A New Adaptive Mesh Approach for Pricing the American Put Option

Transcription

1 A New Adaptive Mesh Approach for Pricing the American Put Option Yang Yang Advised by Professor KISHIMOTO KAZUO Submitted to the Graduate School of Systems and Information Engineering in Partial Fulfillment of the Requirements for the Degree of Master of Engineering at the University of Tsukuba January 00

2 Abstract Various options are now popularly traded all over the world. Thus, the e cient accurate pricing of options is of substantial practical importance. If we assume that the underlying security of an option follows the Black-Scholes dynamics, the price of a European option is given by the general formula, i.e., the expectation of the premium at the maturity date with respect to the equivalent Martingale measure. In the case of American call option, the price is equal to that of corresponding European options, while in the case of American put option, it is known that the pricing is substantially more complicated. Up to now lots of numerical approximation procedures were proposed for pricing American put options. Because of various di culties in calculating the price of American options, however, intensive e orts are still needed for developing new approaches to this problem. In the present paper, based on a trinomial tree approximation, we propose an improved version for pricing the American put option on one underlying asset which follows the Black-Scholes dynamics. We then compare the accuracy of our method with various existing approaches by simulations. The results show that our approach is the most accurate in the case of out-of-the-money. i

3 Acknowledgement This article is the result of research carried out in my second year at the Doctoral Program in Systems and Information Engineering, University of Tsukuba. It is my great pleasure to thank all those who have helped me. First of all, I am most grateful to my instructor Professor Kazuo Kishimoto. His advice, encouragement and enthusiasm have been a great source of inspiration to me. His constructive comments and ideas have significantly shaped this article. I am deeply grateful to Mr. Saito, Mr. Maeda and Mr. Zuo of Doctoral Program in Social and Planning Science for many good advice. Finally, valuable comments are received from Mr. Oda and Mr. Hamada and others who have helped me. Thanks are also due to them. ii

4 Contents 1 Introduction 1 Definition of Options and Their Pricing.1 Definition of Options Black-Scholes Formulation of Option Pricing American Put Option as a Free Boundary Problem Principles of Tree-based Pricing Models Framework of Tree-based Pricing Models Dynamic Programming Procedure for the American Put Option Survey of Existing Tree-based Methods Two-jump process Models CRR, JR and Tian Models SMO and FB Models Three-jump Process Models Boyle, Kamrad and Ritchken Models Adaptive Mesh Method by Gao The Proposal of This Research Exercise Boundary of American Put Option Approximation Error Around the Exercise Boundary iii

5 5.3 Our Proposal Numerical Comparison of Di erent Tree-based Models A General Comparison of Tree-based models by RMS Tests on SMO and FB models Tests on CRR, JR and Tian models Tests on KR and Boyle models Tests on all three-jump process models Conclusion and Future Work Conclusion Future work iv

6 List of Figures 5.1 case: num[i] num[i 1] case: num[i] num[i 1] SMO and FB Models: case of out-of-the-money case of at-the-money SMO and FB Models: case of in-the-money Zigzag Property of SMO Model Zigzag Property of FB Model CRR, JR and Tian Models: case of out-of-money CRR, JR and Tian Models: case of at-the-money CRR, JR and Tian Models: case of in-the-money Five Trinomial models: case of out-of-the-money Five Trinomial models: case of at-the-money Five Trinomial models: case of in-the-money v

7 List of Tables 6.1 RMSE of 54 parameter sets RMSE in the case of out-of-the-money RMSE in the case of in-the-money RMSE in the case of at-the-money RMSE with the same number of calculation nodes Option Value S (0) 100 K 90 r T Option Value S (0) 100 K 100 r T Option Value S (0) 100 K 110 r T vi

8 Chapter 1 Introduction Up to now, no closed-form solution for pricing American put option is known. Thus, numerical approaches are studied extensively. They are classified into the following groups: 1. analytical approximation (ex. MacMillan (1986)),. finite di erence equation (ex. Brennan and Schwartz (1978)), 3. Monte Carlo simulation (ex. Broadie and Glasserman (1996)), 4. tree-based models (ex. Cox (1979)). Tree-based approaches are most flexible, and various versions were proposed. In the present thesis, we focus our attention on the tree-based approach. We survey existing methods and propose a new version. Then, we compare the accuracy of these approaches by numerical simulations. In chapter, we give a brief introduction to the definition of options and their pricing. In chapter 3, framework of tree-based pricing models is reviewed. A survey of existing tree-based methods is made in chapter 4, we present our method in chapter 5. Results of Numerical comparison will be given in chapter 6. Chapter 7 concludes the thesis. 1

