The Pennsylvania State University. The Graduate School. Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO

Transcription

1 The Pennsylvania State University The Graduate School Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO SIMULATION METHOD A Thesis in Industrial Engineering and Operations Research by Shiguang Han 2012 Shiguang Han Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2012

2 The thesis of Shiguang Han was reviewed and approved* by the following: Jose A. Ventura Professor, Department of Industrial and Manufacturing Engineering Thesis Advisor Terry P. Harrison Professor, Department of Supply Chain and Information Systems Paul Griffin Professor, Department of Industrial and Manufacturing Engineering Head of the Department of Industrial and Manufacturing Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Option pricing is among the most important and challenging problems in the modern financial industry. With the development of option markets, various types of options have been created to meet the needs for different investors. The prices of most options do not have a simple closed form solution and efficient computational methods are needed to determine them. The Monte Carlo method has increasingly become a popular computational tool to price complex financial options and the most famous least square Monte Carlo method has been widely used in financial industry since Longstaff and Schwartz created it in They showed that the Monte Carlo method is a simple yet powerful approach for approximating the values of prices and Greeks for complex option types. In this thesis, one type of exotic options, the American-Asian option, is priced by a modified least square Monte Carlo simulation method. This modified approach generates initial asset prices randomly from a carefully chosen distribution and obtains a regression equation for different initial values. This equation can be differentiated analytically to estimate the Greeks, which are very helpful to understand risk sensitivities of options. The Binomial tree method is used as a control method to compare the performance of this modified approach. The extension to multidimensional cases is also discussed.

4 iv TABLE OF CONTENTS LIST OF FIGURES... v LIST OF TABLES... vi ACKNOWLEDGEMENTS... vii Chapter 1 Introduction... 1 Chapter 2 Theory and Algorithms Option types Closed form methods Monte Carlo methods Least square Monte Carlo method Modification of least square method Control method Chapter 3 Valuing American-Asian Options American-Asian option pricing by the modified LSM method A simplified example The fixed strike price American-Asian option example The floating strike price American-Asian option example Multi-dimensional case Chapter 4 Numerical Results Standard American put option results American-Asian option results The fixed strike price case of American-Asian options The floating strike price case of American-Asian options Multi-dimensional results Chapter 5 Conclusion and Future Work REFERENCES APPENDIX A MATLAB Codes APPENDIX B C++ Codes... 52

5 v LIST OF FIGURES Figure 2-1 Single initial price Figure 2-2 Random initial prices Figure 2-3 the Binomial tree model... 16

6 vi LIST OF TABLES Table 3-1 Stock price paths Table 3-2 Cash flows if exercised only at time 3 (the fixed strike price case) Table 3-3 Cash flow if exercised only at time 2 (the fixed strike price case) Table 3-4 Cash flows if exercised only at time 2 and 3 (the fixed strike price case) Table 3-5 Cash flow if exercised only at time 1 (the fixed strike price case) Table 3-6 Cash flows from the option (the fixed strike price case) Table 3-7 Cash flows if exercised only at time 3 (the floating strike price case) Table 3-8 Cash flow if exercised only at time 2 (the floating strike price case) Table 3-9 Cash flows if exercised only at time 2 and 3 (the floating strike price case) Table 3-10 Cash flow if exercised only at time 1 (the floating strike price case) Table 3-11 Cash flows from the option (the floating strike price case) Table 4-1 Standard American put options Table 4-2 American put options prices: LSM versus modified LSM Table 4-3 American-Asian put options (fixed strike price case) Table 4-4 American-Asian call options (fixed strike price case) Table 4-5 American-Asian put options (floating strike price case) Table 4-6 American-Asian call options (floating strike price case) Table 4-7 American-Asian max-put options on two assets... 45

7 vii ACKNOWLEDGEMENTS Here I express thanks to those people who helped me on this Master thesis during the past semester. First, Professor Ventura gave me great guidance on how to carry out this thesis and provided significant technical advice. I gained not only the academic knowledge from every meeting with him but also the careful attitudes toward research and numerous precious advices for my future. Besides, the Dynamic Programming course he taught last spring inspired me to use dynamic methods to solve the option pricing problem in this thesis. Second, Dr. Yao gave me significant assistance to better understand the financial engineering industry and guided my research on simulation methods for option pricing. I appreciate very much Dr. Harrison s time and effort in revising my thesis. I also appreciate the time and effort that Dr. Medeiros, as my simulation course instructor, spent in helping me debug simulation programs. Lastly, I would like to thank everyone else that shared their ideas with me, and gave me their time and comments.

