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