PRICING ASIAN CURRENCY OPTIONS

Transcription

1 PRICING ASIAN CURRENCY OPTIONS A project submitted in the partial fulfillment of the requirements for the degree, Master of Science in Actuarial Sciences. by 'Ogutu Carolyne A. 156/71040/2008 University of Nairobi, School of Mathematics, Chiromo Campus.

2 Declarations This project is my original work and has not been presented for a degree in any other university. ftww OGUTU CAROLYNE A. CANDIDATE This project has been submitt^^lw^^ with our approval university supervisors as Mr. I. Mwaniki i

3 Dedication To my late grandfather George Ogoseah, you will forever be in my heart.

4 Acknowledgements Many people have contributed in various ways to the contents of this project. All of them cannot be mentioned by name, however, their contributions are most sincerely acknowledged. I wish to thank the University of Nairobi for an opportunity to undertake this Masters degree at the institution. My gratitude goes to my fellow classmates and my lecturers for their constant inspiration by their words of criticism and encouragement. To the Head of Actuarial Sciences, Dr. Patrick G.O. Weke, thank you for believing in me and facilitating my accomplishing this worthy course. I miss the right words to thank my academic mentors and supervisors, Mr. C. Achola and Mr. I. Mwaniki for their interest in my work and more so, for the keenness with which they read and discussed the drafts of this project. My heartfelt gratitude and thanks are due to my family for the many hours, days, weeks and months they endured my absence in preparation of this project. To my friends, thank you, for your input was invaluable. It will reward you to learn that the project is as much yours as it is mine. Thank you sincerely. Overally, I thank God for the gift of life and strength to be able to accomplish this work.

5 Abstract Derivatives are financial instruments that whose values depend on some underlying assets. Options give the owner the right but not the obligation to buy or sell the underlying asset at a price called exercise price. Path-dependent Options are options whose payoff depends on the price history of the underlying asset. These options play an important role in financial markets. The Asian Option is one of the prominent examples of path-dependent options. An Asian Option is an option whose payoff depends on the arithmetic average price of the asset. Pricing Asian Options efficiently and accurately has had long-standing research, and numerous approximation methods are suggested in academic literature. In Kenya, though Option pricing has not been introduced, there is considerable trading in Currency Options over-the-counter. In this project, I will explore Asian option pricing on the lattice (binomial and trinomial lattices). A lattice divides the time interval between the option initial date and the maturity date into n equal time steps. Finally, I will use these lattice methods to price Asian currency options in Kenya. iv

6 List of tables and figures List of Tables Table 3.1 Table 3.2 Table 3.3 Pricing US, British and Euro currency call options by the Black Scholes Option Pricing Model 22 Pricing US, British and Euro currency call options by the Binomial Option Pricing Model 30 Comparison of binomial and trinomial lattices with different time steps 40 Table 4.1 Summary of estimated values 50 Table 4.2 Euro asset prices generated with u=1.003 and d= Table 4.3 Euro ordinary currency call values 51 Table 4.4 Euro average asset prices 51 Table 4.5 Euro Asian currency call values 51 Table 4.6 British pound asset prices generated with u=1.003 and d= Table 4.7 British pound ordinary currency call values 52 Table 4.8 British pound average asset prices 52 Table 4.9 British pound Asian currency call values 52 Table 5.0 US dollar asset prices generated with u=1.002 and d= Table 5.1 US dollar ordinary currency call values 53 Table 5.2 US dollar average asset prices 53 Table 5.3 US dollar Asian currency call values 53 Table 5.4 Trinomial Euro asset prices generated with u=1.003 and d= Table 5.5 Trinomial Euro ordinary currency call values 54 V

7 Table 5.6 Trinomial Euro average asset prices 55 Table 5.7 Trinomial Euro Asian currency call values 56 Table 5.8 Trinomial British pound asset prices generated with u= l.()03 and d= Table 5.9 Trinomial British pound ordinary currency call values 57 Table 6.0 Trinomial British pound average asset prices 58 Table 6.1 Trinomial British pound Asian currency call values 59 Table 6.2 Trinomial US dollar asset prices generated with u=1.002 and d=0, Table 6.3 Trinomial US dollar ordinary currency call values 60 Table 6.4 Trinomial US dollar average asset prices 61 Table 6.5 Trinomial US dollar Asian currency call values 62 List of figures Figure 3.1 Pricing US currency call option on the binomial lattice 31 Figure 3.2 Pricing British currency call option on the binomial lattice 32 Figure 3.3 Pricing Euro currency call option on the binomial lattice 33 Figure 3.4 Pricing Asian currency call option on the binomial lattice 36 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Pricing ordinary currency call options on the trinomial lattice by the standard technique 44 Pricing ordinary currency call options on the trinomial lattice by the moment matching technique 45 Pricing Asian currency call options on the trinomial lattice by the standard technique 47 Pricing Asian currency options on the trinomial lattice by moment matching technique 48 vi