9 Chapter Definition of Options and Their Pricing.1 Definition of Options Options are classified either as a call option or a put option. A call(or put) option is a contract which gives the holder the right to buy(or sell) a prescribed asset, known as the underlying asset, by a certain date for a predetermined price. Since the holder is given the right, but not the obligation, to buy (or sell) the asset, he will can make the decision depending on whether the deal is favorable to him or not. The option is said to be exercised when the holder chooses to buy (or sell) the underlying asset actually. If an option can only be exercised on the expiration date, then the option is called a European option, while if the exercise is allowed at any time prior to the expiration date, then it is called an American option. The simple call and put options with no special features are commonly called plain vanilla options. In this thesis, American put vanilla option is the main topic and we call it American put option in short. The counterpart to the holder of the option in the contract is called the writer of the option, unlike the holder, the writer does have an obligation with regard to the option contract. Since the writer of an option has the potential liabilities in the future, he must be compensated by an up-front premium payment of the option by the holder when they enter into the option contract. On the other hand, since the holder is guaranteed to receive a non-negative

10 payo, he must pay a premium in order to enter into the contract. Then a natural question is to decide the fair option premium so that the game is fair to both parties of writer and holder. As to the American option, it is known to be a more complex problem since it depends on the optimal strategy for exercising the option prior the expiration. To proceed with this thesis, it is convenient to employ the following notation: T: time to option maturity ( in years). K: exercise price of option. r: the continuously compounded yearly interest rate. : the variance of the rate of return on the underlying asset(yearly). N: the number of time steps into which the interval of length T is divided. t: T N, length of one time step. S (0): the initial underlying asset value. S (t) or S : the underlying asset value at time t. S (t)u : asset value after an up jump. S (t)d: asset value after a down jump. p u : probability for up jump. p m : probability for horizontal jump. p d : probability for down jump. R : discounted interest rate. P(t) : American put option value at time t. V(t) : European put option value at time t. V cont : continuation value of American put option. h(t): intrinsic value of American put option at time t.. Black-Scholes Formulation of Option Pricing In the standard model, the markets consists of the option, its underlying security, labelled the stock, and a riskless security, labelled the bond. The market is populated by equally 3

11 informed traders. We assume that the market is complete. That is the following conditions are required: trading takes place continuously in time; the riskless interest rate r is known and constant over time; the asset pay no dividend; there are no transaction costs in buying or selling the asset or the option; the assets are perfectly divisible; there are no penalities to short selling and the full use of proceeds is permitted; there are no riskless arbitrage opportunities. The randomness of the continuous stock price process S (t) is assumed to follow the Geometric Brownian motion (GBM): where r is the instantaneous risk-free interest rate, ds (t) rs (t)dt S (t)dz (.1) is the instantaneous volatility of the stock price, dt is an infinitely small increment of time, and dz is a standard wiener process. Consider a portfolio which involves short selling of one unit of a European put option and long holding of units of the underlying asset. The value of the portfolio V is given by S (.) where V V(S t) denotes the European put option price, which is a function of the asset price and time. Here S refers to times S, not infinitesimal change in S. Since both V and are random variables, we apply the Ito lemma to compute their stochastic di erentials as follows: dv V t dt V S ds S V dt (.3) S 4

12 and d dv V ( t V [ t ds S V S )dt ( V S )ds S V S ( V S )rs ]dt ( V S ) S dz (.4) V The stochastic component of the portfolio appears in the last term: ( S ) S dz. If we choose V, then the portfolio becomes a riskless hedge. In equilibrium, the hedged S portfolio must earn the riskless interest rate. Otherwise, suppose the hedged portfolio earns more than the riskless interest rate, then an arbitrager can lock in riskless profit by borrowing as much money as possible to purchase the hedged portfolio. By setting d r dt, we then have d ( V t S V S )dt r(v S V )dt (.5) S and upon rearranging the terms, we obtain V 1 S V t S rs V S rv 0 (.6) This is Black-Scholes partial di erential equation(black and Scholes, 1973). By applying the final payo condition, we can get the formula of European put option value as follows: r(t V(S t) Ke t) N( d ) S N( d 1 ) (.7) where N( ) is the cumulative distribution function for a standardized normal random variable, given by Here, d 1 N(x) 1 x e 1 y dy 1 log S K (r )(T t) T t 5

13 and d 1 log S K (r )(T t) T t.3 American Put Option as a Free Boundary Problem Black-Scholes partial di erential equation follows from an arbitrage argument and from it we can get the formula for European put option. For an American put, the simple arbitrage argument used for the European option no longer leads to a unique value for the return on the portfolio, only to an inequality. For each time t, we must divide the S axis into two distinct regions. If we define S (t) as the critical stock price at t, when S (t) S (t), early exercise is optimal and P 1 P(t) K S (t) t S P S rs P rp 0 (.8) S In the other region, when S (t) S (t), early exercise is not optimal and P 1 P(t) K S (t) t S P S rs P rp 0 (.9) S The boundary condition and the final payo function are: P(S (t) t) max(k S (t) 0) (.10) P(S (T) T) max(k S (T) 0) (.11) Since we do not know a priori where S (t) is, till now no analytical formula has been obtained for this problem. 6