8 Chapter 1 Introduction A financial derivative is a contract that provides its holder with a future payment that depends on the price of one or more primitive asset(s), such as stocks or currencies. In a frictionless market, the no arbitrage principle allows one to express the value of a derivative as the mathematical expectation of its discounted future payment, with respect to a so-called riskneutral probability measure. Options are particular derivatives characterized by nonnegative payoffs. European-style options can be exercised at the expiration date only, whereas Americanstyle ones offer early exercise opportunities to the holder. For simple cases, such as for European call and put options written on a stock whose price is modeled as a geometric Brownian motion (GBM), as studied by Black and Scholes [1], analytic solutions are available for the fair price of the option. However, for more complicated derivatives which may involve multiple assets, complex payoff functions, possibilities of early exercise and stochastic time-varying model parameters, analytic solutions are unavailable. These derivatives are usually priced either via Monte Carlo simulation or via other numerical methods. An important class of options for which no analytic solution is available even under the standard Black-Scholes GBM model is the class of Asian options, for which the payoff depends on the average underlying asset price over a certain time period. An Asian option can be used to hedge the risk exposure of a firm that sells or buys certain types of resources (raw materials, energy, foreign currency, etc.), on a regular basis over some period of time. Because the average in general is less volatile than the underlying asset price itself, these contracts are less expensive than their standard versions (such as standard American options).

9 2 Asian options come in various types. For example, the average can be arithmetic or it can be geometric. One talks of a plain vanilla Asian option if the average is computed over the full trading period and a backward-starting option if it is computed over a right subinterval of the trading period (such as Asian-Bermudan options). This interval usually has a fixed starting point in time. The Asian option can be fixed strike (if the strike price is a fixed constant) or floating strike (if the strike is itself an average). It is called flexible when the payoff is a weighted average, and equally weighted when all the weights are equal. The prices are discretely sampled if the payoff is the average of a discrete set of values of the underlying asset (observed at discrete epochs), and continuously sampled if the payoff is the integral of the asset price over some time interval, divided by the length of that interval. The options considered in this thesis are the most common: equally weighted, discretely sampled Asian options based on arithmetic averaging. European-style Asian options can be exercised at the expiration date only, whereas American-style ones offer earlier exercise opportunities, which may become attractive intuitively when the current asset price is below the current running average for a call option, and when it is above the running average for a put. Here, this thesis focuses on American-Asian put options, whose values are harder to compute than the European-Asian ones because an optimization problem must be solved at the same time as computing the mathematical expectation giving the option's value. American-Asian options are derivative securities with payoffs that depend on the average of an underlying asset price over some specified period and can be exercised any time before maturity. Because of their relatively small exposure to risk, they have become one of the most popular exotic options traded over the counter. There are mainly two types of American- Asian options based on different strike prices: fixed strike price option and floating strike price option. The fixed strike price option is similar to the usual European and American options. The strike price is pre-determined. For the floating strike price option, the strike price is the average value of the underlying asset over a pre-set period of time. In both cases the payoff depends on

10 3 the average underlying asset prices; Asian options are thus one of the basic forms of exotic options. Some relevant works have been done in pricing such exotic options like American-Asian options. Because of the path dependency and early exercise option characters of American-Asian options, finding an accurate and efficient algorithm for pricing such options is always an important and challenging problem. In the past years, Monte Carlo simulation has emerged as the most popular approach in computational finance for determining the prices of American-style options. Some important contributions are those of Tilley [12], Carriere [3], Broadie and Glasserman [2], Tsitsiklis and Van Roy [13], and Longstaff and Schwartz [8]. The most significant improvement in pricing exotic options, especially American-style options, is the wellknown least-square Monte-Carlo (LSM) simulation method developed by Longstaff and Schwartz [7]. In this paper, the authors demonstrate that LSM is robust for many kinds of exotic options. This thesis investigates the theory and practice of pricing two types American-Asian options (both fixed and floating strike price) in one and two dimensions, and their Greeks by LSM. Unlike traditional methods in estimating Greeks, this modified method is inspired by Wang and Caflisch s approach [15], which generates random initial prices for stock price sample paths. This method can exploit the cross-sectional information in the simulated paths at the initial time to infer option value information over a range of initial asset prices. This is done by roughly equating the option value function with the additional conditional expectation function estimated at the initial time. Simple manipulation of this function immediately yields the desired estimates for price and Greeks. The first task has been to implement this method for both fixed and floating strike price cases in the one dimensional condition, and to investigate the convergence behavior of this algorithm over different numbers of basic functions and paths. By doing this, this thesis verifies that LSM converges to the correct value as the numbers of basic functions and paths are increased

