Pricing and hedging of FX plain vanilla options

Transcription

1 Pricing and hedging of FX plain vanilla options An empirical study on the hedging performance of a dynamic Black-Scholes delta hedge with updating implied volatility under the assumption of Heston and Black-Scholes underlying dynamics, respectively, in the interpolation/extrapolation of option prices. Jannik Nørgaard MSc Finance Thesis Supervisor: Elisa Nicolato Department of Business Studies Aarhus School of Business, University of Aarhus August 2011

2 c Jannik Nørgaard 2011 The thesis has been typed with Computer Modern 12pt Layout and typography is made by the author using L A TEX The author wish to thank the following: My supervisor Elisa Nicolato, Researcher at Aarhus School of Business in the Finance Research Group, Aarhus, Denmark for advice. A thanks to Matthias Thul, PhD Candidate in Finance at Australian School of Business, New South Wales, Sydney, Australia for answering questions. I thank the people who ve helped me gain access to the Bloomberg terminals at the University of Aarhus as well as the employees at the Bloomberg service desk for answering my questions. Lastly, thanks to Nordea for providing me the access to the Nordea Markets platform, Nordea Analytics, from where I ve gathered supplementary data.

3 I want to take the opportunity to thank my parents for their unconditional support during my years of study.

4 Abstract The thesis shows evidence against the Black-Scholes assumption of a diffusion process for the log asset price that has stationary and independent normal increments resulting in a log-normal distribution of asset returns by considering a time-series of spot rates on the EURUSD and the USDJPY covering a period of recent years. Observations of distributions exhibiting high peakness and "fat tails" as well as observations of volatility clustering are supported by empirical evidence of heteroscedasticity, implying that the volatility of returns is not constant over time, and evidence of autocorrelation. In order to calibrate The Heston model and the Black-Scholes model to market prices on plain vanilla call options the thesis deals with the foreign exchange specific quoting conventions and considers the difference here between the EURUSD and the USDJPY. A data set of 371 recent trading days are collected from published quotes on Bloomberg where each model is calibrated to a set of option prices on each day to obtain an overall goodness of fit measure that shows the superior performance of the Heston model. In the case of both underlying FX pairs the volatility surface is negatively skew shaped throughout the period considered. Based on the calibrations a large scale hedging experiment is set up where a number of plain vanilla call options with different maturities and strikes is sold on each day. A dynamic BS Delta hedge with updating implied volatility simulated in each of the models results in a better hedging performance when the underlying dynamics follows the Heston model. Furthermore we observe that the hedging error is correlated with the underlying returns.

5 Contents Contents List of Figures List of Tables i iii v 1 Introduction 1 2 Problem Statement Research Approach Delimitation The FX Market FX rate FX forward contract FX options The Black-Scholes model Geometric Brownian Motion The Black-Scholes equation The Garman-Kohlhagen formula Simulation of the Black-Scholes model Empirical facts The distribution of FX returns The Heston model The process The solution Simulation of the Heston model Market data 29 i

6 7.1 Quoting conventions Retrieving the implied volatility Data description 35 9 Calibration of the models Building the market implied volatility surface Calibration of the Heston model Calibration of the Black-Scholes Model Objective Function Calibration results Empirical study on the hedging performances Size of the study Strike levels The hedging portfolio Results Conclusion 55 Bibliography 57 A Retrieving the strike price corresponding to a premium included Delta 60 B Building the market implied volatility surface 63 C Calibration of the Heston model 76 D Calibration of the Black-Scholes model 82 E Simulation of the Heston model 85 F Simulation of the Black-Scholes model 89 G No hedge 92 H Dynamic BS Delta Hedge with updating imp. vol. from the Heston model 97 I Dynamic BS Delta Hedge with updating imp. vol. from the Black-Scholes model 109 ii

7 List of Figures 5.1 Empirical sample frequency for EURUSD Empirical sample frequency for USDJPY Q-Q plot for EURUSD Q-Q plot for USDJPY Daily log returns for EURUSD Daily log returns for USDJPY Autocorrelation for EURUSD Autocorrelation for USDJPY Rolling historic volatility for EURUSD Rolling historic volatility for USDJPY One week moving average of κ One week moving average of θ One week moving average of η One week moving average of ρ One week moving average of v t Call prices 1M on EURUSD 1/4/ Call prices 1Y on EURUSD 1/4/ Imp. vol. 1M on EURUSD 1/4/ Imp. vol. 1Y on EURUSD 1/4/ Call prices 1M on EURUSD 6/1/ Call prices 1Y on EURUSD 6/1/ Imp. vol. 1M on EURUSD 6/1/ Imp. vol. 1Y on EURUSD 6/1/ Call prices 1M on USDJPY 1/4/ Call prices 1Y on USDJPY 1/4/ Imp. vol. 1M on USDJPY 1/4/ Imp. vol. 1Y on USDJPY 1/4/ Call prices 1M on USDJPY 6/1/ Call prices 1Y on USDJPY 6/1/ iii

8 9.20 Imp. vol. 1M on USDJPY 6/1/ Imp. vol. 1Y on USDJPY 6/1/ Development in EURUSD spot rate Development in USDJPY spot rate iv