14 Chapter 3 Principles of Tree-based Pricing Models 3.1 Framework of Tree-based Pricing Models Since there is no closed-form solution for the free boundary problem describing the dynamics of the American put option price, a variety of numerical research has been done. They are classified into the lattice schemes, finite di erence method and Monte Carlo simulation. Simulation is an attractive method for asset pricing because of the generality in the types of assets it can handle and the ease with which it handles multiple state variables, but its major drawback is apparent inability to deal with the free-boundary aspect of American options. Standard simulation procedures are forward algorithms; given a pre-specified exercise policy, a path price is determined. The average over independent samples of path prices gives an unbiased estimate of the security price. By contrast, pricing procedures for assets with early-exercise features are generally backward algorithms. The problem of using simulation to price American put option stems from the di culty of applying the forward-based procedure to a problem that requires a backward procedure to solve. Finite di erence method is used in a wide variety of fields, however, it is not applied to exotic options. Tree-based models have been proven to be very e cient and flexible in the cases where analytic methods fail although they have limitations as well. In this thesis, we will concentrate on numerical methods on American put option, mainly on methods based on 7

15 tree-based models. The main property of tree-based models is that they are constructed in such a way that if the time between two trading dates shrinks to zero, convergence to their continuous counterpart is achieved. By using Ito s Lemma, one can write equation 1 in the following way: or dlog(s (t)) (r )dt dz (3.1) dx! dt " dz (3.) where X log(s (t)) and r. As a result, log(s (t)) follows a generalized wiener process for the time period. Let S (t) and S (t t) denote, respectively, the asset prices at the current time and at on one period t later. In the Black-Scholes continuous model, the asset price dynamics is assumed to follow the Geometric Brownian motion where S (t t) S (t) is lognormally distributed. In the risk neutral world, log S (t t) becomes normally distributed with mean (r " S (t) ) and variance t. By equating the mean and variance of the asset price ratio S (t t) in both continuous and discrete models, we can use tree models S (t) to obtain the numerical value of options: pu (1 p)d e r# t (3.3) pu (1 p)d e r# t e r# t (e$ # t 1) (3.4) where p is the risk-neutral probability for up jump. Let us denote that R e r# t. Thus R d p. By choosing another condition arbitrarily, we can get the values of u d p. In u d sum, for tree-based models, the following two conditions should be fulfilled: Sum of probabilities to be one; Equating the first two moments of the approximating discrete distribution and the corresponding continuous lognormal distribution of the Black-Scholes model. While other remaining conditions can be chosen correspondingly. 8

16 3. Dynamic Programming Procedure for the American Put Option The American option gives its holder greater rights than the European option, via the right of early exercise, potentially it has a higher value. An dynamic programming procedure is required in the tree schemes in order to price an American option. When the early exercise privilege does not exist or is forfeited, risk-neutral valuation leads to the usual formula as: V cont (t) pp t% # u t (1 p)p t%# t d (3.5) R Here, we use V cont (t) to represent the continuation value when we move one time step backward in the tree, P t%# t node at t u and P t% # t d are the put option value of up jump node and down jump t respectively. p is the risk-neutral probability for up jump. When early exercise possibility is taken into account, the continuation value V cont (t) should be compared with the option s intrinsic value h(t) max(k S (t) 0), which is the payo function upon exercise. The following dynamic programming procedure is applied at each node of the tree: P(t) max(h(t) V cont (t)) (3.6) For the case of American vanilla put option, first we build the tree which gives a discrete representation of the stochastic movement of the asset price nodes, then we calculate backwardly in the above way until the initial point to obtain the option value. 9

17 Chapter 4 Survey of Existing Tree-based Methods Since Cox, Ross and Rubinstein(1979) pioneered the lattice approach(crr model), many versions of tree-based methods have been put forward. Although there is no clear standard to classify these methods, in this thesis, we first classify them into two groups according to the jump process as binomial or trinomial methods. In the field of binomial tree methods, according to whether there is adjustment on the nodes for the strike at the maturity or not, we classify them into two subgroups, one includes the SMO model (Leisen,1998) and FB model (Tian,1999), while the other includes CRR model, Jarrow and Rudd (1983) model and Tian (1993) model. Trinomial tree methods consist of two groups. The first is Kamrad and Ritchken model (1991) and Boyle (1988) model, which have one unknown tilt parameter. The second is Gao model (Gao,1999), which has a fixed magnitude of jump. 4.1 Two-jump process Models CRR, JR and Tian Models A binomial tree model was first developed by Cox, Ross and Rubinstein (CRR). The essence of their approach is the construction of a binomial lattice of stock prices where the risk neutral valuation rule is maintained. With a particular selection of binomial parameters including probabilities and jumps, they showed that the CRR binomial model converges to 10