8 TABLE OF CONTENTS Declarations Dedication Acknowledgements Abstract List of tables and figures i ii iii iv v 1. INTRODUCTION Background Options Currency Options Path-dependent Options Asian Options Lattice Approach Statement of Problem Objectives 8 2. LITERATURE REVIEW Pricing Currency Options Pricing Asian Options Analytical Methods Lattice Methods METHODS AND METHODOLOGY Preliminaries Stochastic Processes Arbitrage Pricing Replicating an Option 15

9 Behavior of Arbitragers Risk Neutral Valuation Asian Options Option Pricing Models Black Scholes Option Pricing Model (BSOPM) Pricing Currency Options by BSOPM CRR Binomial Option Pricing Model (The Lattice) Pricing Ordinary Currency Options on the Lattice Pricing Asian Currency Options on the Lattice Hull and White Approximation The Trinomial Lattice Lattice Construction Pricing Asian Options on Trinomial Lattice Calculating Trinomial Probabilities The Standard Technique The Moment Matching Technique Pricing Ordinary Currency Options on the Trinomial Lattice Pricing by the S tandard Technique Pricing by Moment Matching Technique Pricing Asian Currency Options on the Trinomial Lattice Pricing by the Standard Technique Pricing by the Moment Matching Technique DATA AND DATA ANALYSIS Estim ation of parame ters Pricing Asian currency options on the binomial lattice with 3 timesteps The Euro 50

10 4.2.2 The British pound The US dollar Pricing Asian currency options on trinomial lattice with 3 timesteps The Euro The British pound The US dollar Summary of option prices with varying strike prices The US douar The British pound The Euro CONCLUSIONS AND RECOMMENDATIONS 66 References 67 Appendix 70

11 INTRODUCTION LI Background In the past, financial derivatives have played an increasingly important role in the world of finance. Derivatives are financial instruments whose values depend on some underlying assets and are valuable instruments used in hedging. A hedge is a position that offsets the price risk of another position. It reduces the risk exposures or even eliminates them if it provides cash flows equal in magnitude but opposite in directions to those of the existing exposure. Derivatives can be divided into four groups: futures, forwards, options and swaps. Standardized derivatives e.g. futures and options are traded actively in exchanges while non-standardized are traded largely over-the-counter Options Options are financial derivatives that give their buyers the right but not the obligation to buy or sell the underlying assets for a price called the exercise price. The price of purchasing an option is generally much less than the actual price of the underlying asset. This has led to their popularity. They allow hedging risk more cheaply than using only assets, and they allow cheap speculation. There are two types of options Call options - the right but not obligation to buy the underlying asset. Put options - the right but not obligation to sell the underlying asset. Options can also be classified depending on the time of exercise. An American style option can be exercised at any time before the maturity -l-

12 date while a European style option can only be exercised at the maturity date. Exercising an option denotes that the holder exercises the right to buy or sell the underlying asset. An American option contains all the advantages of the European option plus an added advantage of early exercise. Its value is therefore at least as great as that of the European option, all factors held constant. Option pricing theory has a long and illustrious history, but it also underwent a revolutionary change in At that time, F. Black and M. Scholes [1] presented the first completely satisfactory equilibrium option pricing model. This path breaking article formed the basis for many subsequent academic studies. These studies have shown that option pricing is relevant to every area of finance. For example, virtually all corporate securities can be interpreted as portfolios of puts and calls on the assets of a firm. Suppose a firm has a liability of pure discount bonds, the stockholders then have a "call" on the assets of the firm which they can choose to exercise at the maturity date of the debt by paying its principal to the bondholders. Also, the bonds can be interpreted as a portfolio containing a default-free loan with the same face value as the bonds and a short position in a put on the asset of the firm. There are two sides to an options contract. On the one side is the investor who takes the long position (i.e. he buys the option), while on the other side is the investor who takes the short position (i.e. he sells the option). An option holder is given the right of gaining benefit without any obligation. An option is exercised only when it's in the best interest of the holder. For example, in the case of the European style option, the payoff of a long position is given by max (0, ST - X) for call options and max (0, X -

13 ST) for put options. An option can be priced in a discrete time model. The payoff for a European style option at maturity (time n) is Max (Sn - X, 0) for call option Max (X - Sn, 0) for put option where S is the asset price and X is the strike price (exercise price). Options basically attract three kinds of people: hedgers, speculators and arbitrageurs. A hedger is the one who tries to reduce risk by buying or selling the options. A speculator tries to take a position to gain more benefits in the market by forecasting in the future. He longs (or shorts) an option if he believes it is beneficial. An arbitrageur is the one who can gain a risk-less profit if the option value is not "fair". A trading strategy used to gain risk-less profit by taking advantages of mispriced options is called arbitrage. Theoretically, there is a fair price for each option in the market. The market price of an option should be equal to its fair price; otherwise the arbitrageur can take advantage of it Currency Options Currency (foreign exchange) options are options for which the underlying asset is the foreign exchange. That is, they are settled by delivery of the underlying currency. They can either be European style or American style. A European style currency option is the right but not the obligation to exchange two currencies at a fixed rate (the strike rate) on an agreed date in the future (the expiry date). American style contracts on the other hand, can be exercised before expiry. These contracts are either negotiated directly between two parties in an over-the-counter transaction, or traded through an organized futures and options exchange. The advantage of the