11 4 to infinity. In order to have control values with which to compare the estimates, the Binomial tree method developed by Cox, Ross, and Rubinstein [5] has also been implemented. After checking the convergence speed of this algorithm, this thesis moves on to calculating the Greeks. As mentioned above, random initial prices for stock price sample paths are generated and the Greeks are estimated by regression. With these techniques in place and reasonably well tested, this thesis continues with their implementation for a non-trivial max-put option with a payoff depending on two correlated geometric Brownian motions, X and Y, where the underlying asset price is the maximum value of these two assets,. For this two dimensional American-Asian option, this thesis finds estimates for its value for different numbers of paths and basic functions. The rest of this thesis is organized as follows. Chapter 2 provides the general background of option pricing and the theory of LSM, including the convergence properties and the different methods of pricing Greeks and the modification for the LSM method. This chapter also includes a discussion of the control method: Binomial tree method. Chapter 3 first uses a simple numerical example to illustrate the basic idea and logic of this American-Asian option pricing algorithm and then describes LSM and the modified LSM application in American-Asian options. Also the two dimensional case is shown and the performance of this algorithm in higher dimensional cases is presented. Chapter 4 shows the application of this algorithm and provides numerical results for each case. The conclusion of this research work is summarized in Chapter 5.

12 Chapter 2 Theory and Algorithms In this chapter, the general option pricing theories are introduced and the advantages of using Monte Carlo methods to price options, especially American-style options, are emphasized. The most popular least square Monte Carlo (LSM) simulation method used in this thesis is explained in detail and several adjustments and improvements are introduced to make the algorithm have better performance and be easier to apply in practice. To test the performance of the modified LSM, this thesis uses the Binomial tree method to compare the results with those of the modified LSM. The brief outline of Binomial method is provided at the end of this chapter. 2.1 Option types When talking about financial options, there are two basic types of options. A call option gives the holder of the option the right to buy an asset by a certain date for a certain price. A put option gives the holder the right to sell an asset by a certain date for a certain price. The date specified in the contract is known as the expiration date or the maturity date. The price specified in the contrast is known as the exercise price or the strike price. Options can be either American or European, a distinction that has nothing to do with geographical location. American options can be exercised at any time up to the expiration date, whereas European options can be exercised only at the expiration date itself. Most of the options that are traded on the exchanges are American. However, European options are generally easier to analyze than American options.

13 6 Options such as European and American call and put options are what are termed plain vanilla products. They have standard well defined properties and trade actively. Their prices or implied volatilities are quoted by exchanges or by brokers on a regular basis. One of the exciting aspects of the over-the-counter derivatives market is the number of nonstandard products that have been created by financial engineers. These products are termed exotic options. Although they are usually a relatively small part of its portfolio, these exotic options are important to investors because they are generally much more profitable than vanilla options. This thesis emphasizes on one type of such exotic options, which is called Asian option. Asian options are options where the payoff depends on the average price of the underlying asset during some part of the life of the option. If the average value of the underlying asset calculated over a predetermined period is and the strike price is, then the payoff from the average price call is and that from an average price put is. Average price options are less expensive than regular options and are arguably more appropriate than regular options for meeting some of the needs of investors. Another type of Asian option is the average strike option. An average strike call pays off and an average strike put pays off, where is the stock price at time T. Average strike options can guarantee that the average price paid for an asset in frequent trading over a period of time is not greater than the final price. Alternatively, it can guarantee that the average price received for an asset in frequent trading over a period of time is not less than the final price. Such type of Asian option is called floating strike Asian option. 2.2 Closed form methods Determining the right price for options has been an important and a challenging problem in the modern world of finance. An important breakthrough in this regard was made by Black &

14 7 Scholes [1], where using the no-arbitrage principle they derive a partial differential equation (PDE) that helps price certain generic options. Since then, this principle has spurred enormous research in determining the prices of financial securities. Roughly speaking, the no arbitrage pricing principle states that two portfolios of securities that have the same payoffs in every possible scenario should have the same price. Otherwise, by buying low and selling high, sure profit, or arbitrage, can be created from zero investment. Thus, if in a market, $1 can be exchanged for RMB (Chinese currency unit) 6 (assuming zero transaction costs), then the correct exchange rate for $2 is RMB 12. By buying low and selling high, any other rate would lead to an arbitrage. According to the Black & Scholes method, under very general conditions, the no-arbitrage based European option price can be expressed as the mathematical expectation of the payoff from the option discounted at the risk-free rate under a new probability measure referred to as the riskneutral probability measure. This method is a breakthrough in derivative pricing history and makes a significant contribution to the financial industry. However, such closed form solution is not robust for every scenario and sometimes depends on the characteristic of option types. Considering the American put option, which gives the buyer the chance to exercise the option at any time before the expiration date, Black & Scholes formula cannot give a closed form solution easily. For those more complex exotic options, like Asian options, Bermudan options, and multidimensional options, mathematical methods face more challenges. 2.3 Monte Carlo methods When mathematical methods had difficulties in solving those exotic options, Monte Carlo methods were introduced in the financial industry. Monte Carlo methods for evaluating the mathematical expectation of a random variable often involve generating many independent

