ ] ( ) [ ] Notice from that the instantaneous volatility of the stochastic process is This means that by knowing and one can compute. On the other hand, if you know and ; you can find analytically the implied correlation between the log returns of and the log returns of by inverting Equation : The knowledge of the triangular relationship between currencies (where one dependent FX rate can be written as the product of two independent FX rates with a common currency) is used to set up a hedging strategy for illiquid FX options (see section 2.2.4). For more information about structured 11

12 products based on triangular relationships between currencies we refer to the paper of Martin Haugh [5] The Stochastic Intrinsic Currency Volatility Framework The stochastic intrinsic currency volatility framework introduced by Doust [2] is used to set up a hedging strategy for illiquid FX options (see section 2.2.5). In recent years models which assume the volatility to be stochastic instead of constant or a deterministic function of time have become popular. The SABR model [6] of Hagan et al is such a model. The well-known SABR parameters which have to be estimated are: and. In the SABR model the parameter controls whether changes in the quantity being modelled is normally distributed ( ), log-normally distributed ( ) or something in between. The parameter is the stochastic volatility of the quantity being modelled. The parameter is the volatility of and parameter is the correlation between the two Wiener processes corresponding to the quantity being modelled and to its stochastic volatility. The parameter is a small quantity which is used to construct a perturbation expansion. The main result of the SABR model is an approximation formula for the implied BS option volatility. For the case of a log-normal SABR model where (common in FX option trading), the stochastic differential equations for the log-normal SABR model (using to denote the underlying forward quantity being modelled) and the approximation formula for the implied BS option volatility are given by: ( ( ) ) ( ) ( ) The parameters and are calibrated by fitting to market data. To use in Equation one chooses. Doust models the observed FX rate by using the concept of intrinsic currency values. The idea is to model an FX rate as ratios of the intrinsic currency values (foreign) and (domestic), such that 12

13 Notice that in contrast to the FX rate, the intrinsic currency values are not directly observable in the market, but various methods could be used to estimate them [7]. Suppose that are currencies, thus there are totally intrinsic currency values where. Choose then a valuation currency (numeraire) where. Doust shows that with this choice of numeraire and its associated risk-neutral measure, the stochastic processes ( ) produce the right risk-neutral processes for all FX rates for and also contain the required symmetries in terms of change of numeraire and with respect to taking the inverse and product operations. In and is a variable which is the same for all intrinsic currencies, is the risk-free rate in currency, is the intrinsic currency volatility of, is the volatility of, and and are Wiener processes. If we define and to be column vectors whose elements are and, the correlation matrix can be written as ( ) ( ) Considering, the stochastic intrinsic currency volatility framework has the following parameters which need to be estimated: intrinsic currency volatilities. symmetric matrix of correlations between the intrinsic currency values. volatility of volatility variables. symmetric matrix of correlations between the intrinsic currency volatilities. matrix of between all the intrinsic currency values and all the intrinsic currency volatilities Given, the main result of Doust is an approximation formula for the implied BS option volatility in terms of the parameters of the stochastic intrinsic currency volatility framework. Having estimated one can use it to price vanilla FX options with the simple BS option pricing formula. The implied volatility approximation formula for the stochastic intrinsic currency volatility model defined by is given by ( ( ) ) where 13

14 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The basic idea to estimate the parameters of the stochastic intrinsic currency volatility framework is to do a least squares fit between model and market data. To use choose The term is actually a small quantity which was used to construct a perturbation expansion to derive approximation formula defined by the model More details about how we use the stochastic intrinsic currency volatility framework of Doust to set up an hedging strategy for illiquid FX options are given in section

15 2.2 Description of our Approach Volatility Surface Interpolation The (historical) implied volatility surfaces are computed with the following algorithm: -Gather (we use Bloomberg) the historical time series of the following volatilities: at the money volatility, the 25 delta volatility,, the 25 delta volatility,, the 10 delta volatility, and the 10 delta volatility, (see section 2.1.3). - Do this for different given time to maturities (e.g.: 1 week, 2 weeks, 3 weeks, 1 month, 2 months, 3 months, 6 months, 9 months and 1 year). - Interpolate (linearly) at each time step, through all the different volatilities for the given maturities from the previous step. This gives you for each historical day and each time to maturity the corresponding volatility denoted by,, and. -For each historical day and for each compute the by using Equation in combination with the interpolated values from the previous step. - For the volatilities convert the delta space (for calls use the deltas: [ ] and for puts use the deltas: [ ]) in the strike space. This is done by using Equation. For we use the = since. - Interpolate through the volatilities with the spline method for the different strikes computed in the previous step, to get the volatility smile for all possible strike prices on historical day and time to maturity General Introduction Hedging Strategies Assume you have an illiquid FX option at time with a specific strike and expiration date. The option price changes over time due to changes in the input parameters and. We are interested in hedging the illiquid FX option against changes in the underlying spot and volatility without making the option portfolio more illiquid. In the perfect hedge case, one would like (during the ownership of the option) all the potential profits and losses of the illiquid FX option to be exactly offset by taking an opposite position in some replicating portfolio. For simplicity we assume throughout this study that the domestic and foreign risk free rates, and respectively, are equal to and do not change over time. This restriction can be relaxed. We assume that the volatility changes from day to day, across time to maturity as well as across moneyness, In the BS-model the volatility is assumed to be constant and a constant does not change. So there would be no need to hedge the option price against changes in the volatility. However, if we 15

16 compute the implied volatility, we see that it is not a constant. The volatility implied by the market is different for different times different time to maturities and different moneyness values. Denote the sensitivity of an option at time (valued by the BS-model) to small changes in the underlying spot and in the volatility by and respectively. A strategy to hedge a call option against changes in the spot price of the underlying FX rate and against changes in the volatility, could be to construct the following replicating portfolio : - At time compute the current BS-delta value of the call option and invest delta times in the underlying FX rate, that is. Also invest a normalized BS-vega weight =, in a straddle which is delta neutralized, that is ( ), where the call option which needs to be hedged and the ATM straddle (used as hedging instrument) have the same underlying FX rate and the same time to maturity. - Note that the BS-vega value of the ATM call and the ATM put option, needed to construct the ATM straddle, is equal since they have the same strike. Furthermore, an ATM Straddle can be considered as almost delta neutral. - At time (next day), sell your total delta position (computed the day before) at the new spot price of the FX rate, that gives a profit of ( ) ( ) and sell your straddle which was constructed the day before as ATM, that gives a profit of. Then invest again in the underlying FX rate and ( ) in a delta neutralized ATM straddle where =. - Repeat this procedure for every day till maturity. We refer to this analytical model based hedging strategy (derived from the BS option pricing model) as the simple BS-delta-vega hedging strategy. With this hedging strategy, the replicating portfolio has at each (discrete) time step the same delta and vega value as the option we would like to hedge has. By taking the opposite position in the replicating portfolio compared to the position you have in the option which needs to be hedged, you can hedge the option price against changes in the spot price of the underlying FX rate and against changes in the underlying volatility, since the hedged portfolio is then made neutral to delta and vega. In the perfect hedge against changes in the spot price of the underlying and underlying volatility, one would like the change in value of the FX option caused by the changes in those risk factors, to be exactly the same as the change in value of the replicating portfolio ( ). In our study, we want to hedge an illiquid vanilla FX option. Even though the FX option is assumed to be illiquid, one can still (easily) directly invest in the FX rate of the illiquid FX option and hedge the illiquid FX option against changes in the spot price of the underlying FX rate. Since the FX market 16

17 tends to be more liquid for spot transactions than for trades in FX options, the hedging strategy for the illiquid FX option against changes in the spot price of the underlying FX rate is therefore to always use a BS-delta hedge, by investing a weight of in the underlying FX rate. We mainly focus on hedging the illiquid FX option against changes in the underlying volatility. To hedge an illiquid FX option against changes in the volatility, it is in practice not really common to use FX options which have the same underlying FX rate as the illiquid FX option has. Since these options would most of the time also be illiquid and it would make the portfolio only more illiquid. Therefore, to hedge the illiquid FX option against changes in the underlying volatility without creating a more illiquid portfolio, we do not use FX options depending on the same FX rate as the FX rate of the illiquid FX option. Instead we use liquid FX options depending on other underlying FX rates to construct the following financial derivatives as hedging instruments to hedge the illiquid FX option against changes in the underlying volatility: - ATM straddle = delta RR = delta BF = [ ] We want the hedging instruments to be liquid and we want them to have significant explanatory power when using them to model the illiquid FX option price changes. Throughout this study we assume that the illiquid FX option has the underlying FX rate corresponding to currency pair. To select the underlying of the different hedging instruments, we use the following selection criteria: if the illiquid FX option has the underlying FX rate use the underlying FX rate and/or the underlying FX rate to construct the hedging instruments. We use the liquid currency for these hedging instruments since currency pairs containing the dollar value are more heavily traded than the currency pair which contains no dollar value. This gives the hedging instruments a high liquidity. Furthermore note that the product of and is exactly the FX rate We hope thus that the volatilities of those FX rates are (highly) correlated to the volatilities used to price the illiquid FX option. We assume that the illiquid FX option is bought at starting point and always kept till expiration date. The option therefore needs to be hedged all the time between starting point and expiration date and we refer to this time interval as a hedging cycle. The replicating portfolio is rebalanced on a daily basis and therefore the new hedging weights are recomputed daily. If we have a perfect hedge it would imply that the daily of the illiquid FX option is exactly equal to daily of the replicating portfolio. Thus once we have estimated the (daily) new weights of the hedging instruments, the illiquid FX option is then hedged by taking the opposite position in the replicating portfolio compared to the position you have in the illiquid FX option, in order to offset the daily of the illiquid FX option. All the hedging instruments are priced with mid-prices (obtained from the market). So we do not take into account transaction costs. This can be justified by the fact that we hedge with liquid hedging instruments and by the fact that the FX market for spot transaction is already quite liquid. High liquidity usually implies low bid-ask spreads and therefore we use the mid-prices to price each transaction since the bid-ask spread is considered to be small. The illiquid FX option is also priced with mid-prices, since it is bought just once at starting point and then kept till expiration. 17

18 Meanwhile no (daily rebalancing) trades happen and therefore we also use the mid-prices to value the illiquid FX option throughout a hedging cycle. We refer to the different financial derivatives by the following numbers: illiquid FX option; ATM straddle; 25 delta ; 25 delta. With the notation we mean the value of financial derivative, just after having rebalanced at time while the financial derivative is depending on underlying FX rate where With the notation we mean the value of financial derivative, at time just before going to rebalance while the financial derivative is depending on underlying FX rate where The BS-delta value and the BS-vega value at time, of financial derivative depending on underlying FX rate is denoted by and respectively; where. As benchmark strategy we take the simple BS-delta hedging strategy. Considering the simple BS-delta hedging strategy (derived from the BS option pricing model), the value of the replicating portfolio just after having rebalanced at time given by: This means that at time (just after you have rebalanced) you have a position in FX rate. We refer to this analytical model based hedging strategy as the simple BS-delta hedging strategy. Our goal is thus to perform at least as good as the simple BS-delta hedging strategy does perform. This would namely mean that besides of only reducing the spot price risk of the underlying FX rate of the illiquid FX option, you are also able to reduce the volatility risk of an illiquid FX option by using liquid hedging instruments depending on other underlying FX rates than the underlying FX rate of the illiquid FX option (i.e. hedging by not making the illiquid FX option portfolio more illiquid). In this study we also show the results of hedging with the simple BS-delta-vega hedging strategy described in where you (vega) hedge with a delta neutralized ATM straddle (also priced with mid-prices) depending on underlying FX rate. We must notice that in the context of this research, the results of the simple BS-delta-vega hedging strategy are actually not really comparable to the results of other hedging strategies, since our main assumption was that we do not want to hedge volatility with hedging instruments depending on the underlying FX rate because of illiquidity issues assumed to occur for financial derivatives depending on underlying FX rate. Furthermore the results of the simple BS-delta-vega hedging strategy are all based just on mid-prices and do not account for illiquidity issues and high transaction costs which should occur when you daily rebalance the hedging instruments depending on FX rate. We still add the results of this hedging strategy because they can give a nice indication how delta-vega hedging with the underlying FX rate itself 18