9 List of Tables 5.1 Jarque-Bera test on normality Levene s test on equality of variances Premium included Delta Conversion of a Premium Included Delta to Strike Quarterly mean and standard deviation of the goodness of fit of Heston parameters Quarterly mean and standard deviation of the goodness of fit of the Black-Scholes parameter Heston parameter values on 1/4/2010 and 6/1/2010 on EURUSD Heston parameter values on 1/4/2010 and 6/1/2010 on USDJPY Black-Scholes parameter values on 1/4/2010 and 6/1/2010 on EURUSD and USDJPY Number of options under investigation Number of option expirations in quarterly periods Delta level on average of shorted EURUSD call options at initiation Delta level on average of shorted USDJPY call options at initiation Number of EURUSD call options expiring in-the-money Number of USDJPY call options expiring in-the-money The mean profit and loss and standard deviation on the hedging error with Black-Scholes and Heston pricing v

10 1 Introduction In a financial world which have experienced market crashes starting with Black Monday in 1987 the introduction of extreme market movements have given rise to the reconsideration of the assumptions behind the pricing of financial instruments such as options on stocks as well as foreign exchange. In the past market participants and practitioners have relied more on the Black- Scholes model and its assumption about asset returns whereas today the market prices of options do not reflect those predicted by the Black-Scholes model. Instead a family of stochastic volatility models has emerged, with the Heston model being the most well known, with more realistic assumptions about the probability distribution of asset returns today. Still though, the Black-Scholes model are applied by market participants and practitioners in circumvention that avoid its flaws. This thesis incorporates the application of both types of models and tries to uncover pricing misspecifications and, in an empirical study, investigates if one is preferable to the other given a specific pricing and hedging setting. In chapter 3, we start by given an introduction to the FX market and FX plain vanilla options, which are traded over-the-counter (OTC). This fact influences the data collected to represent the market prices, which in this case is retrieved from Bloomberg where an arbitrage free volatility surface is reported from a collection of option quotes from several contributors representing the worlds largest financial institutions. As opposite to exchange traded options that are quoted with a fixed maturity date and with the initiation of new options only on fixed dates, from Bloomberg we are provided with a full set of new options everyday covering the same range of maturities just with the expiration one day later than the previous days quoted options. 1

11 Chapter 4 covers the Black-Scholes (BS) model and its assumptions about lognormally distributed asset returns. With specific interest in the pricing of FX options we present the Garman-Kohlhagen formula, which is a simple extension to the BS model. In this chapter we furthermore introduce the concept of the implied probability density function and risk neutral valuation. Finally we present the simulation of the BS model. In Chapter 5 we analyse the distribution of FX log returns considering a sample of recent years spot FX rates and compare this with the assumption of log-normal distributed returns in the BS model. The findings here inspire to consider different assumptions on the distribution of log returns, which leads us to introduce a stochastic volatility model in the next chapter. Chapter 6 then introduces the process and the closed form solution to the Heston model. In the calibration of the Heston model we calibrate to this closed form solution by numerical integration. Furthermore we present the simulation of the Heston model that is carried out in a mixing solution framework simulated in a Milstein scheme. Before the empirical study we present chapter 7, which explains the very FX specific quoting conventions. More comprehensive than other option markets the FX option market has a wide range of possible conventions which need to be properly handled in order to be able to build a volatility surface based on the quotes in the market. More specifically the volatilities are quoted in trading structures that needs to be converted. Moreover the options are quoted in terms of Delta in the moneyness dimension. Depending on the Delta convention of the specific FX pair, we need to use a numerical estimation technique to retrieve the strike level. Chapter 8 consists of an overview of the data used in the empirical study. In chapter 9 we then calibrate the BS model and the Heston model to each day of 371 trading days in the period from 1/4/ /22/2011. We present the objective function and its inherited weighting scheme that is common for both models. We furthermore analyse the sensitivity of the volatility surface to the change in Heston parameters by looking at two different days. Also a comparison between the ability of the two models to fit the observed market prices is done by calculating the goodness of fit for each model. In chapter 10 we lay out the hedging strategy consisting of a dynamic BS Delta hedge with updating implied volatility simulated in the BS model and simulated in the Heston model. More specifically we hedge a number of shorted call options with different maturities and strike levels. We then identify which elements that change the value of the hedging portfolio. Finally we present the findings of the study comparing the BS model as a tool in the interpolation/extrapolation of the updating 2

12 implied volatility to the Heston model by comparing the hedging performance of the same BS Delta hedge. 3

13 2 Problem Statement In this study we consider the two FX pairs EURUSD and USDJPY. We start by the following introductory research questions: I. How are FX returns distributed considering a period of recent years? II. How does the distribution of FX returns compare to the assumptions about log-normal distributed asset returns in the Black-Scholes model? As pointed out by (Reiswich and Wystrup, 2010), the smile construction procedure and the volatility quoting mechanisms are FX specific and differ significantly from other markets...market participants entering the FX OTC derivative market are confronted with the fact that the volatility smile is usually not directly observable in the market...unlike in other markets, the FX smile is given implicitly as a set of restrictions implied by market instruments. This lead us to the question: III. How do we handle the FX specific quoting conventions in order to end up with market prices on plain vanilla option. In a very recent paper "Applying hedging strategies to estimate model risk and provision calculation" (Elices, 2011) the authors study the hedging performance of the BS model and the Vanna-Volga method by assuming that the market volatility surface is driven by Heston s dynamics calibrated to market for a given time horizon. The hedging strategy is then built in order to neutralize the uncertain factors in the Heston model which consist of the spot and the volatility. In the same way, we rely on a model dependant building of the volatility surface by calibration of the BS model and the Heston model, respectively, to the observed market prices. 4