18 the Black-Scholes model. The parameters of CRR model are: u e$'& # t d e $'& # t p R d u d (4.1) Equations (3 3) and (3 4) provide only two equations for the three unknowns : u d p, the remaining condition has been chosen to be u so that the lattice nodes associated with the binomial tree are symmetrical. In fact, it is a simplified formula but without sacrificing the degree of accuracy. Jarrow and Rudd(1983) 1 relaxed the reconnecting condition u d and chose p 1 as the third condition, then the values for u, d and p are found to be u 1 d e (r )( )# t% $*& # t d e (r *( )# t% $+& # t p 1 (4.) Another version of the binomial model was proposed by Tian (1993), who chose the third condition to be pu 3 (1 p)d 3 e 3r# t e 3 $ # t (4.3) The above relation is derived from matching the third moment of the discrete-time process and the continuous-time process for the asset price ratio. Then the parameters are found to be where Q e$ about S whenever ud, u d RQ [Q 1 Q Q 3] RQ [Q 1 Q Q 3] p R d u d (4.4) # t. Since ud R Q instead of ud 1, the binomial tree loses symmetry 1. For European put options, these three models are proved to converge to the true price with the same order. On the contrary, since the American option valuation problem involves a control, the corresponding results are not available in the American 11

19 option case. BBS(Binomial Black-Scholes) method is identical to CRR method, except that at the time step just before option maturity the Black and Scholes formula replaces the usual continuation value. Breen(1991) proposed the Accelerated Binomial Model, by taking advantage of Richardson extrapolation formula P P(3) 3 5(P(3) P()) 0 5(P() P(1)), where P(3) is the option value allowing exercise at T T 3 and T 3 only. P() is the option value allowing exercise at T and T only. P(1) in fact is the European put option value since option can only be exercised at the maturity. Although this method is faster than the binomial routine for the same N, however, it does not supply enough early exercise opportunities of American put option to obtain accuracy, we will not discuss it in detail in this thesis SMO and FB Models Leisen and Reimer(1998) made a deep analysis on the convergence of CRR, JR and Tian models. They drew the conclusion that in the case of European option, the above three models all converge with order one. Since there is odd-even phenomena when pricing option based on tree models, smooth convergence is thus desirable because more time steps do guarantee more accurate prices. They proposed their SMO model, the main idea of which is to ensure that the strike always lies fixed at a specific node, the center of the tree at the maturity. In order to do it consistently, the refinement N must be even. Thus the new parameter selection u d should fulfill: u e$'& # t e c N d e $ & # t e c N S (0)(ud) N- K (4.5) where c N log(k S (0)) N. The equivalent martingale measure is obtained by setting p R d. Although SMO model enjoys the advantage of smooth convergence, since it does u d not fulfill the local variance properly, it always yields higher errors than other binomial models. In fact, although this model can get smooth result for European put options, we 1

20 cannot expect it to get smooth result for the American put option, maybe only smoother one, since we are starting with a lower initial error. Tian(1999) reconsidered the binomial schemes and developed a flexible binomial (FB) model with a tilt parameter that alters the shape and span of the binomial tree. Then a positive tilt parameter shifts the tree upward while a negative tilt parameter does exactly the opposite. The original CRR binomial model can be regarded as having zero tilt. Tian proposed the FB model as follows: u e$ & # t%/. $ d e $ & # t%0. $ # t # t (4.6) where 1 is an arbitrary constant, called the tilt parameter. Since the FB model is specified only up to an arbitrary constant, it has an extra degree of freedom over the standard binomial model. The arbitrary parameter 1 can be regarded as the degree of tilt in the binomial tree and are decided as follows: First, define u d p be u 0 d 0 and p 0 which are just the same as that of CRR model and using them to construct the tree; u 0 e$ & # t d 0 e $ & # t p 0 e r# t d 0 u 0 d 0 (4.7) Secondly, find the closest node (N j 0 ) to the strike price K at the maturity. Then, j 0 is determined by j 0 [ log(k S (0)) N log(d 0) log( u ] (4.8) 0 ) d 0 where [ ] denotes the closest integer to its argument. To ensure that node (N j 0 ) coincides exactly with the strike price K, 1 is selected as: 1 log(k S (0)) ( j 0 N) N t t (4.9) Finally, the tree is constructed again by the adjusted parameter values of u d p in (4 6) after the tilt parameter 1 is decided. 13