19 would have performed if the FX option was considered to be liquid. Thus if one of the hedging strategies would perform similarly to the simple BS-delta-vega hedging strategy, it would imply that you are able to reduce the same amount of volatility risk, by using hedging instruments depending on other underlying FX rates, as would also have been possible to reduce if the FX option was liquid and hedged with hedging instruments depending on the original underlying FX rate Empirical Hedging Strategies The empirical hedging strategies use a standard linear regression model or a time-varying regression model to estimate the weights needed to invest in the hedging instruments. We denote the FX rate now as. The weights which need to be estimated are denoted by and, depending on which hedging instrument you use (2 = ATM straddle; 3 = 25 delta ; 4 = 25 delta ) and the underlying currency pair. All the options are priced with Equation. For the empirical hedging strategies, we construct the following replicating portfolios where the values of the replicating portfolios just after having rebalanced at time are given by: 1. [ ] where [ ]. BS-delta position in FX rate, position of in a delta hedged ATM straddle based on, position of in FX rate. Referred to as empirical hedging strategy [ ] BS-delta position in FX rate, position of in a delta hedged ATM straddle based on. Referred to as empirical hedging strategy [ ] [ ] where [ ]. BS-delta position in FX rate, position of in a delta hedged ATM straddle based on, position of in FX rate position of in a delta hedged ATM straddle based on. Referred to as empirical hedging strategy [ ] [ ] [ ] [ ] 19

20 where [ ] [ ] and [ ]. BS-delta position in FX rate, position of in a delta hedged ATM straddle based on, position of in a delta hedged 25 delta based on, position of in a delta hedged 25 delta based on, position of [ ] in FX rate. Referred to as empirical hedging strategy [ ] [ ] [ ]. BS-delta position in FX rate, position of in a delta hedged ATM straddle based on, position of in a delta hedged 25 delta based on, position of in a delta hedged 25 delta based on. Referred to as empirical hedging strategy [ ] [ ] [ ] [ ] [ ] [ ] [ ] where [ ] [ ] and [ ]. BS-delta position in FX rate, position of in a delta hedged ATM straddle based on, position of in a delta hedged 25 delta based on, position of in a delta hedged 25 delta based on, position of [ ] in FX rate, position of in a delta hedged ATM straddle based on position of in a delta hedged 25 delta based on, position of in a delta hedged 25 delta based on. Referred to as empirical hedging strategy 6. We want the hedging instruments (ATM straddle, 25 delta, 25 delta ) only to be sensitive against changes in their own volatility. In that way we hope to find empirically a relation between the illiquid FX option price sensitivity (after being delta hedged) and hedging instruments price sensitivities caused by changes in the illiquid FX options volatility and changes in the hedging instruments volatility. That is why we always delta hedge the hedging instruments against changes in the spot prices of their underlying FX rates, since we not want them to be sensitive against these 20

21 changes. Note that an ATM straddle is almost delta neutral, as the delta of an ATM call option is close to and the delta of an ATM put option is close to. Because it not exactly delta neutral, we always delta hedge the ATM straddle, just for certainty. Since the illiquid option value is valued in its domestic currency value we take currency as the numeraire. Therefore we convert the values of all hedging instruments which are not valued in currency such that they are valued in currency. Thus we multiply the values of hedging instruments valued in the currency with the FX rate Because the FX rate is a sensitivity which changes over time, and we do not want to be sensitive against these changes, we also hedge against changes in the spot price of FX rate when we are converting to numeraire currency. Thus we take at time a position of [ ], [ ] and [ ] in to hedge against these changes when we are converting the hedging instruments valued in currency to numeraire currency. The values of the replicating portfolios at time by: just before you are going to rebalance are given 1. [ ] where [ ]. 2. [ ] 3. [ ] [ ] where [ ]. 4. [ ] [ ] [ ] [ ] where [ ] [ ] and [ ]. 5. [ ] [ ] [ ]. 6. [ ] [ ] [ ] 21

22 [ ] [ ] [ ] [ ] where [ ] [ ] and [ ]. The difference between the replicating portfolio value at time just after you have rebalanced and the replicating portfolio value at time just before you are going to rebalance, is the of the replicating portfolio value in the time interval [ ]. Because we rebalance daily, we will refer to it as the day at time For a good hedge we would like that for all times Note that when we are computing day at time of the replicating portfolio, that is, we get for the hedging instruments valued in the currency that: ( [ ] ) ( [ ] ) ([ ] [ ]) ( [ ] ) ( [ ] ) ([ ] [ ]) ( [ ] ) ( [ ] ) ([ ] [ ]) This will be useful when we estimate the unknown hedging instrument weights. and Empirical hedging strategies can be considered as parsimonious regression based hedging strategies while empirical hedging can be considered as the extended regression based strategies, since empirical hedging strategies contain less different hedging instruments (only an ATM straddle) compared to empirical hedging strategies (ATM straddle, 25 delta and 25 delta ). We have chosen to construct the parsimonious regression based hedging strategies with only the ATM straddle (possibly on two different underlying FX rates) as hedging instrument, because 22

23 it is simple to construct a straddle and the at the money property gives the straddle a good liquidity and makes it almost delta neutral. The extended regression based hedging strategies are constructed with the ATM straddle, 25 delta RR and 25 delta BF (also possibly on two different underlying FX rates) as explanatory hedging instruments, since a volatility smile is closely related to these financial derivatives. The algorithm we use for the empirical hedging strategies to come up with the unknown weights needed to invest in the hedging instruments is given below. As said earlier, the weights are recomputed daily since we rebalance on a daily basis, where we use the last historical trading days as window to construct the regressors. A time window scheme of how the empirical hedging strategies are used to recompute the daily hedging instrument weights is shown in Figure. Thus during a hedging cycle, that is from starting point till expiration date, the hedging instrument weights are recomputed on a daily basis from time till time with a historical window of the last trading days. A bigger window would contain more information, but because of volatility clustering we do not want the window to be too long and to look back too much in the past. Figure 2.2: A time window scheme of how the empirical hedging strategies are used to recomupute the daily hedging instrument weights. Algorithm: 1. Starting point 1.1. Illiquid European call FX option with underlying depending on FX rate. The illiquid European call FX option with underlying FX rate is bought at time with value and expires at time. The strike of this illiquid FX option is denoted by ; the volatility at time depending on underlying FX rate, time to maturity and strike is denoted by. The options moneyness at time is defined as. The options analytical BS delta at time is noted as We compute then for the last days, that is at time { } the day 23

24 historical if we would have bought the illiquid FX option on those historical days and sold it one day later. The illiquid FX option is bought at time { }. The historical FX rate spot values { } are all known, as historical strike prices we take { } (such that the historical moneyness is the same as the current moneyness of the illiquid FX option at time ), as volatility we take the historical volatility of that day, corresponding to the current time to maturity and corresponding to the historical strikes you buy the historical options at, that is { }. Furthermore we compute the historical deltas of these [ ] [ ] historical options denoted by { } The bought historical options at time { } are sold one day later at time { }. As input parameters to price the historical options when they are sold, we use the historical FX rate spot values { strikes stay the same, so they are { } the }, the time to maturity decreases with one day to and as the volatility we take { [ ] [ ] }. Having computed these historical one day buy and sell prices, we compute the day historical if you would have bought the option on the historical dates and sold it one day later. Denote the day historical of the illiquid option by the vector. Because we also perform a BS-delta hedge on the illiquid option, we need to compute for the last values of the underlying FX rates, that is { corresponding historical BS-deltas { by. days the historical changes in the spot }, and multiply it with the }. Denote this historical vector of changes 1.2. Liquid European ATM straddle with underlying depending on FX rate. The liquid European ATM straddle with underlying FX rate and it has the same expiration date is constructed at time, with value as the illiquid option has. The strike used to construct the ATM straddle at time is equal to the spot value of the FX rate, that is ; the volatility at time with underlying, time to maturity and strike is denoted by. We compute the call option value and the put option value with those input parameters and the sum of those two prices is the ATM straddle value at time, that is. The BS delta of the ATM straddle, is computed as the sum of the call option delta and the put option delta. Compute then for the last days { } the day historical if you would have constructed the liquid ATM straddle on those historical days and sold it one day later. The ATM straddle is bought at time { }. The historical FX rate spot values { } are all known, as historical strike prices we take { } (because we construct the historical straddles always ATM), as volatility we take the historical volatility of that day, corresponding to the current time to maturity and to the historical strikes you construct the straddles at, that is { }. We also compute the historical deltas of these straddles noted as { } The straddles constructed at time { } are sold one day later at time { } As input parameters to price the historical straddles when being sold we use the historical FX rate spot values { } The strike prices are { }, the time to maturity decreases with one day to and as the volatility we take { }. Having computed these historical day straddle buy and 24

25 sell prices, we compute the day historical if you would have constructed the ATM straddles on the historical dates and sold them one day later, denoted by Because we also perform a BSdelta hedge on the ATM straddles, we need to compute for the last 80 days the historical changes in the in the spot values of the underlying FX rates, that is { multiply it with the corresponding historical BS-deltas { }, and }. Denote this historical vector of changes by. The vector containing ATM straddles is given by: day historical profits of delta hedged. Multiply then the elements of vector with { } and denote this vector by Liquid European ATM straddle with underlying depending on FX rate. Repeat step 1.2; where the underlying FX rate changes from to. Stop when has been computed. 1.4 Liquid European 25 delta with underlying depending on FX rate. The liquid European 25 delta RR with underlying FX rate is constructed at time with value and it expires also at time as the illiquid option does. The used strike prices at time are the 25 delta call strike, denoted by and the 25 delta put strike, denoted by. They are found by using Equation for retrieving the strike when the delta is known. The volatility at time with underlying, time to maturity and strike ( ) is denoted by ( ). These are your input parameters needed to value the 25 delta RR with the BS-formula. We compute the 25 delta call option value and the 25 delta put option value with those input parameters and subtract the 25 delta put option value from the 25 delta call option value to compute the 25 delta value. The BS delta of the 25 delta at time is noted as. Compute then for the last 80 days { } your day historical if you would have constructed the liquid 25 delta on those historical days and sold it one day later. The 25 delta is constructed at time { }. The historical FX rate spot values { } are all known, as historical strike prices we take { } and { } for the call and the put option, respectively. As volatility we take the historical volatility of that day, corresponding to the current time to maturity and to the historical strikes you construct the 25 delta at, that is { } and { } for the call and the put, respectively. We also compute the historical deltas of these 25 delta, noted as { } The constructed 25 delta at time { } are sold one day later at time { } where we use the historical FX rate spot values { }. As strike prices use { } and { } for the call and the put, respectively. The time to maturity decreases with one day to and as the volatility we take { } and { } for the call and the put, respectively. Having computed these historical one day 25 delta buy and sell prices, we compute the day historical if we would have constructed the 25 delta on the historical dates and 25