14 IV. How well does the Black-Scholes and Heston model, respectively, reflect a set of market prices on plain vanilla options over a recent period? Then we use these calibrations in order to investigate how well a pure Delta hedging strategy, with the Delta calculated as a BS Delta, is able to replicate the payoff of a plain vanilla FX call option contract. We create a setting where a set of European plain vanilla FX options with different maturities and strikes are sold every day during a period of 371 trading days. By delta hedging each option contract individually until its expiry, we obtain the hedging error that we express as the difference between the payoff of the option contract and the hedging portfolio. Two experiments are set up where we calculate the BS Delta dynamically with an updating volatility from the Black-Scholes model and an updating implied volatility from the Heston model. This leads to the final research questions: V. Applying a dynamic BS Delta hedge with updating implied volatility under the assumption of Black-Scholes underlying dynamics, what is the standard deviation of the hedging error for each option contract? VI. Applying a dynamic BS Delta hedge with updating implied volatility under the assumption of Heston underlying dynamics, what is the standard deviation of the hedging error for each option contract? VII. Are the outcome of the hedging correlated with the market return? 2.1 Research Approach We point out and argue for our choice of research approach in three areas of the thesis: The inclusion of two different FX pairs, the building of the implied volatility surface and the range of option prices used to build the implied volatility surface. We choose to include both the EURUSD and the USDJPY in the study because of mainly one reason. The quoting conventions for the two pairs are different and by including both we show how to handle these different quoting conventions. In addition to this reason, the volatility surface of these two pairs has historically had different shapes with the EURUSD exhibiting more of a symmetrical smile and the USDJPY exhibiting a step skew (Bossens, Rayee, Skantzos, and Deelstra, 2010), (Beneder and Elkenbracht- Huizing, 2003), (Chalamandaris and Tsekrekos, 2008). Like other studies this is an attempt to cover a different set of market conditions (Bossens, Rayee, Skantzos, and Deelstra, 2010). 5

15 We calibrate to raw data where no interpolation or extrapolation has taken place beforehand. Alternatively we could have used a SVI parametrisation (Gatheral, 2006) or some other functional form to first build the surface and then calibrate to a set of interpolated/extrapolated prices. We calibrate to only a few number of options counting 5 different maturities and 5 different strike levels. This is done because of two reasons. First, we want to calibrate only to raw data that has not yet been interpolated in Bloomberg s own interpolation scheme, which can be seen in (Bloomberg, 2011). Bloomberg s interpolation is based on ATM, 25 Delta and 10 Delta quotes and if available also and 5 Delta (Bloomberg, 2009). This fact ensures us that we only calibrate to raw data. Second, a lot of effort has gone into the development of methods that are able to build the full implied volatility surface with only a few set of option prices (Malz, 1997), (Castagna and Mercurio, 2006), (Reiswich and Wystrup, 2010). On an OTC option market, often only a few prices is available and we want to restrict this study to include only the prices that are most often available. This thesis uses the same range of option prices from the same source as in U. Wystrup and D. Reiswich s article "FX Volatility Smile Construction" (Reiswich and Wystrup, 2010) by using the ATM, 10D RR, 25D RR, 10D VWB and 25D VWB quotes published on Bloomberg. 2.2 Delimitation The thesis is limited in areas where additions would bring more accuracy and detail into the study. In order to test a pricing model for its misspecifications a classical hedging experiment like the one carried out in (Bakshi, Cao, and Chen, 1997) and (Elices, 2011) could be done. Here, they test a model s ability to replicate an option payoff by taking positions in all assets necessary to neutralize risk with that number depending on the assumption of the given pricing model. For the Heston model this implies taking a position in both the underlying and another option in order to obtain a delta-neutral hedge. In this thesis we restrict ourselves to only take a position in one asset, the underlying. So this study cannot be classified under this type of conventional approach. The interest rate setting in this study is simplified. There has been no building of an interest rate term structure to use in the simulation of the option pricing 6