14 over-the-counter market is that large trades are possible with strike prices, expiration dates, and other features tailored to meet the needs of corporate treasurers. The exchange traded market is much smaller than the overthe-counter market. The right to sell one currency is also the right to buy the other currency involved in the contract. For example, suppose the currency option contract conveys the right but not obligation to sell Kshs 700million and to receive US Dollars 70 million in return. This contract is a Kshs put (the right to sell Kenya shillings) and at the same time, a US dollar call (the right to buy US dollars). The strike or exercise price is Kshs/USD 0.1 i.e. each Kenya shilling buys 0.1 US dollars. Currency options are widely used by corporations, institutional investors, hedge funds, traders, commercial and investment bank, central banks and other financial institutions. They can also be used to: Limit the risk of losses resulting from adverse movements in currency exchange rates. Hedge against the foreign exchange risk that results from holding assets such as shares or bonds that are denominated in foreign currencies. Enhance returns on foreign currency investments. Speculate on the movements in currency rates with limited risk. For example, a corporation wishing to hedge a foreign exchange exposure and is due to receive US dollars at a known time in the future can hedge its risk by buying put options on US dollars that mature at that time. The strategy guarantees that the value of the US dollar will not be less than the strike price, while allowing the corporation to benefit from any

15 favorable exchange rate movements. Similarly, corporations due to pay US dollars at a known time in the future can hedge by buying calls on the US dollar that mature at that time. This guarantees that the cost of the US dollar will not be greater than a certain amount while allowing the corporation to benefit from favorable exchange rate movements. Because they offer flexibility, currency options (as other options) can be attractive hedging tools. An option need not be exercised if the buyer of the contract can find a better rate of exchange in the spot market. The drawback is, of course, that buying an option costs a premium Path Dependent Options Due to the rapid growth and deregulation of financial markets, nonstandardized options are created by financial institutions to fit their client's needs. These complex options are usually traded in the rapidly growing over-the-counter market. Most of them depend nontrivially on the price path (history) of the other financial assets. They are called pathdependent options. Path dependent options were first introduced in 1982, and their pricing has relied greatly on the hedging principle of the Black Scholes model. For currency options, path-dependence is on the price path of the foreign exchange. The most well known examples are the lookback options and the Asian options Asian Options An Asian option is an option whose payoff depends on the arithmetic average price of the underlying asset. In some cases the underlying asset of the option is an average; in others, the strike price itself is computed as

16 an average of the underlying assets recent prices. For example, given a European style Asian call, assume that the average price of the underlying asset between the option's initial date and the maturity date is A. Then the option holder has the right to buy (or sell) the underlying asset with price A. This contract is useful to hedge a transaction whose cost is related to the average price of the underlying asset. The price of Asian options is less subject to price manipulation. How? We know that the payoff of a European style ordinary option is determined by the underlying asset's value at the maturity date while the payoff of a European style Asian option is determined by the average price of the underlying asset between the option's initial date and the maturity date. Thus, it's easier to control an asset's price at a specific time point than to manipulate the whole price path. There are no closed form formulae for pricing arithmetic Asian options, thus approximation is used (Lyuu 2004). Some of the approximation methods include Analytical formulae Monte Carlo Simulations The lattice approach The lattice approach is more general than the first two since the analytical formulae and Monte Carlo simulation approximation methods suffer from the inability to price American style options without a bias The Lattice Approach A lattice consists of nodes and edges connecting the nodes. It simulates the (discrete time) price process of the underlying asset from the option's

17 initial date to the maturity date. Assume that the option initiates at year 0 and matures in year T. A lattice divides the time interval [0, T] into n equal time steps. Then the length of each time step is given by T/n. The price of the underlying asset at discrete time step h can be observed on the lattice. The well known Cox-Ross-Rubinstein binomial lattice is described later in the paper. The difficulty with the lattice method in the case of Asian options lies in its exponential nature: since the price of the underlying asset at each time step influences the option's payoff, 2 paths have to be individually evaluated for an n-time step binomial lattice. To solve this, approximation algorithms are employed. 1.2 Statement of the Problem Granted, getting into the foreign exchange business has its risks, as does every other business. It is also clear that investors prefer maximum expected return and less risk. Thus to guarantee success, one has to be able to manage the risks involved. In the foreign exchange business one way to manage foreign exchange risks is to invest in currency options. In Kenya, option pricing has not been fully introduced but there has been considerable trading in currency options over-the-counter. Pricing currency options by considering only the final value can easily be done since there are closed-form formulas for them. The big question then is, "What happens when we take into consideration path dependence?" There have been considerable studies on the pricing of path dependent options especially Asian options. From the literature we have been provided with various solutions to the problem; from analytical solutions to Monte Carlo simulations and even lattice approaches. Undertaking this study will enable me broaden my spectrum as far as option pricing is

18 concerned and also be able to price currency options in Kenya while assuming path-dependence, that is, taking a case of Asian currency options. Asian options are useful for hedging since its price is less subject to price manipulation. However, there are no simple closed-form formulas for the price of an Asian option under the standard continuous time Black Scholes model. In this regard, I will explore the lattice methods available in literature and use the binomial and trinomial lattices to price the path dependent options. I choose to use lattices since it has been established in literature that lattice methods are more efficient as compared to the Monte Carlo simulation methods and also takes into account the early exercise property of American options. 1.3 Objectives The main objective is to price currency options in Kenya, assuming path dependence. The specific objectives are Explore the lattice method of Asian option pricing especially the binomial and trinomial lattices. Price Asian currency options in Kenya on the binomial and trinomial lattice. Compare the results from the two models.