26 sold them one day later. Denote these day historical 25 delta by the vector Because we also perform a BS-delta hedge on the 25 delta, we need to compute for the last days the historical changes in the spot values of the underlying FX rates, that is { { }. }, and multiply it with the corresponding historical BS-deltas Denote this historical vector of changes by. The vector containing the day historical of the delta hedged 25 delta is given by:. Multiply then the elements of vector with { } and denote this vector by. 1.5 Liquid European 25 delta with underlying depending on FX rate. Repeat step 1.4; where the underlying FX rate changes from to. Stop when has been computed. 1.6 Liquid European 25 delta with underlying depending on FX rate. The liquid European 25 delta with underlying FX rate is constructed at time with value and matures also at time as the illiquid option does. The used strike prices at time are equal to the 25 delta call strike, denoted by, the 25 delta put strike, denoted by and to the ATM strike price, denoted by. The volatility at time with time to maturity and strike is denoted by, the volatility at time with time to maturity and strike is denoted by and the volatility at time with time to maturity and strike is denoted by. These are your input parameters needed to value the 25 delta with the BS-formula. Compute the 25 delta call option value, the 25 delta put option value, the ATM call option value and the ATM put option value with those input parameters and subtract then the ATM call and ATM put option values from the 25 delta put option value and the 25 delta call option to compute the 25 delta value. The BS delta of the 25 delta at time is denoted by. Compute then also for the last days { } your day historical if you would have constructed the liquid 25 delta on those historical days and sold it one day later. The 25 delta is constructed at time { }. The historical FX rate spot values { } are all known, as historical strike prices we take { }, { } and { } for the 25 delta call, 25 delta put and ATM call/put, respectively. As the volatility we take the historical volatility of that day, corresponding to the current time to maturity and to the historical strikes you construct the 25 delta at, that is { }, { } and { } for the 25 delta call, 25 delta put and ATM call/put, respectively. We also compute the historical deltas of these 25 delta, that is { } The constructed 25 delta at time { } are sold one day later at time { } where we use the historical FX rate spot values { }. As strike prices we use { }, { } and { } for the 25 delta call, the 25 delta put and ATM call/put, respectively. The time to maturity decreases with one 26

27 day to and as the volatility we take { }, { } and { } for the 25 delta call, the 25 delta put and ATM call/put, respectively. Having computed these historical one day 25 delta buyand sell prices, we compute the day historical if you would have constructed the 25 delta on the historical dates and sold them one day later. Denote these day historical 25 delta profits by the vector. Because we also perform a BS-delta hedge on the 25 delta, we compute for the last days the historical changes in the spot values of the underlying FX rates, that is { }, and multiply it with the corresponding historical BS-deltas { }. Denote this historical vector of changes by. The vector containing the day historical of the delta hedged 25 delta is given by:. Multiply then the elements of vector with { } and denote this vector by. 1.7 Liquid European 25 delta with underlying depending on FX rate. Repeat step 1.6; where the underlying FX rate changes from to. Stop when has been computed. 1.8 Find the optimal weights needed to invest in the hedging instruments at time. As an example we take the replicating portfolio of empirical hedging strategy 6. The next step is to find the optimal replicating portfolio weights at time. This is done by linearly regressing and on. on, The regression equation is: Having found the estimated parameters and via ordinary least squares construct our weights for the hedging instruments at time :, we [ ] 27

28 [ ] [ ] Thus the rebalanced replicating portfolio at time is constructed as follow: BS-delta position in FX rate, position of in a delta hedged ATM straddle depending on underlying, position of in a delta hedged 25 delta depending on underlying, position of in a delta hedged 25 delta depending on underlying, position of [ ] in FX rate, position of in a delta hedged ATM straddle depending on underlying, position of in a delta hedged 25 delta depending on underlying, position of in a delta hedged 25 delta depending on underlying. The illiquid FX option is then hedged by taking the opposite position in the replicating portfolio. When we have estimated the parameters and by, we check with the t-test statistic for each estimated parameter whether it is significant or not. If the estimated parameter is not significant, we set. This means that although empirical hedging strategy 6 is initially defined by hedging with an ATM straddle, 25 delta and 25 delta, the hedging strategy can reduce if one of the estimated parameters is not significant. is to use a regression model with time- Another method to construct the portfolio weights at time varying coefficients following a random walk. State equation: Observation equation: Initial state distribution: 28

29 For empirical hedging strategy 6 we get: ( ) ( ) [( ) ( ) ( ) ( ) ( ) ( ) ] with we mean the element of vector. ( ) [ ] ( ) The mean and variance of the unobserved state vector (dimensions for empirical hedging strategy 6 are ) can be computed by the Kalman filter, given the observations. Writing { } [ ] and, we can derive the updating scheme: 29

30 Given the data vector (assume dimension is ), the unknown parameter vector [ ] corresponding to dataset is estimated by Maximum Likelihood. The model we use is completely Gaussian and the Gaussian loglikelihood is given by ( ) with and constructed from the Kalman filter. Estimation of the unknown parameter vector proceeds by numerically maximizing with respect to. Note that maximizing with the restriction, we get that (matrix) and the time-varying regression model reduces to the standard linear regression model with. In contradiction to the linear regression model with no time-varying coefficients, when we estimate the hedging instrument weights with time-varying coefficients obtained from the, the hedging strategy is not reduced but kept as initially defined. Having run the for the observation vector, our best prediction of is. So at time we compute our forecast ( vector) and construct our weights for the hedging instruments: [ ] [ ] [ ] For more information about the see [8]. 2. Time till time Repeat the whole procedure. At time till time some additional actions happen on a daily basis because of the rebalancing. As an example we will look at day. Thus we repeat the whole procedure. Note that the strike price of the illiquid option does not change over time, it is still. The spot value of the underlying FX rate of the illiquid FX option changes to 30

31 and therefore also the moneyness changes to. The time to maturity decreases with one day to. The volatility is updated for the new day across the new time to maturity and the needed strike. The historical moving window is kept fixed at the last data points. Because the BS-delta and the optimal replicating portfolio weights change over time, we rebalance our replicating portfolio daily according to the new delta and the newly found optimal weights from and/or the. This means we sell our old BS-delta position computed the day before (that is at time ), giving us a day of ( ) and we invest the newly computed delta weight at time in the FX rate, that is. Furthermore we sell our old position in the straddle, bought at time (day before) when it was ATM, giving us a profit of ([ ] [ ] ). The same happens for the 25 delta and the 25 delta, which gives a profit of ([ ] [ ] ) and ([ ] [ ] ) respectively. This is also done for the other hedging instruments depending on FX rate leading to the profits ([ ] [ ]) ([ ] [ ]) and ([ ] [ ]). Finally we sell out position in the FX rate corresponding to currency pair that is [ ] ( ). With the newly found weights and we construct at time our rebalanced replicating portfolio (remember when you are rebalancing to construct the straddle as ATM, the RR as delta and the BF also as delta). Thus the replicating portfolio value at time just after having rebalanced is: [ ] [ ] [ ] [ ] [ ] [ ] [ ] where [ ] [ ] and [ ]. Repeat step 2 till time. 3. Expiration time At maturity time we do not need to compute the optimal weights anymore. Because the illiquid option expires at time and there is no need for hedging anymore. The only actions that happen at time are that we sell our bought positions computed at time. The value is then easily computed as ( ). The value is computed as ( ) ( ), is computed as ( ) ( ) and is computed as 31

32 ( ) ( ) [( ) ( ) ] where refers to the underlying currency pair or Analytical Triangular Hedging Strategy The analytical triangular hedging strategy is based on the triangular relationship between currencies described in section From Equation we know that can be written as From Equation we notice that depends on and. Because of this we can compute the sensitivity of an illiquid FX option (still depending on underlying FX rate ) against changes in and instead of computing the sensitivity against changes in directly. These sensitivities of an illiquid FX option against changes in and are computed as: where is the simple BS-vega of the illiquid FX option. We construct the following replicating portfolio, which is neutral to and and all the underlying spots, where the value of the replicating portfolio just after rebalancing at time is given by: [ ] [ ] where 32

33 [ ]. Thus we have a BS-delta position in FX rate, position of in a delta hedged ATM straddle depending on, position of in FX rate position of in a delta hedged ATM straddle depending on. The hedging instruments we use are the same as we use in empirical hedging strategy 3. However the weights and are now computed analytically based on triangular relationships between currencies and are not empirically estimated with regression. Therefore we refer to this hedging strategy as the analytical triangular hedging strategy. The hedging instrument weights are recomputed on a daily basis during a hedging cycle Notice that and (the BS-vega of an ATM straddle) are computed by taking the sum of the BS-vega ATM put option. As value for the correlation we take the implied correlation of an ATM call option and an such that the illiquid option is still valued with the volatility obtained by the market, which was also the case for the empirical hedging strategies Analytical Intrinsic Hedging Strategy The analytical intrinsic hedging strategy is based on the stochastic intrinsic currency volatility framework of Doust [2]. A summary of his main results was given in section When we use his model to estimate volatility smiles, we look at intrinsic currencies, which are: and. From these currencies we can construct different currency pairs (see section 3.2 for all the possible currency pairs) and estimate their volatility smiles. For each specific day and corresponding time to maturity we have to estimate totally parameters. That is intrinsic currency volatilities for ; volatility of volatility variable (since we use the model where ) and parameters in the correlation matrices and matrix ( parameters in, parameters in and parameters in ). Furthermore each currency pair contains pieces of market data, namely the and. Thus each specific day and corresponding time to maturity has totally pieces of market data. The basic idea to estimate the parameters of the stochastic intrinsic currency volatility framework is to do a least squares fit between model and market data. We will now describe the function which is used by us to estimate the needed model parameters. Note that the function we use is slightly different then Doust describes in his paper, where the biggest difference is the fact that we match 33

34 the market values and to its model values, whereat Doust matches the market values and to its model values. The function which we minimize for each specific day and corresponding time to maturity is defined by ( ) ( ) ( ) ( ) where minimizes the differences between the model and market values, is a maximum entropy term and the encourages a desirable characteristic to. The term is defined as [ ] [ ] [ ] where The notation refers to the observed ATM market volatility for day time to maturity, currency pair and is the corresponding estimated model volatility from Equation where. The notation refers to the observed market volatility for day time to maturity, currency pair and where the strike of this market volatility is found by using equation ; is the corresponding estimated model volatility from Equation. The notation refers to the observed market volatility for day time to maturity, currency pair and where the strike of this market volatility is found by using equation ; is the corresponding estimated model volatility from Equation. The symbol stands for the total number of observed squared errors, thus in our case we have that (since we look at different currency pairs and each currency pair contributes to different market values). The term is defined as 34

35 [ ] [ ( )] where and control the influence of the entropy terms in the minimization process. The weight is taken to be and is taken to be. The term is defined as [ ( )] The term is added to ensure that all are positive which is a desirable characteristic, as value for we take 10. For numerical stability we use the following transformations and optimize across,, and : where, this enforces such that the fitting procedure is more stable. For the same reasons and the cosinus transformation is done because the correlation matrix needs to have values between and. Furthermore we check at each estimation step whether the correlation matrix ( ) is positive definite. Once we have estimated all the parameters for a specific day and time to maturity, we can compute the estimated model volatilities for each strike, that is [ ], from Equation The error between the observed market volatilities and model volatilities is defined as [ ] To price the options with the stochastic intrinsic currency volatility framework we still use the BS-pricing formula, where the input for the volatility is given by [ ], such that the option is priced with its observed market volatility. 35

36 When we want to compute the options price sensitivity according to a specific risk factor, e.g. ; we compute this sensitivity (in the context of the stochastic intrinsic currency volatility framework) numerically by bumping with a value close to zero, that is: ( ) ( [ ] ) ( ) Notice that when pricing the option with the stochastic intrinsic currency volatility framework, the risk factor appears as well as in the model volatility as outside With the notation [ ] we mean the computed model volatility when is bumped with small value, since the model volatility is also depending on, while is just the computed model volatility without bumping with. The error is considered as a constant. Another example is the options price sensitivity according to the risk factor we compute this sensitivity (in the context of the stochastic intrinsic currency volatility framework) numerically by bumping with a value close to zero, that is: ( ) ( [ ] ) ( ) When the option is priced with the stochastic intrinsic currency volatility framework, the risk factor appears only in the model volatility. With the notation [ ] we mean the computed model volatility when is bumped with, while is just the computed model volatility without bumping with. The error is considered as a constant. Thus this is the method we use to compute the options price sensitivity of a financial derivative according to a specific risk factor, when valued with the stochastic intrinsic currency volatility framework. An description of our intrinsic hedging strategy is given below. Equation gives us the stochastic differential equation which is followed by the intrinsic currency volatility. Using Itô s-lemma and Equation, we can show that the stochastic 36