16 models. Nor have we considered option pricing models with stochastic interest rates like in (Bakshi, Cao, and Chen, 1997). Also we disregard the topic of default risk in interest rates which is a warm topic today after the current financial crises. No jump models has been considered like a stochastic volatility plus jump in the underlying (SVJ) model. These types of models are better at reflecting the volatility surface in the short term in comparison with a stochastic volatility model (Gatheral, 2006). Considering the two FX pairs included in this study, and the shape of their respective volatility surfaces, a SVJ model might not even have been able to improve the pricing fit in comparison to a stochastic vol. model. Researchers point out a necessary adjustment of the volatility quote on steply skewed markets (Reiswich and Wystrup, 2010), (Bossens, Rayee, Skantzos, and Deelstra, 2010), (Castagna, 2010). About the quoting conventions on the foreign exchange option market and the importance of the specific adjustment of the vega weighted butterfly (VWB) quote, the following is said:...a market inconsistency that can safely be disregarded in many situations and configurations of prices, but can have a deep impact on the volatility surface building in others. (Castagna, 2010, p. 116). We have excluded the estimations of such an adjustment. Probably the most important limitation of this study is the number of simulations used. This concerns the simulation of the Heston model and the BS model in the hedging experiment set up. The precision of the pricing in the Heston model could be improved by increasing the number of simulations, resulting in an even better hedging performance, assumingly. 7

17 3 The FX Market 3.1 FX rate A foreign exchange rate (FX rate) is the price of one currency in terms of another currency. The two currencies make a currency pair. As an example this could be the currency pair labelled EURUSD. This is the euro/us dollars exchange rate and by the end of the trading day on the 1 th of May 2011 this was quoted at This is the convention on how to quote this particular currency cross, but it is equivalent to USDEUR , which is just the reciprocal value of the first FX rate. The exchange rate EURUSD denotes how many US dollars are worth 1 euro. The domestic (numeraire) currency is the US dollar and the foreign (base) currency is the euro. So generally speaking, the exchange rate is the price of the base currency in terms of the numeraire currency. The last time a US dollar was worth more than a euro was on the 4 th of December 2002 on which day the exchange rate was quoted at Since after the introduction of euro coins and banknotes on the 1 th of January 2002 this has been the only year that the US dollar has been worth more than the euro, reflected in an exchange rate less than FX forward contract The forward contract provides a hedge for someone who wants to lock in the exchange rate for a future transaction. The buyer of a forward contract is then guaranteed a future exchange rate. The forward price is decided as F 0 = S 0 e (rd r f )T (3.1) 8

18 The underlying asset in such contracts is a certain number of units of the foreign currency. The variable S 0 is defined as the spot price in domestic currency of one unit of the foreign currency and equivalently F 0 is the forward price in domestic currency of one unit of the foreign currency. Both domestic and foreign interest rates are the continuously compounded risk-free interest rates per annum Interest rate parity Equation 3.1 is exactly the interest rate parity, which in its continuous compounding form is often equated as F (t, T ) = S t e rf (T t) e rd (T t) (3.2) or by its money market conventions for capitalization and discounting, i.e simple compounding (Castagna, 2010, p. 7) F (t, T ) = S t (1 + r f )(T t) (1 + r d )(T t) (3.3) where r f and r d are the risk-free interest rates per annum and (T-t) follow the time convention of 360 trading days in a year. According to the interest rate parity, the forward exchange rate of a given currency pair is determined by the respective risk-free interest rates. As an example, we consider a holder of one unit of foreign currency. There are two ways that this can be converted into domestic currency at time T. One is by investing it for (T t) years at r f and at the same time selling a forward contract. Then at time T you would be obligated to sell the proceeds from the investment to collect domestic currency. The other possibility is to exchange the foreign currency to domestic in the spot market and then invest these at r d for (T-t) years. In the absence of arbitrage opportunities equation 3.4 should then hold (Hull, 2008, p. 113), which is exactly equation 3.2 rewritten. e rf (T t) F 0 = S 0 e rd (T t) (3.4) The interest rate parity presented here is also called the covered interest rate parity as opposite to the uncovered interest rate parity (Oldfield and Messina, 1977). The former comes from the fact that the trading strategy is risk-free. This is opposite to the latter where you as a holder of the foreign currency still invest in r f, but instead 9

19 of simultaneously entering into a forward contract, you instead keep your position in foreign currency uncovered and exposed to the movement in the exchange rate from t to (T t). Empirical research shows that for developed countries, the covered interest rate parity holds fairly well. Prior to the dismantling of capital controls, and in many emerging markets today (interpreted as political risk associated with the possibility of governmental authorities placing restrictions on deposits located in different jurisdictions), the covered interest rate parity is unlikely to hold (Chinn, 2007). From an option pricing point of view the covered interest parity is an underlying assumption in one of the option pricing models introduced later on here. 3.3 FX options FX options are traded Over-The-Counter (OTC) as opposite to exchange traded options. As a trading platform an exchange serves as a link between a buyer and a seller. The exchange will be providing bid and ask quotes and will be on either one or the other end of the transaction. The market making is in this case carried out by the exchange. In the case of FX options there is no exchange involved in the transaction. A trade will be processed directly between buyer and seller. In one setting, one might think of a buyer being a corporation that is trading from a hedging or speculative point of view and the seller being a bank. On the FX options market one might think of the banks as market makers providing the prices on options and other FX derivatives. In order to hedge a foreign exchange exposure FX options are an alternative to FX forward contracts. The payoff from a long position in a European call option is max(s T K, 0) (3.5) and the payoff from a long position in a European put option is max(k S T, 0) (3.6) with S T being the spot exchange rate at maturity T of the option and K the agreed upon strike price. 10