15 8 samples of the random variable and then taking the average of the sample as a point estimate of the expectation. The accuracy of this method is proportional to the variance of each sample and the number of samples generated. The key advantage of the Monte Carlo methods is that given the value of underlying asset variance, the computational effort (roughly proportional to the number of samples) needed to achieve the desired accuracy is independent of the dimension of the problem. In this respect, it differs from other numerical techniques for evaluating integrals whose performance typically deteriorates as this dimension increases. An alternative approach already discussed above to pricing options is to numerically solve the partial differential equation (PDE) satisfied by their price function. However, if the asset price dynamics is sufficiently complex, a PDE characterizing the option price may even fail to exist. When the underlying dimensionality of the problem is large, numerical techniques (such as finite difference methods) to solve the PDE s may no longer be practical. Thus, for complex options based on multi-dimensional underlying assets, the Monte Carlo method provides a promising pricing approach. However, in many cases the computational requirement to get good accuracy can be prohibitive. To improve the efficiency of Monte Carlo methods to price options, several variance reduction techniques have been proposed in the recent literature. Some other works have been done to make the Monte Carlo methods more practical to use and get more useful results with less computational time. This thesis will introduce one technique with such adjustments Least square Monte Carlo method Recall that an American option can be exercised at any time up to a specified time. Thus, one may associate with such an option an exercise policy, i.e., a prescription that specifies in every scenario whether to hold on to or to exercise the option. Using the no-arbitrage principle,

16 9 the value of the option under each such policy can be expressed as an expectation of a random variable. The rational price for the American option equals that of the policy having the maximum value (otherwise an arbitrage can be created). Finding this policy and hence the value of the option is a difficult problem. It can be seen that the option price as a function of time and state, and satisfies a set of dynamic programming backward recursion equations. A number of Monte Carlo methods have been recently designed to exploit this representation by approximately solving the backward recursion equations. The most famous and popular least square Monte Carlo (LSM) method was created by Longstaff & Schwartz [7]. This LSM method is a simple, yet powerful approach to approximating the value of American options by simulation. By its nature, simulation is a promising alternative to traditional finite difference and Binomial techniques, and has many advantages as a framework for valuing, risk managing, and optimally exercising American options. For example, simulation is readily applied when the value of the option depends on multiple factors. Simulation can also be used to value derivatives with both path-dependent and American-style features. Simulation allows state variables to follow general stochastic processes such as jump diffusions and non-markovian processes. From a practical perspective, simulation is well suited for parallel computing, which allows significant gains in computational speed and efficiency. Finally, simulation techniques are simple, transparent, and flexible. To understand the intuition behind this approach, recall that at any exercise time, the holder of an American option optimally compares the payoff from immediate exercise with the expected payoff from continuation, and then exercises if the immediate payoff is higher. Thus the optimal exercise strategy is fundamentally determined by the conditional expectation of the payoff from continuing to keep the option alive. The key insight underlying the LSM approach is that this conditional expectation can be estimated from the cross-sectional information in the simulation by using least squares. Specifically, this method regresses the ex post realized payoffs from

17 10 continuation on functions of the values of the state variables. The fitted value from this regression provides a direct estimate of the conditional expectation function. By estimating the conditional expectation function for each exercise date, a complete specification of the optimal exercise strategy along each path is obtained. With this specification, American options can then be valued accurately by simulation Modification of least square method After Longstaff & Schwartz [8] published LSM in 2001, a lot of related works trying to assess and improve their algorithm has been done. These works include variance reduction by Chaudhary [4], the stability improvement by Moreno and Navas [9], and Greeks analysis by Stentoft [11]. In this thesis, one modification is made to help LSM better assess Greeks and improve its efficiency. The key insight is shown in Figure 1 and Figure 2. In both figures, we generate the stock price sample paths by a GBM but initial stock prices are set in different ways. In Figure 1, the initial stock price for all the stock price sample paths is 100. However, in Figure 2, the initial stock prices are generated by a specified distribution (with median around 100) for different stock price sample paths. Then the cross-sectional information in the simulated paths at initial time can be obtained to infer option value information over a range of initial asset prices. This is done by roughly equating the option value function with the additional conditional expectation function estimated at the initial time. Simple manipulation of this function immediately yields the desired estimates for price and Greeks. The details of this approach are described below.