37 process ( ) follows the stochastic differential equation (where is the numeraire): ( ) For a full derivation, see Appendix. The numeraire currency is chosen to be currency, thus The intrinsic currency volatilities are not directly observable in the market and have to be estimated. Once we have estimated the intrinsic currency volatilities and all the other parameters of the stochastic intrinsic currency volatility framework, we are able to hedge the illiquid FX option against changes in the stochastic process of and, where we assume the stochastic differential equation of a specific underlying FX rate to have the form given by differential equation of an intrinsic volatility to have the form given by. When we refer to the intrinsic hedging strategy, we use the following hedging instruments: and the stochastic 1. ATM straddle depending on underlying FX rate and multiplied with FX rate such that it is valued in numeraire currency. 2. ATM straddle depending on underlying FX rate. 3. FX rate (independent FX rate). 4. FX rate (independent FX rate). Note that the hedging instruments of the intrinsic hedging strategy are almost the same as we use in empirical hedging strategy 3 and the hedging instrument weights are recomputed on a daily basis during a hedging cycle. The FX rates and are now assumed to be the independent FX rates and is assumed to be the dependent FX rate. Thus the dependent FX rate depends on and and is modeled as It is thus important to notice that is sensitive against changes in and. We denote by the value of hedging instrument (described right above) where ; and by the value of the illiquid FX option. Note that the hedging instruments and the illiquid FX option are valued with the BS-pricing formula, where the input for the volatility is given by [ ]. 37

38 The risk factors we are hedging at are all the intrinsic volatilities of the illiquid FX option and of the hedging instruments, that is and and the independent FX rates and. Denote the risk factors by for, where Denote by the stochastic differential equation followed by, thus ( ) ( ) ( ) ( ) We take the numeraire and furthermore we consider the model where. Using Itô s lemma we get the following stochastic differential equation for : where Furthermore we get the following stochastic differential equation for : 38

39 where To compute the sensitivities and for the illiquid FX option and the hedging instruments we use the method of bumping described in and. To come up with the optimal weights [ ] for the hedging instruments we minimize the criterion function w.r.t. where is defined as: ( ) ( [ ] ) ( ) [( ) ] [ ] where and. To find the minimum of, set the first order derivative of w.r.t. for ; equal to 0 and solve for the weights : By setting Equation equal to we find that: where [ ] [ ] 39

40 [ ] [ ] and [ ] [ ] [ ] [ ] [ ] [ ] In our case, the matrix is given as: [ ] where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Note that for. For the full derivation of matrix see the appendix. Repeating this for each where we get the following analytical solution for the hedging instruments weights when minimizing our criterion function : 40

41 where [ ] and [ ] Hedge Effectiveness Measures The main measures we use to indicate the effectiveness of the hedge during a hedging cycle, i.e. whether the hedge effectively eliminates (volatility) risk associated with a given portfolio, are described below. We assume a hedging cycle takes a period of days. Furthermore the notation stands for the day of the illiquid FX option (thus the derivative we want to hedge) occurred on day and notation is the day of the replicating portfolio occurred on day. The daily on day of a hedged portfolio is thus. - measure The root mean squared error measure is defined as [ ] This measure informs us how close the daily of a hedged portfolio is to the desired value of, during a hedging cycle. The closer the measure to, the better the hedge performed. -Auxiliary regression measure We regress, during a hedging cycle of days, the daily of the illiquid FX option on the daily of the replicating portfolio. The regression equation is From the auxiliary regression we look at the following regression statistics: adjusted perfect hedge, should equal, thus:, and. In a - We want the adjusted of the regression to be as close as possible to. - We want to be as close as possible. The estimator can be interpreted as the average daily amount by which the change in the FX option value differs from the change in the replicating portfolio value (drift term). Therefore a small value is important since a big value would lead to high cumulative profits and losses as time grows. - We want to be as close as possible to (such that the replicating portfolio captures the right dynamics of the illiquid FX option). 41

42 It is important not just to look at one of these regression statistics, but to account for them all simultaneously. -Quantile measures The quantile measure is actually defined as the quantile and the quantile of the different daily occurred during a hedging cycle and is denoted by and respectively. For a good hedge, we want the distribution of the daily to be concentrated around the value. Thus in general we can say that the closer the and measure are to 0, the better the hedge performed. The measures described above we consider as our main hedge effectiveness measures. Another hedge effectiveness measure we also look at is the cumulative at the end of a hedging cycle. For a good hedge we want this measure to be as close as possible to. However, we will not consider this measure as one of our main measures, since this measure contains less information and is not that accurate. It does not provide you the information how the hedge performed throughout the whole hedging cycle, instead it only gives you the total sum of all the past daily realized until the end of a hedging cycle. It could for example happen that in the beginning of a hedging cycle your hedge performed very badly on a specific day, thus the daily of the hedged portfolio on that day is very high. But if the hedge from then on performed perfectly well (that means all zero values for the day ) your cumulative at the end of the hedging cycle will still be very high and indicate a bad hedge, while the hedge was actually only bad for day. Another possibility is that your hedge makes big losses (bad hedge) in the beginning of a hedging cycle and then starts making big profits (bad hedge) such that your cumulative at expiration recovers and ends up being close to. Although this measure it not that accurate, it can still be useful for e.g. traders, who are not really interested in how the hedge performs during a hedging cycle, but are only interested if at the end of the hedging cycle the cumulative is close to. For more information about hedge effectiveness measures we refer to [9] and [10] The Backtest To conclude whether a hedging strategy is in general a good or bad strategy to hedge an illiquid FX option, we performed a backtest. Thus we checked how the hedging strategies performed during a hedging cycle (measured with the hedge effectiveness measures described in the previous section 2.2.6) for many different starting points in the past. When we backtest, the illiquid FX option (vanilla call option depending on FX rate is bought as ATM at initialization for each new starting point and kept till expiration, where the time to maturity at initialization is also the same for each new starting point. Furthermore we assume that we invest at initialization of a hedging cycle always the same amount (valued in the numeraire currency ) in the illiquid FX option portfolio (the portfolio we want to hedge). Thus we buy at each new starting point of a hedging cycle a total of illiquid FX options, where and is the value of the illiquid FX option at starting point (valued in currency ). The main reason why this is done is because at each new starting point of a hedging cycle the value of an ATM illiquid FX option is different. By investing at each new starting point of a hedging cycle the same amount of money we can compare the hedge effectiveness measures consistently over time, since they are then for each new starting point computed relatively to the 42

43 same amount invested at initialization. Although the illiquid FX option is always bought as ATM, the weight changes for each new hedging cycle. Thus the notional amount valued in (foreign) currency at each new starting point is equal to the weight and the notional amount valued in (domestic) currency at each new starting point is the weight multiplied with the FX rate, that is. Finally, it is important to notice that the conclusions we take from the backtest will only hold asymptotically. In this context it means that we can make statistically reliable conclusions about the hedging performance if the corresponding hedging strategy is tested for a lot of different starting days ; and then under the assumption that in the future the hedging strategy will be used to hedge illiquid FX options for a lot of different starting days (thus the hedge will not be used just for a few different starting days ), it would statistically seen, be correct to use the hedging strategy which performed over time the best in the backtest. An illustration of the time scale corresponding to the backtest is given in the next section Data. 3 Data The data we use is gathered from Bloomberg and contains the historical time series of the following volatilities: at the money volatility, the 25 delta butterfly volatility,, the 25 delta risk reversal volatility,, the 10 delta butterfly volatility, and the 10 delta risk reversal volatility,. 3.1 Dataset 1 The first dataset (Dataset ) is used to backtest all the hedging strategies, except for the intrinsic hedging strategy. The volatilities corresponding to the first dataset are gathered for the following tenors: 1 week, 2 weeks, 3 weeks, 1 month, 2 months, 3 months, 6 months, 9 months and 1 year. The first dataset is used for the following underlying currency pairs:,, ; and, and. The spot values of the FX rates depending on these currency pairs are also collected. Each hedging cycle provides us with a hedge effectiveness measure for the corresponding hedging cycle time interval. The full time horizon of all the different starting points used to start a hedging cycle in the backtest based on Dataset is 30/12/ /05/2013, where each new starting point of a hedging cycle is trading day ahead than the starting point of the previous hedging cycle, and thus providing us totally hedge effectiveness measures. Figure shows us the time scale of the backtest corresponding to Dataset. Thus at time which has the date 30/12/2011 we start the backtest by starting the first hedging cycle, that is we buy our first illiquid FX option and hedge it all the time till expiration. This provides us a hedge effectiveness measure for the first hedging cycle with time interval [ ] At time, which is trading day ahead than the starting point of the previous hedging cycle and has the date 02/01/2012, we buy a new illiquid FX option and hedge it all the time till expiration. This provides us a hedge effectiveness measure for the second hedging cycle with time interval [ ] This is repeated times where the starting point of the final hedging cycle has the date 06/05/

44 Figure 3.1: Time scale of the backtest corresponding to Dataset Dataset 2 The second dataset (Dataset ) is used to backtest all the hedging strategies, thus also the intrinsic hedging strategy. The volatilities corresponding to the second dataset are gathered for the following tenors: 1 week, 1 month, 2 months, 3 months, 6 months. The second dataset is used for the following underlying currency pair:, and. The spot values of the FX rates depending on these currency pairs are also collected. The full time horizon of the different starting points used to start a hedging cycle in the backtest based on Dataset is 08/01/ /12/2013, where each new starting point of a hedging cycle is trading days ahead than the starting point of the previous hedging cycle, and thus providing us totally hedge effectiveness measures. The total number of hedge effectiveness measures in Dataset (that is ) is lower than in Dataset (that is ) because we now also consider the intrinsic hedging strategy, which is computationally much more extensive than the other hedging strategies and it takes therefore more time to compute a single hedge effectiveness measure. Since we take less different starting points to start a hedging cycle in the backtest based on Dataset, we have chosen that each new starting point of a hedging cycle is trading days ahead than the starting point of the previous hedging cycle such that the full time horizon of the backtest based on Dataset 2 still covers a wide time scale (almost 1 year). Figure shows us the time scale of the backtest corresponding to Dataset. Thus at time which has the date 08/01/2013 we start the backtest by starting the first hedging cycle, that is we buy our first illiquid FX option and hedge it all the time till expiration. This provides us a hedge effectiveness measure for the first hedging cycle with time interval [ ] At time, which is trading days ahead than the starting point of the previous hedging cycle and has the date 10/01/2013 we buy a new illiquid FX option and hedge it all the time till expiration. This provides us a hedge effectiveness measure for the second hedging cycle with time interval [ ] This is repeated times where the starting point of the final hedging cycle has the date 24/12/2013. Figure 3.2: Time scale of the backtest corresponding to Dataset 2. 44

45 Note that when we hedge with the intrinsic hedging strategy in the second dataset, we estimate the parameters of the stochastic intrinsic currency volatility framework for the following currency pairs: 4 Results In this section we show some estimated volatility smiles from the stochastic intrinsic currency volatility framework of Doust and we show the backtest results of hedging an illiquid FX option. More results can be found in the Appendix. Remember when looking at the Auxiliary regression measure to account for them simultaneously. 4.1 Estimated Volatility Smiles and statistics of the When we use the intrinsic hedging strategy to hedge, we need to estimate the model parameters of the stochastic intrinsic currency volatility framework. To estimate the parameters, we minimize the function given in Once we have estimated the needed parameters, we are able to plot the estimated implied volatility with the approximation formula derived by Doust. Figure and Figure (Appendix ) compare the estimated implied volatility smiles from the stochastic intrinsic currency volatility framework to the implied volatility smile obtained from the market (both Figures use a different historical date). The underlying FX rates of the smiles are given in. From Figure and Figure we notice that the error between the market volatilities and the model volatilities is quite low. Thus we are able to estimate volatility smiles (based on the stochastic intrinsic currency volatility framework) which match the market smiles quite well. Figure and Figure show the market volatility smile, uncensored model volatility smile and the censored model volatility smile depending on FX rate. The uncensored volatility smile is estimated with the assumption that all the market data points, that is and, are available for all currency pairs described in, while the censored volatility smile is estimated with the assumption that for currency pair only the is available from the market. That means that the other currency pairs have all the needed market data available, except currency pair has only available. From Figure and Figure we notice that the censored volatility smile has generally seen a slightly bigger error compared to the error of the uncensored smile. Since the error is not that much bigger, we conclude that even if we would assume that the and volatilities cannot be obtained from the market for currency pair (because of illiquidity issues), we are still able (by using a lot of different currency pairs in the stochastic intrinsic currency volatility framework) to estimate a quite good volatility smile with a low error for. Therefore the stochastic intrinsic currency volatility framework of Doust seems to be a decent model to estimate (illiquid) volatility smiles. 45