20 Assuming we have the pair EURUSD, two counterparties entering into a plain vanilla FX option contract can agree on the following, according to the type of option traded: Type EUR call USD put: The buyer has the right to enter at expiry into a spot contract to buy (sell) the notional amount of EUR (USD), at the strike FX rate level K. Type EUR put USD call: The buyer has the right to enter at expiry into a spot contract to sell (buy) the notional amount of EUR (USD), at the strike FX rate level K. Considering, as an example, the last type listed above, an American company due to receive euro at a known time in the future can hedge its risk by buying put options on euro that mature at that time. This strategy guarantees that the value of the euros will not be less than the strike price while still allowing the company to benefit from any favorable upward movements in the exchange rate. Similarly, if the company where to pay euros in the future they could hedge their expose to upward movements in the exchange rate by buying calls on euros, the first type listed above. whereas forward contracts locks in the exchange rate for a future transaction and guarantees the parties an exchange rate, as described above, an option provides a type of insurance. It costs nothing to enter into a forward contract, whereas options require a premium to be paid paid up front in order to be insured. 11

21 4 The Black-Scholes model This chapter reviews the most well-known option pricing model, The Black-Scholes model (Black and Scholes, 1973), because of its inclusion in the empirical study. Also it remains the building block of present option pricing models, including the Heston model and the Bates model. 4.1 Geometric Brownian Motion Black-Scholes assumes the underlying spot price to follow a geometric Brownian motion generating log-normally distributed returns, the spot price in this case being the exchange rate on any given FX pair. The process is stochastic by including a Wiener process that introduces the randomness to the spot price. ds t = µs t dt + σs t dw = S t (µdt + σdw ) (4.1) The spot price S t depends on S t itself, a constant drift, µ, a constant volatility term, σ, and a standard Wiener process, W t, where dt is denoting a time differential. In order to obtain the explicit solution to this stochastic differential equation (SDE) we consider equation 4.2 the process of logs, i.e. the process describing the log-returns. dlogs t = (µ 1 2 σ2 )dt + σdz (4.2) i.e logs T = logs 0 + (µ 1 2 σ2 )T + σdz (4.3) 12

22 and the explicit solution is then obtained by taking the exponential of logs S T = S 0 e (µ 1 2 σ2 )T +σz T (4.4) 4.2 The Black-Scholes equation With the empirical study of this thesis in mind we have a look at the derivation of the Black-Scholes (BS) equation which is governing the BS option pricing formula. This will tell us the principle of delta hedging. Furthermore we take a look at the necessary adjustments to the Black Scholes equation in order to be able to price FX options in particular. As a note it is not in the interest of this thesis to go through the derivation of the solution to the BS equation that will lead to the BS formula. The Black-Scholes equation can be derived in many alternative ways i.e. using empirically established financial theories such as the CAPM and Arbitrage Pricing theory. The most general derivation assumes an economy with only the underlying asset and a risk-free money market deposit/risk-free bond which together makes up the replicating portfolio of the value of the derivative. Meanwhile, the original derivation uses what is known as the hedging argument, and that is the derivation that we will outline here (Rouah, 2011). The derivation follows from imposing the condition that a risk-free portfolio made up of a position in the underlying asset and the option on that asset must return the same interest rate as other risk-free assets. As a result of this Black and Scholes propose that if it is possible to hedge an option position by dynamically rebalancing a stock position, then the price of a European call option should depend on the underlying spot price, S t (i.e. the FX rate), and the time to maturity on the option, T. In order to perform such a hedge Black and Scholes assumes a set of conditions to hold that they call the ideal market condition: The FX rate, S t, follows the geometric Brownian motion with known constant drift, µ, and volatility, σ. The option can be exercised only at maturity. Trading takes place continuously in time. Money can be borrowed and lend at the same risk-free interest rate. Short selling is allowed. 13

23 Short-term risk-free interest rates (r d and r f ) are known and constant. The underlying asset pays no dividends. (This assumption is relaxed in the case of FX options.) We consider a portfolio made up of a quantity of the risky asset (i.e. the FX pair) and short one option on the FX pair (a put or a call, not yet specified). Let f(s, t) denote the value of the option and Π(t) the value of the portfolio. Π(t) = S f(s, t) (4.5) is chosen at every time t so as to make the portfolio riskless. The self-financing assumption implies that dπ(t) = ds df(s, t) (4.6) In order to decide the quantity to meet this condition we want to know the dynamics of f(s, t). Here we use Ito s Lemma, which is a rule for calculating differentials of quantities dependent on stochastic processes. df(s, t) = f f dt + t S ds σ2 S 2 2 f dt (4.7) S2 and by plugging in 4.7 into 4.6 we get dπ = ds ( f f dt + t S ds σ2 S 2 2 f S )dt 2 ( f )ds ( f S t σ2 S 2 2 f )dt (4.8) S2 observing that the term ds is the only risky element to the portfolio value, we can eliminate this by setting which is satisfied if ( f S ) = 0 = f S (4.9) Then we have constructed a risk-free portfolio with the dynamics given in the last part of 4.8 and by a no arbitrage argument the portfolio must yield the risk-free interest rate, i.e. 14