18 11 Figure 2-1 Single initial price Figure 2-2 Random initial prices The first step in implementing any numerical algorithm to price any American-style option is to assume that time can be discrete. Thus, the algorithm will assume that the derivative expires

19 12 in L periods, and specify the exercise points as = 0 < < < < = T. In practice, of course, many American-style options are continuously exercisable; the modified LSM algorithm can still be applied to these options by taking L to be sufficiently large. Let be a sequence of state variables, where is the stock price at a specified exercisable time point and at a specified stock price sample path. Let be an adapted payout process for the option, satisfying, for a suitable function h. As an example, consider the American-Asian put option from above (use the fixed price American-Asian option case), for which the state variable of interest is the average stock price for a predetermined range of time (e.g, time interval [0,t]),. Then the payout function, where is the strike price. In the traditional LSM algorithm, the state variable is deterministic. While in the modified LSM algorithm, an important adjustment is made by generating the state variable from some predetermined distributions. This means the initial stock prices are random numbers generated from a normal distribution, uniform distribution, or some other distributions. These random initial stock prices have a median around and are different for each stock price sample path. From the payout function, let denote the optimal stopping time for the option from time to the maturity time. Then the algorithm defines the function as the cashflow generated by the option (r is the risk free discount factor), discounted back to time and conditional on no exercising prior to time and on following a stopping strategy from to expiration, which is comparing the exercise value and the holding value of the option. In this formulation, the initial value function at time 0 can be specified as:

20 13 where the maximization is over stopping times, with denoting the set of all stopping times with values in., As shown above, the key to solve this problem is finding the optimal stopping time and the preferred way to solve it is to use the dynamic programming principle. For the American-style option problem, if the holding value is greater than the exercise value, then the option should be held. If the exercise value is greater than the holding value, the option should be exercised. This idea can be written in terms of the optimal stopping times as follows: where is equal to 1 when is true, otherwise the value is 0. Similarly, is equal to 1 when is true, otherwise 0. So the optimal stopping time is either (exercise the option) or (hold the option). Thus the initial value function can be expressed in terms of the optimal stopping times as: where ) represents the expected payout from continuation at time. The key contribution of Longstaff and Schwartz [8] is that they provide a particularly useful method to approximate the, conditional expectations by using least squares regression. The theory on Hilbert spaces [14] tells us that any function belonging to this space can be represented as a countable linear combination of basis vectors for the space. In particular, assuming that belongs to a Hilbert space, it can be written as:,

21 14 where forms a set of basis functions (e.g, ) and are the regression coefficients. The coefficients are generally unknown. Longstaff and Schwartz suggest in their algorithm a procedure for approximating and thus using the first M basis functions and N sample paths for the stock price as a way to solve the least squares minimization problem, where represents the regression coefficient for the basis function when the number of stock price sample paths is N.. The equation above tells us that the initial value function can be approximated by the conditional expected payout from continuation at time 0. This is the essence of the modified LSM algorithm. Compared to the original LSM algorithm, which evaluates the option value at only one point, equation turns out to be a significant step forward because it provides a direct estimation of the option values for a continuous range of stock prices near. In particular, can be obtained simply by taking X to be and get the new equation. Equally important hedging parameters, such as the first order derivative and the second order derivative of the option value, are immediately produced by analytically differentiating the expression:,. As shown in this approach, one does not need to pay attention to the dynamics of sample paths, which means this approach is path independent. Recall those exotic option types mentioned above, such as Asian options, Bermudan options, and multi-dimensional cases, all of these

22 options can be solved by this method. Furthermore, the modification of this approach helps to simplify the calculation of the desired estimates for price and Greeks Control method To test the accuracy and efficiency of the modified LSM, the Binomial tree option pricing method is introduced in this chapter. The Binomial tree approach to price options was first proposed by Cox, Ross, and Rubinstein [5]. First, we consider a simple one-period Binomial tree model, and explain how the no-arbitrage principle guarantees the existence of the risk-neutral measure in this setting and show that the price of an option in this setting is simply the expectation of the discounted pay-off under this risk neutral measure. In this setting, we say that an arbitrage occurs if zero investment leads to no-loss with probability 1 and a positive profit with positive probability, which means that one can make money freely without any investments and without taking any risks.

23 16 Figure 2-3 the Binomial tree model Figure 2-3 shows an asset whose value at time zero equals scenarios, up and down, can occur: the stock price takes value. Suppose that at time 1 two in scenario up and otherwise, where. Suppose that scenario up occurs with probability and scenario down with probability. These probabilities are referred to as market probabilities. Furthermore suppose that in this economy there exists a risk-free asset whose value is 1 at time zero and it becomes in both scenarios at time 1. Assume that an unlimited amount of both these assets can be borrowed or sold without any transaction costs. Here a brief outline of the Binomial tree method for pricing American Asian options will be given. The ideas in this section come primarily from the original paper by Cox, Ross, and Rubinstein [5]. First note that the no-arbitrage principle implies that. Otherwise, if, borrowing amount at the risk-free rate and purchasing the risky security, the investor