46 Figure 4.1.1: The implied volatility smile obtained from market data (blue) compared to the uncensored (green) and censored (red) implied volatility smile estimated by the stochastic intrinsic currency volatility framework of Doust. The underlying FX rate is. On the y-axis the volatility is given in percentage, on the x-axis the strike is given, expressed as the number of for unit of. The range of the strike varies from to. The time to maturity is trading days ( months) and the historical date is 22/05/

47 Figure 4.1.2: The implied volatility smile obtained from market data (blue) compared to the uncensored (green) and censored (red) implied volatility smile estimated by the stochastic intrinsic currency volatility framework of Doust. The underlying exchange rate is. On the y-axis the volatility is given in percentage, on the x-axis the strike is given, expressed as the number of for unit of. The range of the strike varies from to.the time to maturity is trading days ( months) and the historical date is 02/08/ Main Backtest Results for (Dataset 1) In this section we show the main backtest results ( ) of hedging an illiquid FX option where the illiquid FX option is depending on underlying FX rate and the liquid hedging instruments are depending on FX rate and/or on FX rate. The dataset we use is Dataset. We take into consideration all the empirical hedging strategies and the analytical triangular hedging strategy. The illiquid FX option is assumed to be a plain vanilla call option. At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and hold till expiration. The backtest is performed for different starting points (where each new starting point of a hedging cycle is trading day ahead than the starting point of the previous hedging cycle) and for each new starting point we assume that we invest at the initialization of a hedging cycle always the same amount in the illiquid FX option portfolio, which is taken to be. 47

48 By comparing the empirical hedging strategy backtest results in Appendix B, we conclude that the standard linear regression model (where the weights are estimated with ) and the time-varying regression model (where the weights are estimated with the ) perform most of the time quite similarly. The differences are not that significant. A reason for this could be that the estimated variances [ ] in the state equation are mostly a small value close to zero. Then the time-varying regression model reduces to the standard linear regression model. Thus there seems to be not a big necessity to use the more sophisticated time-varying regression model in combination with the. Generally seen, estimating the weights with performs slightly better than estimating with the. We conclude from Appendix B that the best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = ; empirical strategy 2 = ; empirical strategy 3 = ; empirical strategy 4 = ; empirical strategy 5 = ; empirical strategy 6 =. Figures and Table show the main backtest results of each empirical hedging strategy (in combination with the best performing estimation method to estimate the weights), the analytical triangular hedging strategy and the simple BS-delta hedging strategy. From Figures and Figure we conclude that all the hedging strategies hedge (reduce the risk) against changes in the spot price of the underlying FX rate and the underlying volatility, since the and statistics are, compared to case of not hedging at all, significantly closer to. From the Figures and Figure and Table we conclude that all our empirical hedging strategies and the analytical triangular hedging strategy perform better than the simple BS-delta hedging strategy, which is our benchmark strategy. Thus besides of only reducing the spot price risk of the underlying FX rate, we are also able to reduce the volatility risk of an illiquid FX option (without making the hedged portfolio more illiquid) by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. Generally seen we can state the strategies are robust in the sense that performing worse than the simple BS-delta hedging strategy does not happen that often and when it happens it is usually not that much worse. Some strategies perform better than other strategies and we conclude that for this specific illiquid FX option, with underlying FX rate based on currency pair, empirical hedging strategy performs the best. Also the analytical triangular hedging strategy, empirical hedging strategy and empirical hedging strategy perform very well. Figures and Table show the main backtest results of empirical hedging strategy (which was chosen by us as the best performing hedging strategy), the simple BS-delta hedging strategy (benchmark strategy) and the simple BS-delta-vega hedging strategy into more detail. As said earlier, note that the results of the simple BS-delta-vega hedging strategy are not really comparable to the results of other hedging strategies, since our main assumption was that we do not want to hedge volatility with hedging instruments depending on underlying FX rate (now ) because of illiquidity issues. Furthermore the results of the simple BS-delta-vega hedging strategy are all based just on mid-prices and do not account for illiquidity issues / (high) transaction 48

49 costs when you rebalance the hedging instruments. We still add the results of this hedging strategy because they can give a nice indication how delta-vega hedging with the underlying FX rate would have performed if the FX option would not have been illiquid. From the detailed Figures and Table we conclude that empirical hedging strategy performs generally seen over time better than the simple BS-delta hedging strategy. Considering the average value of e.g. the and hedge effectiveness measures from the backtest results in Table we conclude that for this specific illiquid FX option empirical hedging strategy is able to reduce on average additionally about more risk than the simple BS-delta hedging strategy does reduce. Empirical hedging strategy is furthermore quite robust since the strategy almost never performs worse than the simple BS-delta hedging strategy and if the strategy does perform worse, it not that much worse (on average around worse according to the different hedge effectiveness measures). Note that the simple BS-delta-vega hedging strategy performs quite better than as well as the BSdelta hedging strategy as empirical hedging strategy and reduces for this specific illiquid FX option on average additionally about more risk than the simple BS-delta hedging strategy does reduce. Thus under the assumption that the FX option would have been liquid instead of illiquid, it would have been the best to use the simple BS-delta-vega hedging strategy instead of hedging strategies based on hedging instruments depending other underlying FX rates than the FX rate of the FX option itself. 49

50 Zoomed in on the next page Figure 4.2.1: RMSE for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

51 Figure 4.2.2: RMSE for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

52 Figure 4.2.3: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

53 Figure 4.2.4: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity 63 trading days (3 months) and we assume that we invest at initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

54 Figure 4.2.5: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

55 Zoomed in on the next page Figure 4.2.6: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

56 Figure 4.2.7: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

57 Zoomed in on the next page Figure 4.2.8: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

58 Figure 4.2.9: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with the analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

59 Table 4.2.1: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 OLS, 3 OLS, 4 OLS, 5 KF, 6 OLS; and hedged with the analytical triangular hedging strategy. No hedge Simple BSdelta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (OLS) Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (KF) Emp. hedging strategy 6 (OLS) Anal. triangular hedging strategy average std min max average std min max absolute average std min max average std min max average std min

60 max average std min max

61 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

62 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

63 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

64 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

65 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

66 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

67 Table 4.2.2: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategy 3 OLS and hedged with the simple BSdelta-vega hedging strategy. Simple BSdelta hedging strategy Emp. hedging strategy 3 (OLS) Simple BSdelta-vega hedging strategy average std min max average std min max absolute average std min max average std min max average std min max average std min max

68 4.3 Backtest results based on the cumulative at expiration measure for (Dataset 1) In this section we show the backtest results based on the cumulative at expiration measure. The backtest we consider is the same as the backtest which was performed in the previous section 4.2. It is important to notice that we do not consider this hedge effectiveness measure as one of our main measures, since this measure contains less information and is not that accurate. This measure does not give you the information how the hedge performed throughout the whole hedging cycle, instead it only states how good the hedge was at the end of the hedging cycle and has therefore a lot of noise. Although this measure it not that accurate, in practice it can still be useful for e.g. traders, who are not really interested in how the hedge performs during a hedging cycle, but are only interested if at the end of the hedging cycle the cumulative is close to. When we take conclusions from the backtest results based on the cumulative at expiration measure, the main descriptive statistic we look at is the average absolute value of all the different cumulative at expiration. This statistic can be considered as the long run discrepancy between the realized cumulative at expiration and the desired value. We look at the absolute value since our goal is to hedge and not to make a profit, which means that we consider a cumulative at expiration measure of and as an equally well performing hedge. Furthermore it could be misleading to take just the average value (instead of the absolute average) of all the different cumulative at expiration, since a backtest period of really positive cumulative at expiration (bad hedging period) followed by a period of really negative cumulative at expiration (bad hedging period), would give an average cumulative at expiration value close to zero and therefore indicate a good hedge, while the hedging performance was actually quite bad. By comparing the backtest results of the cumulative at expiration measure for all the empirical hedging strategies in Appendix, we conclude that the best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = OLS; empirical strategy 2 = KF; empirical strategy 3 = OLS; empirical strategy 4 = OLS; empirical strategy 5 = KF; empirical strategy 6 = OLS. Figures and Table show the backtest results based on the cumulative at expiration measure of each empirical hedging strategy (in combination with the best performing estimation method to estimate the weights), the analytical triangular hedging strategy and the simple BS-delta hedging strategy. From Figure we can nicely observe that when you hedge, the cumulative at expiration is over time more stable (less volatile) and closer to than if you would not hedge at all. Thus if you do not hedge your cumulative at expiration is quite volatile. Note that if you do not hedge the maximum profit at expiration is unrestricted, while the maximum loss is limited to, since that was the amount initially invested in the illiquid FX option portfolio. From Figure and Table we conclude (considering only the cumulative at expiration 68

69 measure) that all our empirical hedging strategies perform generally seen over time better than just the simple BS-delta hedging strategy. All the empirical strategies have namely a lower absolute average value for the different realized cumulative at expiration. However for the analytical triangular hedging strategy, the absolute average value is slightly higher than the absolute average value for the simple BS-delta hedging strategy. This would imply that according to the cumulative at expiration measure the analytical triangular hedging strategy performs worse than just the simple BS-delta hedging strategy. You add extra volatility risk instead of reducing volatility risk. This conclusion is not expected (since according to the main hedge effectiveness measures in the previous section the analytical triangular hedging strategy performed better than the simple BS-delta hedging strategy) and therefore it indicates the weakness and inaccuracy of this hedge effectiveness measure. We conclude that (according to the cumulative at expiration measure) for this specific illiquid FX option, with underlying FX rate based on currency pair, empirical hedging strategy performs the best. Figure and Table show the backtest results based on the cumulative at expiration measure for empirical hedging strategy, the simple BS-delta hedging strategy and the simple BS-delta-vega hedging strategy into more detail. From Figure and Table we conclude that empirical hedging strategy performs generally seen over time better than the simple BS-delta hedging strategy and is quite robust (thus does not exceed the absolute cumulative measure of the simple BS-delta hedging strategy often with a high value). The simple BS-delta-vega hedging strategy performs over time better than as well as the BS-delta hedging strategy as the empirical hedging strategy. Thus if the FX option would have been liquid it would have been the best to use the simple BS-delta-vega hedging strategy instead of hedging strategies based on hedging instruments depending on other underlying FX rates than the FX rate of the FX option itself (conclusion based on only the cumulative at expiration measure). 69

70 Zoomed in on the next page Figure 4.3.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 KF (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

71 Figure 4.3.2: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 KF (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

72 Figure 4.3.3: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 KF (red line) and hedged with the simple BS-deltavega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

73 absolute average Table 4.3.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 KF, 3 OLS, 4 OLS, 5 KF, 6 OLS; and hedged with the analytical triangular hedging strategy. No hedge Simple BS-delta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (KF) Cum. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (KF) Emp. hedging strategy 6 (OLS) Anal. triangular hedging strategy std min max Table 4.3.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, empirical hedging strategy 2 KF and hedged with the simple BSdelta-vega hedging strategy. Simple BSdelta hedging strategy Emp. hedging strategy 2 (KF) Simple BSdelta-vega hedging strategy Cum. absolute average std min max