24 dπ = rπdt (4.10) Plugging the risk-free dynamics of the option value in 4.8 and the first equation 4.5 into 4.10 and rewrittin, we get the BS equation in ( f t σ2 S 2 2 f f )dt = r( S f(s, t))dt S2 S f t 1 2 σ2 S 2 2 f f = r( S f(s, t)) S2 S f t σ2 S 2 2 f S + r f S rf = 0 (4.11) 2 S The derivation stipulates that in order to hedge the single option, we need to hold a quantity of the FX pair, which turns out to be the quantity f. This is the S principle behind delta hedging. Any price of a derivative with the same assumed process for the underlying as in equation 4.1 has to follow the BS equation.the equation has many solutions for the derivative price, f, where the particular price that is obtained depends on the payoff function of the given derivative. In the case of a European call/put the solution is obtained in the BS formula, but for more complex payoff functions accompanied by more exotic options the analytical solution may be hard to obtain. 4.3 The Garman-Kohlhagen formula In the same year 1973 as the Black and Scholes paper was published the pricing model was quickly adjusted to include dividend paying stocks by Merton (1973). Robert C. Merton further concludes in this paper that the assumption of lognormally distributed returns and continuous trading is critical to the model. Without these, the delta hedge would not give a perfect hedge, thus making the arbitrage argument invalid. Many years later after the FX options was first listed on the Philadelphia Stock Exchange in 1982 (Exchange, 2004), the pricing model was adjusted to also be able to price FX plain vanilla options (Garman and Kohlhagen, 1983). Under similar assumptions as in Black-Scholes, that it is possible to operate a perfect local hedge between a FX option and underlying foreign exchange, Garman and Kohlhagen derive a PDE. One of the insights is that the risk-free interest rate of foreign currency r f has the same impact on the FX option price as the continuous dividend yield on the stock option. The main contribution is to combine the Black-Scholes model with the interest rate parity theory, as presented in the 15

25 beginning of this thesis. More precisely, by assuming the covered interest rate parity to hold and the underlying FX rate to follow a geometric brownian motion, the logarithmic difference between the forward, F (t, T ), and the spot, S(t), FX rates can be explained by the spread between the domestic risk-free interest rate, r d, and the foreign risk-free interest rate, r f. The resulting pricing formula for a call option in equation 4.12 is presented in its forward rate form, where the forward rate is explicitly present in the formula. This is a Black model (Black, 1976) (adjusted to price FX options), which is a variation of the original BS model and can be generalized into a class of models known as log-normal forward models. The adaption of the covered interest rate parity into the option pricing formula becomes apparent when we compare the calculation of the forward rate in Equation 4.12 to Equation 3.2. c = e rd (t,t )τ) [F (t, T )φ(d 1 ) Kφ(d 2 )] (4.12) d 1 = F (t,t ) ln( ) + 1 K 2 σ2 τ σ τ d 2 = d 1 σ τ F (t, T ) = S t e rf (t,t )τ e rd (t,t )τ with the the equivalent spot rate form of the Garman-Kohlhagen formula c = S 0 e rf (t,t )τ φ(d 1 ) Ke rd (t,t )τ φ(d 2 ) (4.13) d 1 = ln( S 0 K ) + (rd (t, T ) r f (t, T ) σ2 )τ σ τ d 2 = d 1 σ τ The foreign and domestic interest rates are risk-free and constant over the term of the option s life. All interest rates are expressed as continuously compounded rates Implied Probability Density Functions In order to establish a link between the observed option prices in the market and the characteristic shapes of the volatility surface we mention the implied risk-neutral density function (RND). 16

26 The RND in the Black-Scholes model is assumed to be lognormal with mean (r d r f v 2 /2)(T t) and variance v 2 (T t). The price of an undiscounted call option is given by C(S 0, K, T ) = E[max{S T K, 0}] (4.14) = K (s K) φ(s; T, S 0 )ds (4.15) where φ(s; T, S 0 ) in (4.15) is the probability density function of S T. This is a general pricing formula independent of the choice of pricing model. Pricing an option in this framework requires the knowledge of the probability density function, which is the distribution of the future spot prices. (Breeden and Litzenberger, 1978) found that provided a continuum of European call options with same maturity and a strike range going from zero to infinity written on a single underlying FX pair, we can recover the RND in a unique way by differentiating (4.15) with respect to K twice Risk-neutral valuation C K = φ(s; T, S 0 )ds (4.16) K 2 C K = φ(s; T, S 0)ds (4.17) 2 Another approach to find the price of a derivative is by risk neutral valuation or equivalently by the Martingale approach. The equivalence between the PDE approach and the risk neutral valuation is guaranteed by Feynman-Kac by establishing a link between PDEs and stochastic processes. The solution to the Garman-Kohlhagen equation can also be expressed in terms of an expectation. By the Feynman-Kac theorem we have V (S t, t) = E Q [ ] e T t rs dds V (S T, T ) (4.18) where S t is the solution to the SDE (4.1) with µ = r d r f. The drift is risk neutral and consists of the continuously compounded domestic interest rate net of the foreign interest rate. What (4.18) says is that the value of a contingent claim (a claim that is dependant on the underlying value) like a European option, can be calculated by finding the risk neutral expectation of the discounted terminal payoff. The terminal payoff is discounted by the domestic interest rate and the risk neutral 17