24 would earn at least in time period one, where his liability would be. Thus with zero investment he is guaranteed sure profit (at least with positive probability if ), 17 violating the no arbitrage condition. Similarly, if, then by short-selling the stock, the investor would gain amount, which he would invest in the risk-free asset. At time 1 he gains while his liability is at most (the price at which he can buy back the risky security to close the short position), thus leading to arbitrage. This is the basic logic of the Binomial tree method. The application of this method in option pricing is shown below. Assume and the expected stock price grows at the risk-free rate. In order to match the risk-neutral no arbitrage behavior, the probability of an up move is calculated as, where is the time interval between each Binomial tree node [10]. In the Binomial tree model, the parameters are chosen so that this evolution matches a geometric Brownian motion with volatility and drift r. First divide the time T over N smaller steps of length. After the time-step =n, the stock price can move up to or down to with probabilities p and (1-p), respectively. Note that at each time-step n, the tree has n+1 nodes, labeled. Then with the dynamic programming backward recursion, it is easy to price back to. In the following chapters, this thesis puts emphasis on American-Asian options pricing. In order to price such exotic options, the only change from the original Binomial tree method is modifying the payoff function. In the American-Asian option case, the payoff is correlated with the average stock prices over the time period. While the disadvantage of this method is that if one wants to price more complex options, or 2 or more dimensional options, the computation time increases exponentially and the algorithm should be changed a lot to fit certain situations. In the single asset Binomial tree model, we all know the number of tree nodes in time period n is.

25 18 However, in the higher m-dimensional case, the tree nodes in time period n would be. Thus, the computational effort grows exponentially. And for complex options, like options with a jump diffusion process, the Binomial tree method is not appropriate to price them, because the jump diffusion process is not allowed in the Binomial tree model. For this reason, it is inferior to the LSM method for pricing multi-dimensional options and some other more complex options.

26 Chapter 3 Valuing American-Asian Options 3.1 American-Asian option pricing by the modified LSM method Compared to the standard American option, the American-Asian option is a more exotic path-dependent option. In order to use the modified LSM to price this exotic options, the pricing steps discussed in Section should be accomplished (this thesis uses the fixed strike price American-Asian option as an example to explain the steps of this approach). The modified LSM method uses randomly distributed initial prices to generate the stock price sample paths. A few experiments have been conducted to investigate the relationship between the initial distribution and the corresponding results and the results turn out to be very robust to the choice of initial distribution, which coincides with the intuitional guess (the initial stock prices do not affect the inside logic of the entire algorithm). Specifically, for consistency and simplicity, the initial distribution (t=0) for all examples is set to:, where follows the standard normal distribution N(0,1), σ and T are the corresponding stock volatility and time to expiration, and could be used to adjust the variance of the distribution. The median of these randomly distributed stock prices is expected to be. Once the stock prices sample paths are generated, the modified algorithm uses the same technique as mentioned in Section to price the American-Asian options. Suppose that the underlying asset price follows a geometric Brownian motion [1]:, where r is the risk-free interest rate, represents the volatility, r and are positive constants, and is a standard Brownian motion under the risk-neutral measure.

27 20 However, the strike prices in Asian-style options are path-dependent and are calculated by the average prices over a predetermined time horizon. Let, the path variable, denote the average asset price for the period up to t; so from time 0 to nt, there are nt+1 values:, where n is number of exercisable points in one time unit. The two types of payoffs at a specified time point for the American-Asian options are: (1) fixed strike price (take the put option as an example):, (2) floating strike price (take the put option as an example):. In the circumstance above, the original algorithm should be revised for replacing the current stock prices by the average prices. This means, when comparing the exercise value with the expectation holding value of an option, the exercise value is no longer, but (in the fixed strike price case). By solving the problem backward, the last step in applying the modified LSM approach to price American-Asian options is regressing the option values on the initial stock prices, the first four power polynomials are used as the set of basis functions in the regression. The option value function is approximated as follows [15]: Direct substitution and differentiation would yield estimates for price, (delta, first order derivative of ) and (gamma, second order derivative of ): It is straightforward to add additional basis functions as explanatory variables in the regression if needed. Using more than four basis functions, however, causes little change to the

28 numerical results; four basis functions are adequate to obtain effective convergence of the algorithm in this thesis A simplified example As it is explained in Longstaff and Schwartz paper [7], a simple numerical example is shown in this section to illustrate the basic idea of the simulation methods for both the fixed strike price and the floating strike price American-Asian options. Consider a three-year American-Asian put option on a share of non-dividend paying stock that can be exercised at the end of year 1, year 2 and year 3. The current stock price is 1.00 and the fixed strike price is The risk-free rate is 6% per annum (continuously compounded). For simplicity, we illustrate the algorithm by using only eight stock price sample paths and the initial prices are produced from a uniform distribution on [0.90, 1.10]. (This is for illustration only; in practice many more paths would be sampled and other distributions could be used to generate initial prices.) The entire sample paths are constructed in Table 3-1. Each stock price sample path follows the geometric Brownian motion process:, where is the stock price at time t, r is the risk-free interest rate which equals 6% per annum, represents the volatility which equals 0.1, and is a geometric Brownian motion under the risk-neutral measure which is generated from the standard normal distribution N(0,1).