19 initial date to the maturity date. Assume that the option initiates at year 0 and matures in year T. A lattice divides the time interval [0, T] into n equal time steps. Then the length of each time step is given by T/n. The price of the underlying asset at discrete time step h can be observed on the lattice. The well known Cox-Ross-Rubinstein binomial lattice is described later in the paper. The difficulty with the lattice method in the case of Asian options lies in its exponential nature: since the price of the underlying asset at each time step influences the option's payoff, 2 n paths have to be individually evaluated for an n-time step binomial lattice. To solve this, approximation algorithms are employed. 1.2 Statement of the Problem Granted, getting into the foreign exchange business has its risks, as does every other business. It is also clear that investors prefer maximum expected return and less risk. Thus to guarantee success, one has to be able to manage the risks involved. In the foreign exchange business one way to manage foreign exchange risks is to invest in currency options. In Kenya, option pricing has not been fully introduced but there has been considerable trading in currency options over-the-counter. Pricing currency options by considering only the final value can easily be done since there are closed-form formulas for them. The big question then is, "What happens when we take into consideration path dependence?" There have been considerable studies on the pricing of path dependent options especially Asian options. From the literature we have been provided with various solutions to the problem; from analytical solutions to Monte Carlo simulations and even lattice approaches. Undertaking this study will enable me broaden my spectrum as far as option pricing is

20 LITERATURE REVIEW 2.1 Pricing Currency Options Biger and Hull (1983) provided direct derivation of valuation formulas for European put and call foreign exchange options using the Black Scholes methodology. They showed that these formulas can be derived by assuming Expectation Theory 1 of exchange rates and the Capital Asset Pricing Model. Jabbour, Onayov and Petrescu (2006) valued currency options by incorporating higher order moments via an Edgeworth series expansion function. They calculated the option values and compared their results to those of the Black Scholes method. Their results indicated significant differences especially for the out-of-the money call options and at-themoney put options. 2.2 Pricing Asian Options The major problem in pricing Asian Options is that we do not know much about the distribution of the underlying asset's average price A(T). A(T) can be viewed as the sum of lognormal random variables, and the density function of a sum of lognormal random variables is currently unavailable. For this reason, there are no closed-form solutions for pricing Asian options. The existing approximation methods are thus grouped in three categories, that is, analytical methods, Monte Carlo simulation and lattice (and related PDE) approach. 1 See Appendix A. 1-9-

21 2.2,1 Analytical Methods The analytical approach shows that the option value is approximated by semi-closed or closed form formulae. In 1994, Bouaziz, Briys and Crouchy derived a closed-form solution for the valuation of European Asian options whose strike price is an average. They considered both plain vanilla average rate options and forward-starting average options. They compared their results to those obtained by Monte Carlo simulations and found that their proposed solution was fairly accurate even for high levels of volatility. Barraquand and Pudet (1996) showed that the problem of pricing pathdependent contingent claims lead to solving a degenerate diffusion PDE in the space augmented with the path-dependent variables. They described a numerical technique called the forward shooting grid method for pricing contingent claims. This technique was as accurate as Monte Carlo simulation but with faster execution time. The forward shooting grid was also able to into account the early exercise property of American options. In 1995, Rogers and Shi used the change of numeraire to reduce the dimension of the PDE for both the floating and fixed strike Asian options. This reduced the problem of pricing Asian options to a problem of solving a parabolic PDE in two variables. Vecer (2002) provided a simpler and unifying approach for pricing Asian Options for both continuous and discrete arithmetic average options. He introduced a one dimensional PDE which was easily implemented to give fast and accurate results and had stable performance for all volatility. In addition, in 2002, Forsyth et al carried out research showing the convergent nature of the various methods given for pricing path dependent options. They confirmed that the forward shooting grid by Barraquand and Pudet (1996) and the Hull and White (1993) approximation converge to the true solution when interpolation is -10-