21 4. Three-jump Process Models 4..1 Boyle, Kamrad and Ritchken Models In an attempt to improve the accuracy and reliability of option valuation schemes, more than two-jump process were introduced. Trinomial model allows greater freedom in the selection of parameters to achieve certain objectives, like avoiding negative probabilities, attaining a faster rate of convergence, etc. The tradeo is the loss of e ciency in general since a trinomial scheme requires more computational steps compared to that of a binomial scheme. A direct link between the approximating process of the asset price movement and the arbitrage strategy is not essential. In fact, any contingent claim can be valued by computing an appropriate conditional expectation. If such conditional expectation is di cult to evaluate, one may use an approximation discrete process to approximate the underlying asset price movement and di erent approximating procedures will lead to di erent numerical schemes. Boyle(1988) proposed the following three-jump process for the approximation of the asset price movement over one period. Let S be the current asset price. Over one period, the stock price rises to us with probability p 1, remains to be S with probability p, goes down to ds with probability p 3. By setting sum of probabilities to be one p u p m p d 1 (4.10) Equating the first two moments of the approximating discrete distribution and the corresponding continuous models: p u u p m p d d e r# t (4.11) p u u p m p d d e r# t e$ # t (4.1) The remaining two conditions were chosen by Boyle to be ud 1 14

22 and The explicit expressions for p u and p d are: where W R e$ p u # t. u e. $ & # t 1 is a free parameter (W R)u (R 1) p (u 1)(u d 1) (W R)u (R 1)u 3 (4.13) (u 1)(u 1) In the risk neutral world, log S # t S is normally distributed with mean (r ) t and variance t. Alternatively, we may write log S (t t) log S (t) "34 (4.14) where 3 is a normal random variable with mean (r ) t and variance t. Kamrad and Ritchken (1991) proposed to approximate 3 by the following discrete random variable with the distribution where v 516 t and 1 3 v with probability p u 0 with probability p m v with probability p d 1. The corresponding values for u m and d in the trinomial scheme are: u e v m 1 and d e v. By equating the mean and variance of discrete and continuous models while keeping the condition that sum of the probabilities is one, we obtain p u p d 1 1 (r $ ) t 17 1 p m (r $ ) t (4.15) 16 By choosing di erent values for 1, a range of probability can be obtained as well. 15

23 4.. Adaptive Mesh Method by Gao It is known that tree-based models often have a peculiar even-odd convergence property, which causes the approximation error to alternate between two quite di erent values as the number of time steps in the tree goes from even to odd and back to even. To solve this problem, Gao(1999) put forward the Adaptive Mesh method to minimize the approximation error of nodes around the strike price. In his trinomial tree, X(t) is denoted to be the meanadjusted value of the log of the asset price, that is, X(t) is the deviation of log(s t ) from its expected value as of time 0. Over the next time step the underlying asset price is allowed to move to one of three values, designated as up(u), down(d), and middle(m). Associated with these branches are three risk-neutral probabilities, p u p d and p m. The middle node value is set to be no change and the up and down moves to be of equal magnitude. That is, over one time period, X goes to X h with probability p u, to X h with probability p d and remains unchanged with probability p m. Besides satisfying the three conditions about equating mean, variance and sum of probability to be one, this model can also set the kurtosis in the tree equal to that of the corresponding continuous one. We obtain the parameters as follows by solving the equations. 1 p u 6 p m 3 p 1 d 6 h 3 t (4.16) Setting the fine price steps to be h, therefore, the time steps to be t can guarantee that fine 4 tree nodes will overlap coarse-mesh nodes one step before expiration. We call the version with no fine tree Gao model, while the version with fine tree around the strike AMM in short. There are still some random tree-based models, such as random-time binomial model (Leisen,1999) and the random tree model proposed by Broadie-Glasserman(1996). By these models we can only obtain the scope of the option value and the numerical results depend greatly on the quality of the simulated random variables, we will not give a detail analysis on these models here. 16

24 Chapter 5 The Proposal of This Research 5.1 Exercise Boundary of American Put Option Since American put option can be exercised at any time before maturity, the holder must make a decision whether to exercise or not at any time. We define S (t) the optimal exercise price when the intrinsic value equals to the continuation value at time t. At each step t, when S (t) S (t), it is better not to exercise the option. When S (t) 8 S (t), it is better to exercise the option. The collection of these critical price for all times constitutes a curve, which is called optimal exercise boundary. As we have mentioned before, since we do not know the boundary as a priori, we find two neighboring nodes at each step t where the sign of S (t) S (t) changes. The critical price falls between the two nodes at that step. 5. Approximation Error Around the Exercise Boundary As we know, the main reason of the odd-even phenomena for pricing European put option by tree-based models is that there is nonlinearity error around the strike because the true option value does not change proportionally as the asset price changes between two nodes bracket the strike. As to the case of American put option, we must take approximation error around the exercising boundary into account as well.when holding value becomes larger than the intrinsic value, the payo function no longer follows the linearity property as it 17