74 4.4 Main Backtest Results for (Dataset 1) In this section we show the main backtest results ( ) of hedging an illiquid FX option where the illiquid FX option is depending on underlying FX rate and the liquid hedging instruments are depending on FX rate and/or on FX rate. The dataset we use is Dataset. We take into consideration all the empirical hedging strategies and the analytical triangular hedging strategy. The illiquid FX option is assumed to be a plain vanilla call option. At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and hold till expiration. The backtest is performed for different starting points (where each new starting point of a hedging cycle is trading day ahead than the starting point of the previous hedging cycle) and for each new starting point we assume that we invest at the initialization of a hedging cycle always the same amount in the illiquid FX option portfolio, which is taken to be. By comparing the empirical hedging strategy backtest results in Appendix C, we conclude that the standard linear regression model (where the weights are estimated with ) and the time-varying regression model (where the weights are estimated with the ) perform most of the time quite similarly (as was also the case for underlying ). There seems to be seems not a big necessity to use the more sophisticated time-varying regression model in combination with the. Generally seen, estimating the weights with performs slightly better than estimating with the. We conclude from Appendix C that the best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = OLS; empirical strategy 2 = OLS; empirical strategy 3 = OLS; empirical strategy 4 = OLS; empirical strategy 5 = KF; empirical strategy 6 = OLS. Figures and Table show the main backtest results of each empirical hedging strategy (in combination with the best performing estimation method to estimate the weights), the analytical triangular hedging strategy and the simple BS-delta hedging strategy. From Figures and Figure we conclude that all the hedging strategies hedge (reduce the risk) against changes in the spot price of the underlying FX rate and the underlying volatility, since the and statistics are, compared to case of not hedging at all, significantly closer to. Our main conclusion from the Figures and Figure and Table is the same as our main conclusion for the underlying, that is that all our empirical hedging strategies and the analytical triangular hedging strategy perform better than the simple BS-delta hedging strategy. Thus besides of only reducing the spot price risk of the underlying FX rate, we are also able to reduce volatility risk of an illiquid FX option (without making the hedged portfolio more illiquid) by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. We conclude that for this specific illiquid FX option, with underlying FX rate based on currency pair empirical hedging 74

75 strategy performs the best (as was also the case for underlying ). Furthermore the analytical triangular hedging strategy and empirical hedging strategy perform very well. Figures and Table show the backtest results of empirical hedging strategy, the simple BS-delta hedging strategy and the simple BS-delta-vega hedging strategy into more detail. From the detailed Figures and Table we conclude that empirical hedging strategy performs generally seen over time better than the simple BS-delta hedging strategy. Considering the average value of different hedge effectiveness measures from the backtest results in Table we conclude that for this specific illiquid FX option empirical hedging strategy is able to reduce on average additionally about more risk than the simple BS-delta hedging strategy does reduce. Empirical hedging strategy is furthermore very robust. Performing worse than the simple BS-delta hedging strategy barely happens and when it occurs it is on average just a few percent worse. Also now it holds that if the FX option would have been liquid it would have been the best to use the simple BS-delta-vega hedging strategy (reduces for this specific illiquid FX option on average additionally about more risk than the simple BS-delta hedging strategy does reduce) instead of using hedging strategies based on hedging instruments depending on other underlying FX rates than the FX rate of the FX option itself. 75

76 Zoomed in on the next page Figure 4.4.1: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

77 Figure 4.4.2: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

78 Figure 4.4.3: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

79 Figure 4.4.4: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

80 Figure 4.4.5: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

81 Zoomed in on the next page Figure 4.4.6: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

82 Figure 4.4.7: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

83 Zoomed in on the next page Figure 4.4.8: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

84 Figure 4.4.9: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

85 Table 4.4.1: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 OLS, 3 OLS, 4 OLS, 5 KF, 6 OLS; and hedged with the analytical triangular hedging strategy. No hedge Simple BSdelta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (OLS) Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (KF) Emp. hedging strategy 6 (OLS) Anal. triangular hedging strategy average std min max average std min max absolute average std min max average std min max average

86 std min max average std min max

87 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

88 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

89 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

90 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

91 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

92 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 3 OLS (light blue line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

93 Table 4.4.2: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategy 3 OLS; and hedged with simple BS-deltavega hedging strategy. Simple BSdelta hedging strategy Emp. hedging strategy 3 (OLS) Simple BSdelta-vega hedging strategy average std min max average std min max absolute average std min max average std min max average std min max average std min max

94 4.5 Backtest results based on the cumulative at expiration measure for (Dataset 1) In this section we show the backtest results based on the cumulative at expiration measure. The backtest we consider is the same as the backtest which was performed in the previous section 4.4. Thus the illiquid FX option is depending on underlying FX rate. Like said earlier, we do not consider this hedging performance measure as one of our main measures, since this measure contains less information and is not that accurate. By comparing the backtest results of the cumulative at expiration measure for all the empirical hedging strategies in Appendix, we conclude that the best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = OLS; empirical strategy 2 = OLS; empirical strategy 3 = OLS; empirical strategy 4 = OLS; empirical strategy 5 = KF; empirical strategy 6 = OLS. Figures and Table show the backtest results based on the cumulative at expiration measure of each empirical hedging strategy (in combination with the best performing estimation method to estimate the weights), the analytical triangular hedging strategy and the simple BS-delta hedging strategy. From Figure we observe that by hedging the cumulative at expiration is over time more stable (less volatile) and closer to than if you would not hedge at all. If you do not hedge your cumulative at expiration is quite volatile. From Figure and Table we conclude (considering only the cumulative at expiration measure) that all our empirical hedging strategies and the analytical triangular hedging strategy perform generally seen over time better than just the simple BS-delta hedging strategy. All the empirical strategies have namely a lower absolute average value for the different realized cumulative at expiration. For this specific illiquid FX option, with underlying FX rate based on currency pair, the analytical triangular hedging strategy performs the best. Notice that empirical hedging strategy has actually a lower absolute average mean value (see Table 4.5.1), but we have not chosen empirical hedging strategy as the best performing hedging strategy since a comparison with the analytical triangular hedging strategy has shown that the absolute cumulative values of the analytical triangular hedging strategy exceed the absolute cumulative values of the simple BSdelta hedging strategy less often and also with a lower value. Therefore the analytical triangular hedging strategy is more robust compared to the empirical hedging strategy, even though the absolute average value of the different realized cumulative at expiration is (slightly) lower for empirical hedging strategy. Figure and Table show the backtest results based on the cumulative at expiration measure for the analytical triangular hedging strategy, simple BS-delta hedging strategy and simple BS-delta-vega hedging strategy. 94

95 From the plots and the table we conclude that analytical triangular hedging strategy performs generally seen over time better than the simple BS-delta hedging strategy and is quite robust (thus does not exceed the absolute cumulative value of the simple BS-delta hedging strategy often with a high value). Notice furthermore that according to the cumulative at expiration measure the simple BSdelta-vega hedging strategy compared to the analytical triangular hedging strategy does not perform over time that much better as expected. Thus this implies that if the FX option would have been liquid one could hedge with the analytical triangular hedging strategy (which is a hedging strategy where the hedging instruments are depending on other underlying FX rates than the underlying FX rate of the FX option which needs to be hedged) and get almost the same benefits as hedging with the simple BS-delta-vega hedging strategy (which is a hedging strategy where the hedging instruments are depending on the same underlying FX rate as the underlying FX rate of the FX option which needs to be hedged). This conclusion confirms our belief that the cumulative at expiration measure is not a really accurate hedging performance measure. Zoomed in on the next page Figure 4.5.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

96 Figure 4.5.2: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); and hedged with analytical triangular hedging strategy (brown line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

97 Figure 4.5.3: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), analytical triangular hedging strategy (brown line) and hedged with the simple BS-delta-vega hedging strategy (dotted red line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

98 absolute average Table 4.5.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 OLS, 3 OLS, 4 OLS, 5 KF, 6 OLS; and hedged with the analytical triangular hedging strategy. No hedge Simple BS-delta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (OLS) Cum. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (KF) Emp. hedging strategy 6 (OLS) Anal. triangular hedging strategy std min max Table 4.5.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, analytical triangular hedging strategy and hedged with the simple BS-delta-vega hedging strategy. Simple BSdelta hedging strategy Anal. triangular hedging strategy Simple BSdelta-vega hedging strategy Cum. absolute average std min max

99 4.6 Main Backtest Results for (Dataset 2) In this section we show the main backtest results ( ) of hedging an illiquid FX option where the illiquid FX option is depending on underlying FX rate and the liquid hedging instruments are depending on FX rate and/or on FX rate. The dataset we use is dataset 2 and the main difference with sections is that we now also take into consideration the intrinsic hedging strategy, which was left out in the previous sections. Our main attention is to examine whether the intrinsic hedging strategy performs better than the simple BS-delta hedging strategy and to compare the performance of the intrinsic hedging strategy to the performance of the empirical hedging strategies and the analytical triangular hedging strategy. The illiquid FX option is assumed to be a plain vanilla call option. At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and hold till expiration. The backtest is performed for different starting points (where each new starting point of a hedging cycle is trading days ahead than the starting point of the previous hedging cycle) and for each new starting point we assume that we invest at the initialization of a hedging cycle always the same amount in the illiquid FX option portfolio, which is. By comparing the empirical hedging strategy backtest results in Appendix D, we conclude that estimating the weights with seems to perform most of the time slightly better than estimating with the. The best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = OLS; empirical strategy 2 = OLS; empirical strategy 3 = OLS; empirical strategy 4 = OLS; empirical strategy 5 = KF; empirical strategy 6 = OLS. Figures and Table show the main backtest results of each empirical hedging strategy (in combination with the best performing estimation method to estimate the weights), the analytical triangular hedging strategy, the intrinsic hedging strategy and the simple BS-delta hedging strategy. From Figures and Figure we conclude that all the hedging strategies reduce the risk against changes in the spot price of the underlying FX rate and the underlying volatility, since the and statistics are significantly closer to compared to the case of not hedging. The main conclusion from Figures and Figure and Table is that the intrinsic hedging strategy performs over time better than just the simple BS-delta hedging strategy. Thus besides of only reducing the spot risk, we are also able to reduce the intrinsic volatility risk of an illiquid FX option (without making the hedged portfolio more illiquid) by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. Furthermore we can conclude that all the empirical hedging strategies and the analytical triangular hedging strategy also perform better than the simple BS-delta hedging strategy. Considering all the empirical hedging strategies and the analytical triangular hedging strategy, it 99

100 seems that empirical hedging strategy hedging strategy also performs very well. performs the best, although the analytical triangular Figures and Table show the backtest results of empirical hedging strategy, the intrinsic hedging strategy, the simple BS-delta hedging strategy and the simple BS-deltavega hedging strategy. From the detailed Figures and Table we can clearly see that empirical hedging strategy and the intrinsic hedging strategy perform generally seen better than the simple BS-delta hedging strategy and are furthermore quite robust. Comparing empirical hedging strategy to the intrinsic hedging strategy it is hard to tell which hedging strategy performs the best. Generally seen it seems that the intrinsic hedging strategy (which reduces for this specific illiquid FX option on average additionally about more risk than the simple BS-delta hedging strategy does reduce) performs slightly worse than that empirical hedging strategy (which reduces for this specific illiquid FX option on average additionally about more risk than the simple BS-delta hedging strategy does reduce). Furthermore we notice that empirical hedging strategy performs a bit better for the first half of the dataset, while the intrinsic hedging strategy seems to perform a bit better for the second half of the dataset. Finally we also conclude that if the FX option would have been liquid it would have been the best to use the simple BS-delta-vega hedging strategy (which reduces for this specific illiquid FX option on average additionally about more risk than the simple BS-delta hedging strategy does reduce) instead of hedging strategies based on hedging instruments depending on other underlying FX rates than the FX rate of the FX option itself. 100

101 Zoomed in on the next page Figure 4.6.1: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line), intrinsic hedging strategy (orange line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

102 Figure 4.6.2: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

103 Figure 4.6.3: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

104 Figure 4.6.4: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

105 Figure 4.6.5: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

106 Zoomed in on the next page Figure 4.6.6: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line), intrinsic hedging strategy (orange line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