22 used in the backward induction. The results also showed that the PDE method is convergent in the continuous time limit for Asian Options. Fujiwara (2006) proposed a fast, accurate and simple numerical method for pricing American Asian options. He combined three methods developed in computational fluid dynamics to do this. The results showed that the method was consistent with other analytical solutions The Lattice Methods In 1999, Aingworth, Motwani and Oldham presented an asymptotic fullypolynomial approximation scheme for pricing Asian options on the lattice, in both European and American versions. They pruned paths that exceeded a fixed value and aggregated unpruned paths with similar values. This permitted their scheme to run in polynomial time. The result was that their findings were as good as the existing techniques and performed well across a wider range of parameters. Dai, Huang and Lyuu (2004) introduced a lattice approach for pricing Asian options by exploiting the method of Lagrange multipliers to minimize approximation (especially Hull and White (1993) approximation) error. The result was that, given the same convergence rate with other pricing models; their approach gave better results and performed much faster. In 2008, Lo, Wang, Hsu, developed a modified Edge worth binomial model with higher moment consideration for pricing American Asian options. They used the lognormal distribution as a benchmark and compared their results to those of Hull and White (1993) and Chalasani et al (1999). The end result was that their results were better than Chalasani et al but similar to those of Hull and White (1993). Also, their analysis showed that the modified Edgeworth binomial model can value American Asian options with greater accuracy and speed given higher moments in the underlying distribution. -li-

23 Jabbour, Kramin, Kramin and Young (2002), elaborated an n-order multinomial lattice approach to value derivative instruments on asset prices characterized by a lognormal distribution. They used a moment matching technique to do this by nonlinear optimization. Overally, they developed a straightforward explanation of the underlying treo building procedure for which numerical efficiency is a motivation for actual application. In addition, Yamada and Primbs (2006) analyzed properties of multinomial lattices that model general stochastic dynamics of the underlying stock by taking into account any given cumulants (or moments). First, they provided a parameterization of multinomial lattices, and demonstrated that mean, variance, skewness and kurtosis of the underlying may be matched using five branches. They then investigated the convergence of the multinomial lattice when the basic time period approaches zero, and proved that the limiting process of the multinomial lattice that matches annualized mean, variance, skewness and kurtosis is given by a compound Poisson process. In 1988, Boyle who first introduced the Trinomial Option Pricing Model extended the lattice approach by Cox-Ross-Rubinstein (1979) on a single asset to a case of two state variables. Boyle's trinomial model was based on a moment matching methodology. The mean and variance of the discrete distribution were equated to those of the continuous lognormal distribution. He also introduced a numerically optimized parameter A, to ensure non-negativity of the risk-neutral probabilities. Dai and Lyuu (2002) proposed a multiresolution (MR) trinomial lattice for pricing European- and American- style arithmetic Asian options. The lattice used the notion if integrality of asset prices and multiple resolution (the lattice consisted of nodes with precision and others without precision) to make an -12-

24 exact pricing algorithm realizable and practical. During their analysis they realized that their algorithm exhibited a reduction in running time and also errors in the backward induction while calculating option prices. As a follow up study, Dai and Lyuu (2004) realized that the multiresolution lattice worked for up to n = 160 but exponential time algorithms can't work with n that large. So, they introduced a new trinomial lattice structure for pricing Asian options and the result was a first exact and convergent lattice algorithm breaking the exponential time barrier. -13-

25 METHODS AND METHODOLOGY 3.1 Preliminaries Stochastic Processes A stochastic process is a variable that changes over time in an uncertain way. Economical variables, like the asset's price and the exchange rates, are usually modeled as stochastic processes in academic models. The randomness of these processes are usually governed by some fundamental stochastic processes like the Brownian motion. Define {Bt} as a Brownian motion where Bs denotes the process value at time s. Then the following properties hold 1. Normal increments: Bt B8 has normal distribution with mean 0 and variance t - s. 2. Independent increments: Bt - Bs is independent of the past, that is, of Bu, where 0 < u < s. 3. Continuity of paths: Bt, t > 0 are continuous functions of t. In this paper, we assume the financial asset's price is a lognormal stochastic process. Suppose the time interval is [0,T], define S(t) as the price of a financial asset at year t. The asset price process follows the continuous-time diffusion process 2 as where r is the domestic risk-free interest rate per annum, rr is the foreign risk-free interest rate per annum and o is the annual volatility. 2 See Appendix A.2 for the proof. -14-

26 3.1.2 Arbitrage-Free Pricing An arbitrage opportunity is one that, without any initial investment, generates non-negative returns under all circumstances and positive returns under some circumstances. In an efficient market, such opportunities should not exist. In financial markets, an option buyer pays an amount called the option premium to a seller at the option initial date. This premium can be viewed as the fair price of the option. The fair option price can be determined by arbitrage free pricing. Arbitrage free option pricing can be done by either replicating an option, observing the arbitrageurs behavior or risk-neutral valuation Replicating an option An option is said to be replicated by a portfolio A if A can be constructed so that the future payoff of A, that is at time T, is always equal to the payoff of the option at time T. The fair price of the option should be equal to the cost of constructing portfolio A (at inception) since the payoffs of A and the option are the same. The idea is to construct a portfolio at time 0 that replicates exactly the option's terminal payoff at time T. Let ^ = = (a0,/70) e K 2 denote a portfolio of an investor with a short position in one call option. Let and VT{$) denote the wealth of this portfolio at dates t = 0 and t = T. Therefore, V0(t) = a0s+fi0 VT{t) = a0st+p0{l + r) -15-