25 does in the exercising region any more. Although the approximation error here is relatively smaller than that around the strike price, we know, since we calculate backwardly, it has greatly e ects on the nodes of the previous step. So does it until the initial node. In order to deal with this approximation error, it is reasonable to construct the fine tree just around this exercise boundary while not increase too much nodes as possible as we can. The problem is how to construct a proper structure to transmit the valuation information properly around the boundary between the fine tree and the coarse ones. 5.3 Our Proposal Let us look at Figure 5 1. When we try to construct fine tree from step N 1 to step N, we must calculate the values of seven nodes from C 1 to C 7. Here the values of nodes A 1 to A 7 can be calculated by the original coarse tree. Among them, nodes C 1 C C 3 are just below the boundary, which means, it is better not to exercise at these nodes. Meanwhile, it is easy to find that the values of C 4 C 5 C 6 can be calculated by taking advantage of the fine tree constructed from step N 1. The most di cult one is how to deal with the nodes such as C 7. If we try to construct fine tree from C 7 again, not only it makes the calculation more complex, but also in every step, since such kinds of nodes have relation with the nodes of next step that we have not obtained its value at that step, the mainstream of backward calculation is destroyed as well. Here we denote that C 7 go up to sub 15 with probability p 1, go up to sub 13 with probability p, go down to sub 11 with probability p 3 and go down to sub 9 with probability p 4 as it shows by the dotted line of Figure 5 1. By equating the mean and variance of the four-jump discrete model and standard continuous model, we can obtain : p 1 p p p 3 Thus the holding value of node C 7 can be obtained by: 3 48 (5.1) V cont e rt (p 1 P sub15 p P sub13 p 3 P sub11 p 4 P sub9 ) (5.) 18

26 Also we can take advantage of the position relation between the neighboring boundary points to specify some cases in order to give precise numerical results. When the critical point falls between the jth and j 1th node at step i, we use num[i] j to memorize the position of this critical stock point at i. Cases are classified as: num[i] num[i 1] 0, num[i] num[i 1] 1, num[i] num[i 1], otherwise. For example, when num[i] num[i 1] as shown in Figure 5 1, the values of C 4 C 5 and C 6 can be calculated by taking advantage of the fine tree indicated by the dash lines. While in the case of num[i] num[i 1] 1, see Figure 5, the value of C 3 C 4 and C 5 nodes can be valued properly by the fine tree as well. There is little possibility for the case num[i] num[i 1] 0 to happen since in generally the boundary has a decreasing property from the maturity. The purpose for us to classify the case according to the position relation between the neighboring boundary points is to give precise value for the nodes as possible as we can and avoid or lessen the number of nodes calculated by the way using fourjump process mentioned above since it does not accord with the mainstream of three-jump structure to some extent. In the original three-jump process, the third moment is fulfilled besides mean and variance, however, in the case of the above four-jump process, the third moment can not be satisfied from the discrete to the continuous mode. Although we try to build the fine tree along the boundary, it does not mean that we can build it till the initial point. In fact, the fine tree can only be constructed when 4 num[i] i 3, that is, there is some nodes below and over the critical points is a necessary condition. With respect to the number of nodes, we only increase less than 40N nodes, which is less than the number of nodes when the number of step is increased by 0 for trinomial models. Since the main idea of our method is to build fine tree around the boundary, we call it FTAB model in short here. 19

27 S S R R R R O Q Q R S S R R O BCE B0= BC> BC? BCD BC@ B/A Q 9;: 9;< K E Q K = 9;E 9= K > 9;> K? LNM 9;? K D 9;D K A 9;@ 9 A FHGJI A > FHGJI A? FHGJI A D FHGJI FHGJI A A FHGHI A P FHGHI : FHGJI FHGJI E < FHGJI = FHGHI > FHGJI? FHGHI D FHGJI A BCE B0= BC> BC? BCD BC@ B/A C 7 C 6 C 5 9;: 9V< 9VE 9W= 9V> 9V? 9VD 9V@ 9XA S* C 4 C 3 C C 1 FHGJI A > FHGJI A? FHGJI A D FHGJI FHGJI A A FHGJI A P FHGHI : FHGHI < FHGJI E FHGHI = FHGJI > FHGJI? FHGJI D FHGJI A Figure 5.1: case: num[i] T num[i T 1] U Figure 5.: case: num[i] T num[i T 1] U 1 0