29 22 Table 3-1 Stock price paths Path t=0 t=1 t=2 t= The fixed strike price American-Asian option example Let us start with the fixed strike price case. First, we apply the LSM method to these sample paths to obtain the optimal stopping rule that maximizes the value of the option at each point along each path. The LSM method is recursive in nature and we need to work backwards one step at a time. Conditional on not exercising the option before the final expiration date at time 3, the cash flows realized by the option holder at time 3 are given in Table 3-2. The last column shows the average stock prices from t=0 to t=3. In Table 3-2, for example, the average stock price from t=0 to t=3 of Path 4 is 0.95, which is calculated based on the stock prices in Table 3-1,. Thus, the cash flow if exercised only at time 3 is 0.15, which is the strike price minus the average stock price. We can tell from Table 3-2, Paths 3, 4, 6, and 7 are in-the-money because they have positive cash flow at t=3. Now we consider if the option can be exercised both at t=2 and t=3. Then if the option is inthe-money at time 2, the holder must decide whether or not to exercise. As Table 3-3 shows, for example, the average stock price from t=0 to t=2 of Path 4 is 0.96, which is calculated based on the stock prices in Table 3-1,. Thus, the cash flow if

30 23 exercised only at time 2 is 0.14, which is the strike price minus the average stock price. We can tell from Table 3-3, there are five paths for which the option is in-the-money at time 2 and they are Paths 1, 4, 6, 7, and 8. The algorithm uses only these in-the-money paths since it is better to estimate the conditional expectation function. The reason for including only in-the-money paths in the regression is primarily due to numerical considerations. This is clearly explained and numerically backed up in Longstaff and Schwartz [8], more than two or three times as many basis functions may be needed to obtain the same level of accuracy as obtained by the estimator based on in-the-money paths. Table 3-2 Cash flows if exercised only at time 3 (the fixed strike price case) Path t=1 t=2 t=3 K Table 3-3 Cash flow if exercised only at time 2 (the fixed strike price case) Path t=1 t=2 K

31 24 Let X denote the stock prices at time 2 and Y the corresponding discounted cash flows from continuation. We combine Table 3-1 and Table 3-3, the five in-the-money observations on X are 1.08 (Path 1), 0.97 (Path 4), 0.77 (Path 6), 0.84 (Path 7), and 1.22 (Path 8), and the corresponding values for Y from Table 3-2 are 0.00, 0.15, 0.27, 0.16, and Then regressing Y on a constant, X and yields the estimated conditional expectation function:. Because there are only eight stock price sample paths in this numerical example and higher order basis function does not improve the regression accuracy a lot here, we only use X and as basis function. The of this regression is 82% and the p-value is In real experiments, the regression results will be more significant when the number of stock prices sample paths increases. With this conditional expectation function, the algorithm compares the value of immediate exercise with the value from continuation to find that it is optimal to exercise the option at time 2 for paths 4, 6, 7 and 8. This leads to the matrix in Table 3-4, which shows the cash flows received by the option holder conditional on not exercising prior to time 2. For example, in Path 1, the exercise profit at time 2 is 0.02 while the expected value from the expectation function is 0.03, which is calculated based on the expectation function. Thus, we should hold this option at time 2 and wait for better chance to exercise it. However, in the other 4 paths, the expected values from the function are all lower than the exercise profits, so we should exercise the option at time 2. Proceeding recursively, next consider the paths in-the-money at time 1. As shown in Table 3-5, there are paths 1, 4, 5, 6, 7, and 8. For example, the average stock price from t=0 to t=1 of Path 4 is 0.96, which is calculated based on the stock prices in Table 3-1,. Thus, the cash flow if exercised only at time 1 is 0.14, which is the strike price minus the average stock price.