27 A portfolio <I> replicates the option's terminal payoff whenever Kr(^)=Cr. where CT is the option's terminal payoff Behavior of Arbitrageurs In an efficient market, the market price of an option should be equal to the fair price otherwise arbitragers can gain riskless profit by taking advantage of the mispriced options. Suppose the market price for a currency call today V is larger than the fair price B. The arbitrager can short a call and buy the portfolio A. He will earn V - B > 0 today. At maturity, he will neither win nor lose anything since the final payoffs of the call and A are equal. Thus he gains V - B > 0 without paying anything or suffering any risk! Thus the market price quickly goes back to B, since such trades causes inconsistency and hence inefficiency in the market. If the option value V is lower than B, an arbitrager can construct an arbitrage strategy by longing a currency call and shorting the portfolio A. By the same argument, the market price of the option will finally go back to the fair price B. Thus the market price of the option is equal to cost of the replication under arbitrage free considerations Risk Neutral Valuation This is the valuation of an option or other derivative assuming the world is risk-neutral. Risk neutral valuation gives the correct price for a derivative in all worlds not just in a risk-neutral world. In a risk-neutral world, it is assumed that all investors are indifferent to risk. Risk neutral valuation is derived from one key property of the Black Scholes Merton differential equation, that is, that the equation doesn't involve any variables that are affected by the risk preferences of investors

28 The present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value at the risk free rate. The assumption that the world is risk-neutral does, therefore, considerably simplify the analysis of derivatives. Consider a derivative that provides a payoff at one point in time. It can be valued using risk neutral valuation by using the following procedure: Assume that the expected return from the underlying asset is the risk-free interest rate r. Calculate the expected payoff from the derivative. Discount the expected payoff at the risk-free interest rate. To find the value of an option, we adjust the probabilities for risk and discount by the risk-free rate. The risk-neutral probabilities should be the same probabilities that would give us today's stock price if we discount the expected stock payoff by the risk-free rate. Note that the option value evaluated by using replication is the same as the value evaluated by taking expectation, but the latter is simpler. In a continuous time model, the value of a European-style option under risk neutral valuation can be expressed as,f' r^[max(s(r)-x,0)] for call options e- rr (? [max(x-5(r),0)] forputoptions where X is the strike price and S is the asset price. In a discrete time model, the value is obtained by changing S(T) into Sn. Thus we have -17-

29 e~ rt E Q [ max for call options e' rt E Q [max(at -5b,0)] for put options Asian Options An Asian option is an option whose payoff depends on the average price of the underlying asset during a specific period. Define the average price of the underlying asset from year 0 to year t as A(t) = * Then the payoff for a European style Asian option at maturity date is Max (A(T) - X, 0) Max (X - A(T), 0) for call options for put options In discrete time model, the average price of the underlying asset is redefined as The payoff thus becomes Max (An - X, 0) for call options Max (X- An, 0) for put options -18-

30 By applying risk-neutral valuation we have the payoff in the discrete time model as e' rt E Q [max (4, -^,0)] for call options e~' T E Q [max for put options 3.2 Option Pricing Models Though pricing an option at expiration is easy, pricing at any prior moment is anything but easy. The no-arbitrage principle, albeit valuable in deriving various bounds is insufficient to pin down the exact option value without further assumptions on the probabilistic behavior of the asset prices. The major obstacle towards an option pricing model is that it seems to depend on the probability distribution of the underlying asset's price and risk-adjusted interest rate used to discount the option's payoff. Neither factor can be observed directly. The two main models in option pricing include the Black Scholes Option Pricing Model and the CRR Binomial Option Pricing Model (the Lattice Approach). The Black-Scholes model came close to solving the distribution and interest rate problem since it allows the price to move to any infinite number of prices in any finite amount of time. The mathematics involved in this model was very time consuming though. The Binomial model on the other hand limits the price movement to two choices in a period, therefore simplifying the mathematics tremendously at some expense of realism. However, because the Binomial Option Pricing Model converges to the Black Scholes Option Pricing Model as the period length goes to zero, all is not lost. -19-

31 3.2.1 Black Scholes Option Pricing Model In the early 1970s F. Black, M. Scholes and R. Merton developed what came to be known as the Black Scholes (or Black Scholes Merton) model. The model has had a huge influence on the way that traders price and hedge options. The backbone of this model is the Black Scholes Merton differential equation. This is an equation that must be satisfied by the price of any derivative dependent on a non-dividend paying asset. The assumptions underlying the differential equation include Asset price follows generalized Wiener process with drift n and volatility o as constants. Short selling of securities with full use of proceeds is permitted. No transaction costs or taxes, all securities are perfectly divisible. There are no dividends paid during the life of the derivative. There are no riskless arbitrage opportunities. Security trading is continuous. The risk-free rate of interest, r, is constant and same for all maturities. The Black Scholes Merton differential equation is given by V. CV 1 2c2d 2 / f + fs + 7 = rf dt ds 2 OS 2 ' The above differential equation has various solutions corresponding to all the different derivatives that can be defined with S as the underlying variable. The particular derivative that is obtained when the equation is solved depends on the boundary conditions used. For example, in the case of a European call option, the boundary condition is -20-