27 expectation and the Q measure involves the process of S T to evolve not as original but risk neutrally. To recapitulate the general pricing framework above, there is a connection between the existence of a replication portfolio replicating the final value of the option, and the existence of a equivalent martingale measure. They both guarantee an arbitrage-free price. This can be calculated as the current value of the replication portfolio, or as the expected value of the discounted terminal payoff of the option calculated under the risk-neutral probability measure. 4.4 Simulation of the Black-Scholes model We consider the risk neutral process in Equation (4.19) and compute the risk neutral expectation of the terminal payoff as suggested by the Feyman-Kac theorem. ds t = (r d t r f t )S t dt + σ t S t dw (4.19) 18

28 5 Empirical facts 5.1 The distribution of FX returns Empirically we observe a departure from the normality assumption in the Black- Scholes model when we have a look at the distribution of log returns on EURUSD and USDJPY. In figures 5.1 and 5.2 the frequency distributions of two samples of daily log returns from 1/6/2006-5/3/2011 is pictured. A lognormal distribution with the same mean and standard deviation as the implied distribution is depicted by the solid line. The empirical distributions are highly peaked compared to the normal distribution. Furthermore from figures 5.3 and 5.4, which depict a Q-Q plot of the log returns vs. a normal distribution, we can observe that the empirical distributions of log returns does in fact exhibit fat tails and clearly deviates from the normality assumption. From the visual evidence of a highly peaked and fat tailed distribution (leptokurtic), we can conclude that small and large movements in the empirical samples occur more likely compared to normally distributed log returns. By looking at figures 5.5 and 5.6, where we plot the daily log returns of EURUSD and USDJPY, we see that large moves follow large moves (both up and down) and small moves follow small moves (both up and down). This is the so-called volatility clustering, where we observe that high and low volatility is clustered around certain time periods. This observation indicates autocorrelation, which is confirmed in Figures?? -??. Here the autocorrelations of absolute returns are estimated where all lags included is significantly positive. In addition to this, Figures 5.9 and 5.10 demonstrates mean reversion in the log returns by showing how volatility evens out when measured over a longer horizon. 19

29 Sample frequency Daily log- return EURUSD Sample frequency Daily log- return USDJPY Figure 5.1: Empirical sample frequency for EURUSD Figure 5.2: Empirical sample frequency for USDJPY 0.04 QQ Plot of Sample Data versus Standard Normal 0.06 QQ Plot of Sample Data versus Standard Normal Quantiles of Input Sample Quantiles of Input Sample Standard Normal Quantiles Standard Normal Quantiles Figure 5.3: Q-Q plot for EURUSD Figure 5.4: Q-Q plot for USDJPY Daily log return 0.06 EURUSD Year Daily log return 0.06 USDJPY Year Figure 5.5: Daily log returns for EU- RUSD Figure 5.6: Daily log returns for USD- JPY Sample Autocorrelation Function Sample Autocorrelation Function Sample Autocorrelation Sample Autocorrelation Lag Lag Figure 5.7: RUSD Autocorrelation for EU- Figure 5.8: Autocorrelation for USD- JPY 20

30 Historic vola,lity 0.3 EURUSD Year 3 month 1 year Historic vola,lity 0.35 USDJPY Year 3 month 1 year Figure 5.9: Rolling historic volatility for EURUSD Figure 5.10: Rolling historic volatility for USDJPY Jarque-Bera To confirm our results and to find further evidence against the normality assumption underlying the Black-Scholes model we make use of the Jarque-Bera test (Jarque and Bera, 1987). Based on the sample kurtosis and skewness we test the null hypothesis that the data is drawn from a normal distribution. The null hypothesis is a joint hypothesis of the skewness being 0 and the excess kurtosis being 0, which in the latter case is the same as a kurtosis of 3. The overall conclusion by looking into tabel 5.1, when considering the full sample of log returns, is that we clearly reject the null hypothesis, that the sample data is from a normal distribution, in both the EURUSD and USDJPY case. This conclusion comes with a high degree of certainty with a significance level below 0.1%. When we then have a look at the separate years considering first the EURUSD, we are able to reject in 3 out of 6 years at a significance level of 5.0%, whereas for the USDJPY case this is 4 out of 6 years. When looking into the estimates of the overall skewness and kurtosis and comparing the two pairs, one observes that in terms of skewness the EURUSD deviates the most from a normal, whereas in terms of kurtosis it is the USDJPY that deviates the most from the normal. These differences in skewness and kurtosis between the two pairs is somewhat visual in figures 5.1 and 5.2 from before. Comparing the tails of the frequency distributions one might see that the EURUSD log returns has a longer right tail exhibiting more positive skewness whereas the USDJPY log returns has a longer left tail exhibiting more negative skewness (Even though apparently not enough for the full sample to be negatively skewed). Both distributions though are on an overall scale slightly positively distributed meaning that most values are concentrated on the left of the mean, with extreme values to the right (as opposite to negatively skewed distributions, where most values are concentrated on the right of the mean, with extreme values to the left). The difference in the kurtosis of the two pairs of log returns is also somewhat visual from the figures 5.3 and 5.4 from before, where the USDJPY 21