32 25 Table 3-4 Cash flows if exercised only at time 2 and 3 (the fixed strike price case) Path Expected Value at t=2 Exercise Profit at t=2 Decision at t= Hold Exercise Exercise Exercise Exercise Path t=1 t=2 t= Table 3-5 Cash flow if exercised only at time 1 (the fixed strike price case) Path t=1 K

33 26 Similarly as we explained at time 2, X represents the stock price at time 1 and Y the discounted value of subsequent option cash flows. We combine Table 3-1 and Table 3-5, the six in-the-money observations on X are 1.09 (Path 1), 0.93 (Path 4), 1.11 (Path 5), 0.76 (Path 6), 0.92 (Path 7), and 0.88 (Path 8), and the corresponding values of Y from Table 3-4 are 0.00, 0.14, 0.00, 0.29, 0.18, and The linear regression returns the estimated conditional expectation function:. The of this regression is 76% and the p-value is This gives the value of continuing at time 1 for paths 1, 4, 5, 6, 7, and 8 as 0.01, 0.11, 0.00, 0.26, 0.12, and 0.15, respectively. The value of immediate exercise is 0.03, 0.14, 0.04, 0.27, 0.14, and This means that we should exercise at time 1 for paths 1, 4, 5, 6, 7, and 8. For example, in Path 1, the exercise profit at time 1 is 0.03 while the expected value from the expectation function is 0.01, which is calculated based on the expectation function. Thus, we should exercise this option at time 1. For the other five paths, the results are same. Thus, we should exercise the option at time 1 for all these six paths. Table 3-6 summarizes the cash flows assuming that early exercise is possible at all three times. Having identified the cash flows generated by the American-Asian put option at each date along each path, one can estimate the option value as a function of the initial stock prices by conducting a linear regression at time 0. For this all the simulated paths should be considered, and define X as initial stock prices for each path and Y the corresponding discounted payouts. Eight observations on X from Table 3-1 are 1.05, 1.07, 1.02, 0.99, 1.01, 0.91, 1.00, and 0.95, the values of Y from Table 3-6 are 0.03, 0.00, 0.01, 0.14, 0.04, 0.27, 0.14, and 0.19 for the fixed strike option case. Finally, regressing Y on a constant X and results in our desired option value function,. The of this regression is 87% and the p-value is 0.02.

34 27 Table 3-6 Cash flows from the option (the fixed strike price case) Path Expected Value at t=1 Exercise Profit at t=1 Decision at t= Exercise Exercise Exercise Exercise Exercise Exercise Path t=1 t=2 t= The floating strike price American-Asian option example After the discussion about the fixed strike price American-Asian option, we introduce the pricing process of the floating strike price case. All the parameters used in this case are identical with those in the fixed strike price case. And we still use Table 3-1 as the stock price sample paths table. Similar as we explained in the fixed strike price case, conditional on not exercising the option before the final expiration date at time 3, the cash flows realized by the option holder at time 3 are given in Table 3-7. The last column shows the average stock prices from t=0 to t=3

35 28 and is the stock price at time 3, which is also the strike price in the floating strike price case. In Table 3-7, for example, the average stock price from t=0 to t=3 of Path 4 is 0.95, which is calculated based on the stock prices in Table 3-1,. Thus, the cash flow if exercised only at time 3 is Because the strike price is less than the average stock price, the payoff function shows that the option has no value at time 3. We can tell from Table 3-7, Paths 1, 2, 5, 6, 7, and 8 are in-the-money because they have positive cash flow at t=3. Now we consider if the option can be exercised both at t=2 and t=3. Then if the option is inthe-money at time 2, the holder must decide whether or not to exercise. As Table 3-8 shows, for example, the average stock price from t=0 to t=2 of Path 4 is 0.96, which is calculated based on the stock prices in Table 3-1, exercised only at time 2 is 0.01, which is the strike price. Thus, the cash flow if minus the average stock price. We can tell from Table 3-8, there are four paths for which the option is in-the-money at time 2 and they are Paths 2, 4, 5, and 8. Table 3-7 Cash flows if exercised only at time 3 (the floating strike price case) Path t=1 t=2 t=

36 29 Table 3-8 Cash flow if exercised only at time 2 (the floating strike price case) Path t=1 t= Let X denote the stock prices at time 2 and Y the corresponding discounted cash flows from continuation. We combine Table 3-1 and Table 3-8, the four in-the-money observations on X are 1.26 (Path 2), 0.97 (Path 4), 1.56 (Path 5), and 1.22 (Path 8), and the corresponding values for Y from Table 3-7 are 0.28, 0.00, 0.22, and Then regressing Y on a constant, X and yields the estimated conditional expectation function:. The of this regression is 62% and the p-value is With this conditional expectation function, the algorithm compares the value of immediate exercise with the value from continuation to find that it is optimal to exercise the option at time 2 for paths 4, 5, and 8. This leads to the matrix in Table 3-9, which shows the cash flows received by the option holder conditional on not exercising prior to time 2. For example, in Path 2, the exercise profit at time 2 is 0.10 while the expected value from the expectation function is 0.15, which is calculated based on the expectation function. Thus, we should hold this option at time 2 and wait for better chance to exercise it. However, in other 3 paths, the expected values from the function are all lower than the exercise profits, so we should exercise the option at time 2.