32 / = max (5-^,0) where S is the underlying asset price and K is the strike price. The Black Scholes formulas for European calls and puts are given below c = S0N(dO ~ Ke~ rt N(d2) where d2 = dx - crvf r is the risk-free interest rate, So is the asset price at time 0 and K is the strike price are derived by either solving the Black Scholes Merton differential equation subject to the boundary conditions European call: / = mar (5 K% 0) when t = T European put: / = max(k S, 0) when t = T Or by risk neutral valuation Pricing Ordinary Currency Options by BSOPM Currency options can be valued by the standard Black Scholes Option Pricing Model by making some adjustments to the pricing formula. Define -21-

33 So to be the spot exchange rate (i.e. So is the value of one unit of the foreign currency in domestic currency). The owner of foreign currency receives a yield equal to the risk-free interest rate, rr, in foreign currency. The bounds for the call and put prices are given by c 5 0 e" r / T - Ke^ p ^ Ke~ rt S0e~ r f T and the pricing formulas are given by c = Soe-^Nid^ - Ke-^NQ) V = ffe-^vf-da) where,n (!) + ( r - r ' + ff2 / 2 )r d2 = d1- ovf The domestic interest rate, r, and the foreign interest rate, rr, are the rates for a maturity T. For example, given the following exchange rate prices and strike prices for the US dollar, British pound and Euro, we can find the call prices by applying the Black Scholes Option Pricing Model (BSOPM). Note that, the risk-free rate is 5%, the foreign interest rate is 7%, volatility is 20% and the time to maturity is 1 year. The results 3 are given in the table 3.1 below. J Results developed by DerivaGem Software. (See Appendix) -22-

34 Currency Asset Price Strike Price Call Price US dollar British pound G Euro Table 3.1 Pricing US, British and Euro currency call options by the Black Scholes Option Pricing Model CRR Binomial Option Pricing Formula (The lattice approach) This was developed by Cox Ross and Rubinstein in 1979 [2]. Assuming that the asset follows a multiplicative binomial process over discrete time periods, then the rate of return on the asset over each period has two possible values. That is, u with probability p and d with probability 1-p. If the current level of asset is S then at the end of the time period we will have it to be either us or ds. Assume also that the interest rate is constant. That is, one can borrow or lend freely at this interest rate. -23-

35 Let r = 1 + r, we must have u > r > d 4 for no arbitrage. If this doesn't hold then we would have profitable riskless arbitrage opportunities involving the asset and riskless lending and borrowing. Suppose we have a call (with one period left to expiration) on this asset. The payoff of a call is given by C = max{0,st - K) Let C be the current value of the call, Cu be the value when the asset goes to us and Cd the value when the asset value goes to ds. Since we only have one period to expiry, exercising this gives us Cu = max{0,us - K] Cd = max{0,cls K] Suppose the two mainly traded assets in the market are i. A risky asset ii. A risk-free asset Let a = the proportion of the portfolio held in the risky asset. 8 = the proportion of the portfolio held in the risk-free asset. * See Appendix A. 4 for proof of inequality. -24-

36 If we purchase a portfolio with this proportion, it will cost as + P At the end of the period, the value of the portfolio is Since we can choose the proportions as we please, suppose we have the terminal payoff of the portfolio equal to the option payoff. We therefore have aus + r/]= Cu ads + fp = Cd Solving this for B and a we have that a ~ Cu-d)s and n _ ucd-dcu P (u-d)r Choosing a and 6 in this way gives us a hedging portfolio. For no arbitrage we have, Cu Q uq dcu -25-

37 And this implies, T. f d * u f Let us define q = - and 1 Q = - then we have, ^ u-d ^ u-d C=j[qCu + (l-q)cd] This is the exact value formula for the value of a call one period prior to the expiration in terms of S, K, u, d and r. From the formula above, we note the following: The probability p doesn't appear in the formula. That is, even if different investors have different subjective probabilities about upward and downward movement in the asset, they would still agree on the relationship of C to S, K, u, d, and r. The value of the call doesn't depend on the investors' attitude towards risk. This is because the only assumption made about his behavior is whether he prefers more wealth to less wealth. The only random variable on which the value of the call depends is the asset price S. Note that, q is a probability and is in fact the value p would take in equilibrium if the investors were risk-neutral. Thus, the value of the call can be interpreted as the expectation of its discounted future value in a risk neutral world. Suppose now we have two periods to expiration. Our asset price is then represented as follows, -26-

38 And the call as, Thus since there are two periods left to expiration, we note that l r Cu =j[qcuu + (l-q)cud] 1 Cd=J LlCud - CI - q)cdd] -27-

39 Note, to be able to maintain the portfolio for one more period, the value of the portfolio at the end of the current period will always be exactly sufficient to purchase the portfolio we would want to hold over the last period. Therefore, we would have to readjust the proportions in the hedging portfolio, but we would not have to put up any more money. This is termed as the portfolio being self-financing 5. Replacing these values to the former equation we have, c = ^[q 2 Cuu q)cud + (1 - q) 2 Cdd] r This is a recursive procedure for finding the value of a call with any number of periods to go. By starting at expiry and working backwards we have the general valuation formula for any n as, C = pi Y - qr-'maxio.uun-js-k) By limiting the number of upward moves in the next n periods to be able to finish in the money to "a", this equation simplifies to n 1 Y n! 2~tjl (n /*)! )=a q*(l q) n ~J[ufd n ~JS K] where by opening the bracket and simplifying this further we have, s See Appendix A.3-28-