107 Figure 4.6.7: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

108 Zoomed in on the next page Figure 4.6.8: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line), intrinsic hedging strategy (orange line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

109 Figure 4.6.9: for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 KF (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line) and the intrinsic hedging strategy (bold orange line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

110 Table 4.6.1: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 OLS, 3 OLS, 4 OLS, 5 KF, 6 OLS; hedged with the analytical triangular hedging strategy and the intrinsic hedging strategy. No hedge Simple BSdelta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (OLS) Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (KF) Emp. hedging strategy 6 (OLS) Anal. triangular hedging strategy Intrinsic hedging strategy average std min max average std min max absolute average std min max average std min max average std min

111 max average std min max

112 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

113 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

114 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

115 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

116 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

117 Figure : for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

118 Table 4.6.2: Summary statistics of the backtest results for:, and ; if hedged with the simple BS-delta hedging strategy, empirical hedging strategy, intrinsic hedging strategy and hedged with the simple BS-delta-vega hedging strategy. Simple BSdelta hedging strategy Emp. hedging strategy 2 (OLS) Intrinsic hedging strategy Simple BSdelta-vega hedging strategy average std min max average std min max absolute average std min max average std min max average std min max average std min max

119 4.7 Backtest results based on the cumulative at expiration measure for (Dataset 2) In this section we show the backtest results based on the cumulative at expiration measure. The backtest we consider is the same as the backtest which was performed in the previous section 4.6. We take into consideration all the empirical hedging strategies, the analytical triangular hedging strategy and the intrinsic hedging strategy. By comparing the backtest results of the cumulative at expiration measure for all the empirical hedging strategies in Appendix, we conclude that the best performing estimation method for the weights of each empirical hedging strategy is: empirical hedging strategy 1 = OLS; empirical strategy 2 = OLS; empirical strategy 3 = OLS; empirical strategy 4 = OLS; empirical strategy 5 = OLS; empirical strategy 6 = OLS. From Figure we observe that when you hedge, the cumulative at expiration is over time more stable and closer to than if you would not hedge at all. From Figure and Table we conclude (considering only the cumulative at expiration measure) that, except for empirical hedging strategy (which performs a bit worse than the simple BS-delta hedging strategy), all the other empirical hedging strategies, the analytical triangular hedging strategy and the intrinsic hedging strategy perform generally seen over time better than just the simple BS-delta hedging strategy. They have all a lower absolute average value for the different realized cumulative at expiration. We conclude that of all strategies (according to the cumulative at expiration measure) for this specific illiquid FX option, with underlying FX rate based on currency pair, empirical hedging strategy performs the best. Figure and Table show the backtest results based on the cumulative at expiration measure for empirical hedging strategy, the intrinsic hedging strategy, the simple BS-delta hedging strategy and the simple BS-delta-vega hedging strategy more into detail. From the plots and the table we notice that (according to the cumulative at expiration measure) empirical hedging strategy and the intrinsic hedging strategy perform generally seen over time better than the simple BS-delta hedging strategy. Empirical hedging strategy performs a bit better than the intrinsic hedging strategy and is more robust. Finally we notice that the simple BS-delta-vega hedging strategy performs better than both the intrinsic hedging strategy and empirical hedging strategy. 119

120 Zoomed in on the next page Figure 4.7.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 OLS (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line), intrinsic hedging strategy (orange line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

121 Figure 4.7.2: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategies: 1 OLS (green line), 2 OLS (red line), 3 OLS (light blue line), 4 OLS (purple line), 5 OLS (yellow line), 6 OLS (black line); hedged with analytical triangular hedging strategy (brown line), intrinsic hedging strategy (bold orange line) and not hedged at all (dotted green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

122 Figure 4.7.3: Cumulative (upper picture) and ABSOLUTE cumulative (lower picture) at expiration for different starting points if the illiquid FX European call option is hedged with the simple BS-delta hedging strategy (bold dark blue line), empirical hedging strategy 2 OLS (red line), the intrinsic hedging strategy (orange line) and hedged with the simple BS-delta-vega hedging strategy (dotted black line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

123 Table 4.7.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, empirical hedging strategies: 1 OLS, 2 OLS, 3 OLS, 4 OLS, 5 OLS, 6 OLS; hedged with the analytical triangular hedging strategy ant the intrinsic hedging strategy. No hedge Simple BS-delta hedging strategy Emp. hedging strategy 1 (OLS) Emp. hedging strategy 2 (OLS) Emp. hedging strategy 3 (OLS) Emp. hedging strategy 4 (OLS) Emp. hedging strategy 5 (OLS) Emp. hedging strategy 6 (OLS) Anal. triangul ar hedging strategy Intrinsic hedging strategy Cum. absolute average std min max Table 4.7.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with the simple BS-delta hedging strategy, empirical hedging strategy 2 OLS, intrinsic hedging strategy and hedged with the simple BS-delta-vega hedging strategy. Simple BSdelta hedging strategy Emp. hedging strategy 2 (OLS) Intrinsic hedging strategy Simple BSdelta-vega hedging strategy Cum. absolute average std min max

124 5 Conclusion and Recommendations In this study we carried out an extensive empirical analysis on the hedging performance of alternative hedging strategies specially developed for illiquid FX options. We mainly focused on hedging the illiquid FX option against changes in the underlying volatility without making the hedged portfolio more illiquid. To hedge the illiquid FX option against changes in the underlying volatility we did not use hedging instruments depending on the same underlying FX rate as the illiquid FX option is depending on, since it should be assumed that these hedging instruments also suffer from similar illiquidity issues and this would make the hedged portfolio only more illiquid. Instead, we used liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. Our hedging strategies could be divided into empirical hedging strategies and into analytical model based hedging strategies. The empirical hedging strategies were based on a standard linear regression model (weights estimated with ) or on a time-varying regression model (weights estimated with ). While the analytical model based hedging strategies were derived from the triangular relationship between currencies and from the stochastic intrinsic currency volatility framework of Doust. Our benchmark strategy was the simple analytical BS-delta hedging strategy. The main conclusion from the backtest results is that all our hedging strategies performed generally seen over time better than just the simple BS-delta hedging strategy. In order to find the best performing hedging strategy for a specific illiquid FX option one would like to hedge, it is needed to perform a backtest for that specific illiquid FX option and to analyze the backtest results carefully. Considering the average value of the different hedge effectiveness measures from our backtests we conclude that for all the specific illiquid FX options we have hedged, we were able to reduce on average additionally about till more risk than the simple BS-delta hedging strategy did reduce. Thus besides of only reducing the spot price risk of the underlying FX rate, we were also able to reduce volatility risk of an illiquid FX option (without making the hedged portfolio more illiquid) by using liquid hedging instruments depending on other underlying FX rates than the FX rate of the illiquid FX option. Generally seen we can state that the hedging strategies are quite robust in the sense that performing worse than the simple BS-delta hedging strategy did not happen that often and when it happened it was usually not that much worse, approximately about worse. From the backtest results we can furthermore conclude that when we looked at the empirical hedging strategies, the standard linear regression model (where the weights are estimated with ) and the time-varying regression model (where the weights are estimated with the ) performed most of the time quite similarly. However, the standard linear regression model seemed to perform a bit better and therefore one can just use the simple method to estimate the weights instead of the more sophisticated method. From the stochastic intrinsic currency volatility framework of Doust we derived the intrinsic hedging strategy, which is different from the other strategies in the sense that the intrinsic hedging strategy focuses on hedging against changes in the intrinsic currency volatility instead of focusing on traditional hedging against changes in the traditional BS implied volatility (thus not intrinsic volatility) of the FX option. The intrinsic hedging strategy did perform better than the BS-delta hedging strategy and reduced for the specific illiquid FX option we considered on average additionally about more risk than the simple BS-delta hedging strategy did reduce, however when we compared the performance of the intrinsic hedging strategy to the performance of the best performing hedging strategy which was taken from the group of the empirical hedging strategies and 124

125 the analytical triangular hedging strategy, the intrinsic hedging strategy performed slightly worse than this best performing strategy taken from the group of empirical hedging strategies and the analytical triangular hedging strategy (which reduced on average additionally about more risk than the simple BS-delta hedging strategy did reduce). A reason for this could be that the intrinsic hedging strategy is very sophisticated and estimating the correct estimators is therefore quite cumbersome. Finally we conclude that if the FX option would have been liquid and did not suffer from illiquidity issues it would have been the best to use the simple BS-delta-vega hedging strategy, based on hedging instruments depending on the same underlying FX rate as the FX option you need to hedge has, instead of using hedging strategies based on hedging instruments depending other underlying FX rates. It remains therefore a challenge to find a hedging strategy based on hedging instruments which are depending on other underlying FX rates than the FX rate of the FX option itself, while being able to reduce the same amount of volatility risk as would have also been possible to reduce if the FX option was liquid and simply BS-delta-vega hedged with hedging instruments depending on the same underling FX rate as the FX rate of the FX option itself. The different hedging strategies we used were the same throughout a hedging cycle. For further research it would be interesting to study the concept of an adaptive hedging strategy, where you do not use just one specific hedging strategy throughout the whole hedging cycle but based on some criterion measure, e.g. the lowest average statistic of the last five days, intermediately switch from hedging strategy to the currently best performing strategy according to your criterion measure. It is also worth to further investigate and improve the new concept of intrinsically hedging against changes in the intrinsic volatility, since there is not yet much literature found how to use the stochastic intrinsic currency volatility framework of Doust to hedge. One could try to use for example other hedging instruments or define some other criterion function/approach to come up with the hedging instrument weights. 125

126 References [ ] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3), (1973). [ ] P. Doust, The stochastic intrinsic currency volatility framework, Applied Mathematical Finance, 19 (5), (2012). [ ] M. Garman and S. W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (3), (1983). [ ] U. Wystup, FX options and structured products, Risk Books, Wiley Finance, [ ] M. Haugh, Foreign exchange and quantos, Financial Engineering Syllabus: Continuous-Time Models, Columbia University, [ ] P.S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, Managing smile risk, Wilmott Magazine, , (2002). [ ] J. Chen and P. Doust, Improving intrinsic currency analysis: using information entropy and beyond, RBS research article, November [ ] J. Durbin and S.J. Koopman, Time series analysis by state space models, Oxford University Press, [ ] J. Hanley and J. Cotter, Re-evaluating hedging performance, Journal of Futures Markets, 26 (7), (2006). [ ] J. Finnert and D. Grant, Testing hedge effectiveness under SFAS 133,

127 Appendix A Estimated Volatility Smiles A.1 Estimated Volatilty Smiles for historical date 22/05/

128 Figure A.1: The implied volatility smiles obtained from market data (blue) compared to the implied volatility smiles estimated by the stochastic intrinsic currency volatility framework (green). The plotted smiles, read in horizontal order from left to the right, correspond respectively to the currency pairs given in. On the y-axis the volatility is given in percentage, on the x-axis the strike is given, expressed as the number of the domestic currency for unit of the foreign currency. The range of the strike varies from to. The time to maturity is trading days ( months) and the historical date is 22/05/2013. A.2 Estimated Volatilty Smiles for historical date 02/08/

129 Figure A.2: The implied volatility smiles obtained from market data (blue) compared to the implied volatility smiles estimated by the stochastic intrinsic currency volatility framework of Doust (green). The plotted smiles, read in horizontal order from left to the right, correspond respectively to the currency pairs given in. On the y-axis the volatility is given in percentage, on the x-axis the strike is given, expressed as the number of the domestic currency for unit of the foreign currency. The range of the strike varies from to. The time to maturity is trading days ( months) and the historical date is 02/08/

130 Appendix B vs. : Main Backtest Results for (Dataset 1) B.1 Empirical Hedging Strategy 1 B.1.1 Backtest Results Figure B.1.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