28 Chapter 6 Numerical Comparison of Di erent Tree-based Models 6.1 A General Comparison of Tree-based models by RMS In this chapter, we make numerical comparison of tree-based models mentioned above. We fix the initial stock price as 100, and consider three cases of the strike price , 110, which correspond to out-of-the-money, at-the-money and in-the-money. Parameter on interest rate, volatility and maturity are also changed. The 54 combinations include strike price of and 110, maturities of years, volatilities of , and interest rate of Tables and 6 4 show us the root mean squared errors for the computed values calculated by cy program. We first calculate the option values numerically by using Gao model with 0000 time steps. We regard them as true values of the corresponding option prices. Then lattices with time steps are examined. The approximation error is measured by the relative root-mean-squared(rms) error: RMS 1 m m iz 1 e i where e i ( ˆP i P i ) P i are relative errors, P i is the true option value. ˆP i is the estimated option value. m is the number of parameter sets. 1

29 Table 6.1: RMSE of 54 parameter sets Model RMSE FTAB AMM BBS Boyle FB GAO CRR JR Tian SMO KR From Table 6 1, we observe that our proposal (FTAB) increases the accuracy of computation. We also observe that solutions of our method enjoy smooth property. Especially, it performs well in the case of out-of-the-money. For example, our method with 50 time steps is comparable in accuracy to CRR model with 150 time steps. Let N be the number of time steps. Binomial model has N 3N nodes in total while trinomial model has N N nodes. That of AMM model is N N 40 while in our model the total number is nearly N 4N 160. We choose to be the total number of nodes for each model and the corresponding number of step for binomial, trinomial, AMM and our FTAB is respectively. We use the above parameter sets again and obtain the RMSE as shown in Table 6 5. From Table 6 5, it is easy to find that our method is relatively a good one in the case of out-of-the-money with regard to the same number of calculation nodes when compared with other tree-based models. On the other hand, there seems to be no authority model with regard to the number of calculation nodes for all cases.

30 Table 6.: RMSE in the case of out-of-the-money Model RMSE FTAB AMM BBS Boyle FB GAO CRR JR Tian SMO KR Tests on SMO and FB models The computed option values by SMO and FB are equal to each other in the case of at-themoney. We can check it in Figure 6. Although the main idea of SMO and FB are similar, in the case of in-the-money, SMO model converges to the true value from the above, while FB model converges to the true value from below. FB model converges faster than SMO model, because the main idea of FB model is to adjust the value of the nearest point around the strike price at maturity to equal to the strike price and make the tilt parameter not so large as SMO model. On the other hand, although these two models enjoy the property of relatively smooth convergence, not the same as the case of European put options, the zigzag problem still exists as shown in Figures 6 4 and 6 5. Another problem about SMO model is that in the case of deep-in-the-money, when c N becomes negative, in order to maintain the condition that u 8 R 8 d, thus the inequality ( t) 1 log( K ) S(0) T must be fulfilled as well. Meanwhile, we point out that the variance of SMO model is not r 3

31 Table 6.3: RMSE in the case of in-the-money Model RMSE FTAB AMM BBS Boyle FB GAO CRR JR Tian SMO KR equal to that of continuous model. Number of steps to be even is also required. 6.3 Tests on CRR, JR and Tian models For European type option, convergence of price is ensured very easily from weak convergence of the process. Things are more complicated for the American put option, because in general convergence of maximum over expectations on functions of the process cannot be derived from weak convergence only. Thus, we can only say from the results of numerical tests, there is no obvious indication that one is better than the other two. Figures and 6 8 show the option valued calculated by these three models. BBS model is not good when the number of time steps is not large enough although it is relatively a better method than other binomial models in the case of European put options. The merit of BBS method is that using it we can avoid the problem of nonlinear error around the strike at the maturity, but in the case of American put option, the model has ignored that there is still possibility of exercising for some nodes from next-to-maturity step to maturity. 4

32 Table 6.4: RMSE in the case of at-the-money Model RMSE FTAB AMM BBS Boyle FB GAO CRR JR Tian SMO KR Table 6.5: RMSE with the same number of calculation nodes Model RMSE in the money at-the-money out-of-the-money FTAB AMM KR Boyle FB CRR Tian GAO SMO BBS