31 EURUSD Table 5.1: Jarque-Bera test on normality USDJPY period skewness excess kurtosis JB sign. level skewness excess JB sign. level > % % % < 0.100% < 0.100% < 0.100% % % > % < 0.100% > % % < 0.100% < 0.100% log returns seems to exhibit the most kurtosis. The test statistic JB is defined as JB = n 6 (S K2 ) (5.1) where n is the number of observations, S is the sample skewness in Equation 5.2 and K is the sample excess kurtosis in Equation 5.3. S = ˆµ 3 ˆσ 3 = 1 n n i=1 (x i x) 3 ( 1 n n i=1 (x i x) 2 ) 3 2 (5.2) K = ˆµ 4 ˆσ 4 3 = 1 n ( 1 n n i=1 (x i x) 4 n i=1 (x i x) 2 ) 3 (5.3) 2 where ˆµ 3 and ˆµ 4 are the estimates of the third and fourth central moments, respectively, x is the sample mean and ˆσ is the estimate of the second central moment, the variance. 22

32 5.1.2 Levene Excess kurtosis might indicate heteroscedastic returns, where homoscedastic returns is the assumption underlying the Black & Scholes model. We therefore perform the Levene s test of homoscedatic returns, where the null hypothesis is that the variance of two successive subsamples are equal as well as the variances of all subsamples. Considering the latter we strongly reject the hypothesis that the variance in the subsamples are constant thus violating the assumption in the Black Scholes model. Comparing the individual successive yearly subsamples, in the case of the EURUSD we are able to reject in 2 out of 5 cases at a significance level of 5%. In the case of the USDJPY this is 4 out of 5 cases in correspondence with the superior excess kurtosis compared to the EURUSD case. Table 5.2: Levene s test on equality of variances EURUSD USDJPY period 1 period 2 volatility 1 volatility 2 Levene sig. level volatility 1 volatility 2 Levene sig. level % 6.16% 0.859% 7.83% 9.62% 1.244% % 13.78% 0.000% 9.62% 16.18% 0.000% % 12.03% 9.691% 16.18% 12.68% 1.659% % 11.76% % 12.68% 10.36% 2.458% % 9.85% 7.890% 10.36% 9.87% % % 0.000% 23

33 6 The Heston model The most well-known and popular of all stochastic volatility models is the Heston model (Gatheral, 2006) and was presented in (Heston, 1993). 6.1 The process The process followed by the underlying asset in the Heston model is with ds t = µs t dt + v t S t dw (1) t (6.1) dv t = κ(v t θ)dt + η v t dw (2) t (6.2) dw (1) t dw (2) t = ρdt where κ is the rate of reversion of v t to the long run variance, θ, η is the volatility of volatility and ρ is the correlation between the two stochastic increments of the processes dw (1) t and dw (2) t. The process of the underlying in (6.1) is the same process assumed in the Black Scholes model presented in (4.1) only now the volatility is stochastic. That is, another random factor is introduced by dw (2) t. What defines the specific process of the underlying in the Heston model compared to the general case of stochastic volatility models is dv t = α(s t, v t, t)dt + ηβ(s t, v t, t) v t dw (2) t (6.3) α(s t, v t, t) = κ(v t θ) β(s t, v t, t) = 1 24

34 where the process followed by the instantaneous variance, v t, can be categorized as a version of the square root process (CIR) in (Cox, Ingersoll Jr, and Ross, 1985). Given that the Feller condition in equation (6.4) is satisfied the variance process is always strictly positive. (Anderson, 2005) shows that this condition is often violated when calibrating the Heston model to market data. 2κθ η 2 (6.4) What makes the Heston stochastic volatility model stand out from other stochastic volatility models can be adressed to two reasons. First, the volatility process is non-negative and mean reverting which is what we observe in the market. Secondly, The Heston model has a semi-analytical closed form solution for European option, which is fast and relatively easy to implement. The closed form solution is especially useful when calibrating the parameters in the model to the observed vanilla option market. This efficient computational ability of the model is characterised as the greatest advantage of the model over other potentially more realistic SV models (Janek, Kluge, Weron, and Wystup, 2010). Furthermore, after adapting the model to a FX setting, the model is described as being particular useful in explaining the volatility smile found in FX markets often characterised by a more symmetrical smile when comparing to equity markets where the structure is a strongly asymmetric skew as a consequence of the leverage effect on these markets(janek, Kluge, Weron, and Wystup, 2010). 6.2 The solution The PDE of the Heston model can be derived using the same approach as when we derive the PDE for the BS model where standard arbitrage arguments is used. In addition to the replication portfolio used to derive the BS model another asset in the form of an option is added in order to hedge the randomness introduced by the stochastic volatility. The following PDE can then be derived V t vs2 2 V S 2 + ρηvs 2 V v S η2 v 2 V v 2 V + rs S rv +{κ(θ V ) λ(s, v, t) v} V v = 0 (6.5) where λ(s, v, t) is the market price of volatility risk. The closed-form solution of a European call option on an FX pair for the Heston model is S t P 1 Ke (r d r f )(T t) P 2 (6.6) 25