131 B.1.2 Auxiliary Regression Backtest Results Figure B.1.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

132 Figure B.1.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

133 Figure B.1.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

134 B.1.3 Quantile Measure Backtest Results Figure B.1.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

135 Figure B.1.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

136 B.1.4 Summary Statistics Main Backtest Results Table B.1.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

137 B.2 Empirical Hedging Strategy 2 B.2.1 backtest results Figure B.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

138 B.2.2 Auxiliary Regression Backtest Results Figure B.2.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

139 Figure B.2.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

140 Figure B.2.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

141 B.2.3 Quantile Measure Backtest Results Figure B.2.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

142 Figure B.2.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

143 B.2.4 Summary Statistics Backtest Results Table B.2.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

144 B.3 Empirical Hedging Strategy 3 B.3.1 backtest results Figure B.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

145 B.3.2 Auxiliary Regression Backtest Results Figure B.3.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

146 Figure B.3.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

147 Figure B.3.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

148 B.3.3 Quantile Measure Backtest Results Figure B.3.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

149 Figure B.3.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

150 B.3.4 Summary Statistics Backtest Results Table B.3.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

151 B.4 Empirical Hedging Strategy 4 B.4.1 Backtest Results Figure B.4.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

152 B.4.2 Auxiliary Regression Backtest Results Figure B.4.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

153 Figure B.4.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

154 Figure B.4.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

155 B.4.3 Quantile Measure Backtest Results Figure B.4.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

156 Figure B.4.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

157 B.4.4 Summary Statistics Backtest Results Table B.4.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

158 B.5 Empirical Hedging Strategy 5 B.5.1 Backtest Results Figure B.5.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

159 B.5.2 Auxiliary Regression Backtest Results Figure B.5.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

160 Figure B.5.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

161 Figure B.5.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

162 B.5.3 Quantile Measure Backtest Results Figure B.5.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

163 Figure B.5.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

164 B.5.4 Summary Statistics Backtest Results Table B.5.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

165 B.6 Empirical Hedging Strategy 6 B.6.1 Backtest Results Figure B.6.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

166 B.6.2 Auxiliary Regression Backtest Results Figure B.6.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

167 Figure B.6.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

168 Figure B.6.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

169 B.6.3 Quantile Measure Backtest Results Figure B.6.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

170 Figure B.6.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

171 B.6.4 Summary Statistics Backtest Results Table B.6.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

172 Appendix C vs. : Main Backtest Results for (Dataset 1) C.1 Empirical Hedging Strategy 1 C.1.1 Backtest Results Figure C.1.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

173 C.1.2 Auxiliary Regression Backtest Results Figure C.1.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

174 Figure C.1.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

175 Figure C.1.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

176 C.1.3 Quantile Measure Backtest Results Figure C.1.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

177 Figure C.1.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

178 C.1.4 Summary Statistics Backtest Results Table C.1.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

179 C.2 Empirical Hedging Strategy 2 C.2.1 Backtest Results Figure C.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

180 C.2.2 Auxiliary Regression Backtest Results Figure C.2.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

181 Figure C.2.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

182 Figure C.2.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

183 C.2.3 Quantile Measure Backtest Results Figure C.2.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

184 Figure C.2.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

185 C.2.4 Summary Statistics Backtest Results Table C.2.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

186 C.3 Empirical Hedging Strategy 3 C.3.1 Backtest Results Figure C.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

187 C.3.2 Auxiliary Regression Backtest Results Figure C.3.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

188 Figure C.3.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

189 Figure C.3.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

190 C.3.3 Quantile Measure Backtest Results Figure C.3.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

191 Figure C.3.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

192 C.3.4 Summary Statistics Main Backtest Results Table C.3.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

193 C.4 Empirical Hedging Strategy 4 C.4.1 Backtest Results Figure C.4.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

194 C.4.2 Auxiliary Regression Backtest Results Figure C.4.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

195 Figure C.4.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

196 Figure C.4.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

197 C.4.3 Quantile Measure Backtest Results Figure C.4.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

198 Figure C.4.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

199 C.4.4 Summary Statistics Backtest Results Table C.4.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

200 C.5 Empirical Hedging Strategy 5 C.5.1 Backtest Results Figure C.5.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

201 C.5.2 Auxiliary Regression Backtest Results Figure C.5.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

202 Figure C.5.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

203 Figure C.5.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

204 C.5.3 Quantile measure Backtest Results Figure C.5.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

205 Figure C.5.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

206 C.5.4 Summary Statistics Backtest Results Table C.5.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

207 C.6 Empirical Hedging Strategy 6 C.6.1 Backtest Results Figure C.6.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

208 C.6.2 Auxilary Regression Backtest Results Figure C.6.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

209 Figure C.6.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

210 Figure C.6.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

211 C.6.3 Quantile Measure Backtest Results Figure C.6.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

212 Figure C.6.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

213 C.6.4 Summary Statistics Backtest Results Table C.6.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

214 Appendix D vs. : Main Backtest Results for (Dataset 2) D.1 Empirical Hedging Strategy 1 D.1.1 Backtest Results Figure D.1.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

215 D.1.2 Auxiliary Regression Backtest Results Figure D.1.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

216 Figure D.1.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

217 Figure D.1.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

218 D.1.3 Quantile Measure Backtest Results Figure D.1.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

219 Figure D.1.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

220 D.1.4 Summary Statistics Backtest Results Table D.1.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

221 D.2 Empirical Hedging Strategy 2 D.2.1 Backtest Results Figure D.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

222 D.2.2 Auxiliary Regression Backtest Results Figure D.2.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

223 Figure D.2.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

224 Figure D.2.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

225 D.2.3 Quantile Measure Backtest Results Figure D.2.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

226 Figure D.2.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

227 D.2.4 Summary Statistics Backtest Results Table D.2.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

228 D.3 Empirical Hedging Strategy 3 D.3.1 Backtest Results Figure D.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

229 D.3.2 Auxiliary Regression Backtest Results Figure D.3.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

230 Figure D.3.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

231 Figure D.3.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

232 D.3.3 Quantile Measure Backtest Results Figure D.3.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

233 Figure D.3.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

234 D.3.4 Summary Statistics Backtest Results Table D.3.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

235 D.4 Empirical Hedging Strategy 4 D.4.1 Backtest Results Figure D.4.1: RMSE for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

236 D.4.2 Auxiliary Regression Backtest Results Figure D.4.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

237 Figure D.4.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

238 Figure D.4.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

239 D.4.3 Quantile Measure Backtest Results Figure D.4.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

240 Figure D.4.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

241 D.4.4 Summary Statistics Backtest Results Table D.4.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

242 D.5 Empirical Hedging Strategy 5 D.5.1 Backtest Results Figure D.5.1: RMSE for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

243 D.5.2 Auxiliary Regression Backtest Results Figure D.5.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

244 Figure D.5.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

245 Figure D.5.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

246 D.5.3 Quantile Measure Backtest Results Figure D.5.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

247 Figure D.5.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

248 D.5.4 Summary Statistics Backtest Results Table D.5.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

249 D.6 Empirical Hedging Strategy 6 D.6.1 Backtest Results Figure D.6.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

250 D.6.2 Auxiliary Regression Backtest Results Figure D.6.2.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

251 Figure D.6.2.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

252 Figure D.6.2.3: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

253 D.6.3 Quantile Measure Backtest Results Figure D.6.3.1: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

254 Figure D.6.3.2: for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/

255 D.6.4 Summary Statistics Backtest Results Table D.6.4: Summary statistics of the backtest results for:, and ; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) average std min max average std min max absolute average std min max average std min max average std min max average std min max

256 Appendix E vs. : Cumulative at expiration measure for (Dataset 1) E.1 Empirical Hedging Strategy 1 Figure E.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

257 Table E.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) Cum. absolute average std min max E.2 Empirical Hedging Strategy 2 Figure E.2: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

258 Table E.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) Cum. absolute average std min max E.3 Empirical Hedging Strategy 3 Figure E.3: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

259 Table E.3: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) Cum. absolute average std min max E.4 Empirical Hedging Strategy 4 Figure E.4: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

260 Table E.4: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) Cum. absolute average std min max E.5 Empirical Hedging Strategy 5 Figure E.5: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

261 Table E.5: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) Cum. absolute average std min max E.6 Empirical Hedging Strategy 6 Figure E.6: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

262 Table E.6: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) Cum. absolute average std min max

263 Appendix F 1) vs. : Cumulative at expiration measure for (Dataset F.1 Empirical Hedging Strategy 1 Figure F.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

264 Table F.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) Cum. absolute average std min max F.2 Empirical Hedging Strategy 2 Figure F.2: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

265 Table F.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) Cum. absolute average std min max F.3 Empirical Hedging Strategy 3 Figure F.3: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

266 Table F.3: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) Cum. absolute average std min max F.4 Empirical Hedging Strategy 4 Figure F.4: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

267 Table F.4: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) Cum. absolute average std min max F.5 Empirical Hedging Strategy 5 Figure F.5: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

268 Table F.5: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) Cum. absolute average std min max F.6 Empirical Hedging Strategy 6 Figure F.6: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days ( months) and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 30/12/ /05/

269 Table F.6: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) Cum. absolute average std min max

270 Appendix G 2) vs. : Cumulative at expiration measure for (Dataset G.1 Empirical Hedging Strategy 1 Figure G.1: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.1: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 1, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 1 (OLS) Emp. hedging strategy 1 (KF) Cum. absolute average std min max

271 G.2 Empirical Hedging Strategy 2 Figure G.2: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.2: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 2, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 2 (OLS) Emp. hedging strategy 2 (KF) Cum. absolute average std min max

272 G.3 Empirical Hedging Strategy 3 Figure G.3: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.3: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 3, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 3 (OLS) Emp. hedging strategy 3 (KF) Cum. absolute average std min max

273 G.4 Empirical Hedging Strategy 4 Figure G.4: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.4: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 4, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 4 (OLS) Emp. hedging strategy 4 (KF) Cum. absolute average std min max

274 G.5 Empirical Hedging Strategy 5 Figure G.5: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.5: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 5, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 5 (OLS) Emp. hedging strategy 5 (KF) Cum. absolute average std min max

275 G.6 Empirical Hedging Strategy 6 Figure G.6: Cumulative at expiration for different starting points if the illiquid FX European call option is hedged with empirical hedging strategy (blue line) and hedged with empirical hedging strategy (green line). At each new starting point the illiquid FX option is bought as ATM with time to maturity trading days and we assume that we invest at the initialization of a hedging cycle always the same amount, that is, in the illiquid FX option portfolio. The time horizon of the different starting points for the backtest is 08/01/ /12/2013. Table G.6: Summary statistics of the backtest results for the cumulative at expiration measure; if hedged with empirical hedging strategy 6, estimation method for hedging instruments: OLS and KF. Emp. hedging strategy 6 (OLS) Emp. hedging strategy 6 (KF) Cum. absolute average std min max

276 Appendix H S.D.E. of Exchange Rate derived from Intrinsic Currency Values Using Itô s-lemma and Equation, we show that the stochastic process ( ) follows the stochastic differential equation given in, that is ( ) where is the numeraire currency. The steps to derive Equation are given below. ( ) ( ) [ ] Use Itô calculus: [ ] [ ] ( ) ( ) [ ] [ ] 276

277 Thus ( ) 277

278 Appendix I Derivation of Matrix In this appendix we show the full derivation of each element of matrix. As numeraire we take and furthermore we consider the model where. ( ) [ ] ( )( ) [ ] [ ] ( )( ) ( ) ( ( ) ) [ ( ) ] ( ) ( ) ( ( ) ) [ ( ) ] ( ) ( ) 278

279 [ ] ( )( ) [ ] ( ) ( ( ) ) [ ( ) ] ( ) ( ) ( ( ) ) [ ( ) ] ( ) ( ) [ ] ( ) ( ( ) ) [ ( ) ] ( ) ( ) ( ( ) ) 279

280 [ ( ) ] ( ) ( ( ) ) [ ( ) ] ( ) ( ( ) ) ( ( ) ) [ ( ) ] ( ) ( ( ) ) [ ( ) ] ( ) 280