33 nml.55.5 SMO and FB models SMO and FB models SMO FB option value SMO FB option value * (N 1) refinement * (N 1) refinement Figure 6.1: SMO and FB Models: case of out-of-the-money option value SMO and FB models * (N 1) refinement SMO FB Figure 6.: case of at-the-money `_ b`c `_ b` `_ b[ c `_ `a bd c stu `a bd olpqr `a b_ c `a b_ `a ba c `a ba `a b^ c [ \ [ ] [ ^ [ ef ghi f jhf ik _ [ `[ [ `\ [ `] [ Figure 6.3: SMO and FB Models: case of in-the-money Figure 6.4: Zigzag Property of SMO Model 6.4 Tests on KR and Boyle models One of the features of the CRR approach is that by assuming that the stock price can move only either upward or downward, the relationship between the hedging argument and the mathematical development is especially clear. Their approach provides a powerful demonstration that options can be priced by discounting their expected values in a risk-neutral world. If we know the distributional assumptions and are assured that a risk-neutral valuation procedure is appropriate, then other types of discrete approximation can be used. We can regard the option valuation problem as a a problem in numerical analysis and replace the 6

34 ˆ { }x ~ y { }x ~ x { }x ~ w { }x ~ { }x z z Ž { }x z y Š Œ { }x z x { }x z w { }x z { }x z { }x y v w v x v ƒ y H v ƒ z v {v v {w v option value CRR, JR and Tian models CRR JR Tian * (N 1) refinement Figure 6.5: Zigzag Property of FB Model Figure 6.6: CRR, JR and Tian Models: case of outof-money continuous distribution of stock prices by a suitable discrete process as long as the discrete distribution tends to the appropriate limit. Thus, it is reasonable to take consideration on trinomial and even more branches tree-based models. Comparing with trinomial models, binomial models such as the CRR model lack flexibility in the sense that the jump size of the binomial tree is fixed for a given set of option parameters and time step. Trinomial trees, on the other hand, are more flexible because the jump size is only fixed up to an arbitrary parameter (called the dispersion parameter) 1. The advantage of the trinomial method over the binomial is that it provides another degree of freedom since the move spacing can be set independently of move timing. With respect to the dispersion parameter 1, from numerical results we know: 1 When 1 1, Boyle model reduces to the CRR model; When 1 1, KR model also reduces to a binomial scheme; can be decided case by case to obtain desirable value. 7

35 6.14 CRR, JR and Tian models 7.8 CRR, JR and Tian models 6.1 CRR JR Tian 7.75 CRR JR Tian option value option value * (N 1) refinement * (N 1) refinement Figure 6.7: CRR, JR and Tian Models: case of atthe-money Figure 6.8: CRR, JR and Tian Models: case of inthe-money 6.5 Tests on all three-jump process models The trinomial model put forward by Gao is di erent from the Boyle and KR models although all of these models allow the underlying asset price to move to one of three values, designated as up, down and middle. However, in Gao model it is not necessary to choose the value of 1 and the up and down moves are allowed to be of equal magnitude. Thus it is easy to construct fine tree to join with the coarse one. In his paper, Gao put forward the method to construct fine tree around the strike at the maturity to decrease the error when pricing American put option. We mention it as AMM method in this thesis. As mentioned in last chapter, the method put forward by this research is try to construct fine tree around the boundary because of the approximation error around the boundary. Based on the numerical results, we can find our method is relatively better than the AMM method with the same time steps. Especially when the number of steps is not large enough the improvement is obvious. Figures and 6 11 show that our FTAB model is relatively smooth among these five threejump process models. The option value of Figures and 6 11 are listed in Tables and 6 8. The parameters are S (0) 100 r T 1 and the strike price is and 110 respectively. True American put option values are , and in the corresponding case derived by using Gao model 0000 tree-step. Lattices with 10, 0, 40, 80, 160, 30, 640, 180, 560 time 8

36 steps are examined trinomial trees model Boyle KR GAO AMM 1 FTAB trinomial tree models 6.09 option value option value Boyle KR GAO AMM FTAB * (N 1) refinement * (N 1) refinement Figure 6.9: Five Trinomial models: case of outof-the-money Figure 6.10: Five Trinomial models: case of atthe-money 1 trinomial tree models Boyle KR GAO AMM FTAB option value * (N 1) refinement Figure 6.11: Five Trinomial models: case of inthe-money 9

37 Table 6.6: Option Value S (0) U 100 K U 90 r U 0 05 U 0 T U 1 N SMO FB CRR JR Tian BBS N Boyle trikr GAO AMM FTAB Table 6.7: Option Value S (0) U 100 K U 100 r U 0 05 U 0 T U 1 N SMO FB CRR JR Tian BBS N Boyle trikr GAO AMM FTAB

38 Table 6.8: Option Value S (0) U 100 K U 110 r U 0 05 U 0 T U 1 N SMO FB CRR JR Tian BBS N Boyle trikr GAO AMM FTAB