40 C = 5 7=a 7 -tfr z-n n f u J d n^ vu r The latter expression in brackets is a binomial distribution function denoted by 0[am,q] while the former expression in brackets is also a binomial distribution with the function denoted by 0[a; n,q 1. Where, (t)<7 and 1 - q = (1 - q) Suppose now trading takes place more frequently say, one hour or even one minute. Cox-Ross-Rubinstein (1979) established that by choosing 6 u = and Where t is the fixed length of calendar time to expiration (e.g. 1 day), n the number of periods of length h (the elapsed time between successive asset price changes), the binomial option pricing formula converges to the classical Black-Scholes Option Pricing formula, when t is divided into more and more subintervals and f, u, d and q chosen in a way that the multiplicative binomial probability distribution of asset prices goes to lognormal (see CRR 1979 [2]). 6 See Appendix for estimation of o and p -29-

41 Pricing Ordinary Currency Options on the Lattice Currency options can be valued using the binomial tree (binomial option pricing model) by using the following formulas: u = e ffyfrt and d = i u P a-d = ^d Where a = e( r-r /) At For example, using the same asset dynamics as the example for the BSOPM, we get the new call values 7 as Currency Asset Price Strike Price Call Price with time steps = 4 US dollar ,25 British pound Euro Table 3.2 Pricing US, British and Euro currency call options by the Binomial Option Pricing Model. 7 Developed by DerivaGem Software -30-

42 TT*US Dolar At each node: Uppervalue = Underlying Asset Price Lower vaiue= Option Price Values in red are a result of early exercise. Strike price=75 Discount factor per step Time step, dt= years, days Figure 3.1: Pricing US currency call option on the binomial lattice -31-

43 The Sterling Pound At each node: Upper value = Underlying Asset Price Lower value = Option Price Values in red area result of early exercise. Strike price = 112 Discount factor per step = Time step, dt = years, days Figure 3.2: Pricing British currency call option on the binomial lattice -32-

44 The Euro At each node: Upper value = Underlying Asset Price Lower value = Option Price Values in red a re a result of early exercise. Strike price = 100 Discount factor per step = Time step, dt = years, days Figure 3.3: Pricing Euro currency call option on the binomial lattice Pricing Asian Currency Options on the Lattice Asian options can be priced on the lattice; however, the pricing algorithm is much more complex as the value of Asian options is influenced by the historical average price of the underlying asset. For most nodes, there is more than one possible option value at a node since there is more than one price path reaching this node and most of these price paths carry distinct historical average prices. -33-

45 Let S^Si,...,Sn denote the prices of the underlying asset over the life of the option and X be the exercise price. The (arithmetic) Asian call has a terminal value given by (see Lyuu 2002 [6]) and the put has a terminal value of max (*~ntlz 5 '' 0 ) At initiation, Asian options cannot be more expensive than the standard European options under the Black Scholes Option Pricing Model. Asian options are hard to price, for example, given Take the terminal price SQu 2 d. Different paths to it such as (So,Sou,Sou 2,Sou 2 d) and (So,Sod,Sodu,Sodu 2 ) may lead to different averages and hence payoffs -34-

46 max (S0 -f SQu + SQu 2 + S0u 2 d) X,0 and max (50 + S0d + S0du + 50du 2 ) X,0 respectively. For example, suppose we have a 3-time-step lattice with the asset price as 100, strike price as 100, the domestic interest rate as 7%, and foreign riskfree interest rate as 5% and the volatility as 20%. The resultant lattice is given as

47 Asset Price loo Strike Price loo Domestic Risk-free Interest rat* 796 Foreign Risk-free Interest Rate 596 Volatility 2096 Term to maturity 1 Time steps 3 Time per time step Up movement Down movement Probability of up movement Probability of down movement Dy >>;i mi < s I I 7Q Ail* 11 Option I >y rni c ' Figure 3.4: Pricing Asian currency call options on the binomial lattice -36-

48 Note that the binomial tree for the averages therefore does not combine. A straightforward algorithm is to enumerate the 2" price paths for an n- period binomial tree and average the payoffs. However, the exponential complexity makes this impractical. Thus Monte Carlo method and approximation algorithms are the alternatives left. Hull and White (1993) [3] developed an approximation method for Asian options based on the binomial tree, and is described below Hull and White Approximation Consider a node at time j with the underlying asset price equal to SoWd' i. Let such a node be denoted by N(i,j). The running sum Sm=o^m at this node has a maximum value of S0 < 1 + u + u u' + u l d+... +u l dj~ Divide this by j + 1 and call it Amax(j,i)- Similarly the running sum has a minimum value of Divide this by j+1 and call it Amin(j,i). Note that both are running averages. Although the possible running averages at N(i,j) are far too many, all lie between AminQ,i) and AmaxO,i)' Pick k+1 equally spaced values in this range and treat them as the true and only running averages, which are -37-