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Effect of basis functions in least-squares Monte Carlo (LSM) for pricing options

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Effect of basis functions in least-squares Monte Carlo (LSM) for pricing options

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Content EFFECT OF BASIS FUNCTIONS IN LEAST-SQUARES MONTE CARLO (LSM) FOR PRICING OPTIONS by Hao Wu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (APPLIED MATHEMATICS) April 2014 Copyright 2014 Hao Wu Dedication I dedicate this thesis to my advisor Prof. Jianfeng Zhang, for his inspiration and patience. To my thesis committee members Prof. Jin Ma and Prof. Robert Sacker, for their support and presence. To all my friends for their encouragement and help. Hao Wu ii Contents Dedication ii List of Figures v Abstract vi 1 Introduction 1 1.1 European options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Least-Squares Monte Carlo (LSM) . . . . . . . . . . . . . . . . . . . 8 2 Pricing American options using LSM 9 2.1 Pricing American put options on one share of stock . . . . . . . . . 9 2.2 Pricing American put options in multidimensional models . . . . . . 12 2.2.1 High-dimensional underlying stocks . . . . . . . . . . . . . . 13 2.2.2 Pricing American put options with 5-dimensional stocks . . 17 2.2.3 Pricing American put options with 10-dimensional stocks . . 19 2.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Pricing 10-D European options using LSM 22 3.1 LSM combined with Black-Scholes PDE . . . . . . . . . . . . . . . 22 3.1.1 Flexible PDE method to obtain the explicit solution . . . . . 22 3.1.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 24 3.2 LSM combined with modified price PDE . . . . . . . . . . . . . . . 28 3.2.1 Modified PDE method to obtain the explicit solution . . . . 28 3.2.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 29 4 Conclusions 31 Reference List 33 iii A Pricing American options on one share of stock by LSM 34 A.1 MyAmericanLSM_test_ex1 . . . . . . . . . . . . . . . . . . . . . 34 A.2 MyAmericanLSM_test_ex2 . . . . . . . . . . . . . . . . . . . . . 37 B Pricing 10-D European options by LSM combined with PDE 40 iv List of Figures 1.1 European call option . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 European put option . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Pricing American put options on one share of stock . . . . . . . . . 10 2.2 Standard error of pricing 5-D American put options . . . . . . . . . 17 2.3 Mean of pricing 5-D American put options . . . . . . . . . . . . . . 18 2.4 Standard error of pricing 10-D American put options . . . . . . . . 19 2.5 Mean of pricing 10-D American put options . . . . . . . . . . . . . 20 3.1 Numerical results by using different basis functions . . . . . . . . . 26 3.2 Option price by using different basis functions . . . . . . . . . . . . 26 3.3 Standard error by using different basis functions . . . . . . . . . . . 27 3.4 Numerical results by using different basis functions . . . . . . . . . 30 3.5 Option price by using different basis functions . . . . . . . . . . . . 30 v Abstract In modern financial world, it is one of the most challenging problems to valuate American-style options. Finite difference methods could be used only if the dimen- sions of derivatives are no more than three. In order to overcome the restriction, a simple and powerful approach, known as Least-Squares Monte Carlo (LSM), appeared in our sight, which was firstly proposed by Longstaff & Schwartz (2001). This approach is really easy to implement, because only simple least-squares is essentially required. Besides, it could be widely applied to more complex and gen- eral options, and LSM has its advantage of dimensional insensitiveness. Nowadays,theLeast-SquaresMonteCarlo(LSM)approachhasdefinitelybecome a powerful method for pricing options. This backward method considerably sup- ports the growing interest in financial models that involve multiple assets, in these situations, traditional finite difference method fades certainly. However, one would face an important and hard choice in the step of regression by using Least-Squares Monte Carlo (LSM), which in fact, is the choice of basis functions. This paper aims at the effect of different basis functions in high-dimensional cases for pricing American and European options. Particularly, for European options, a flexible method with PDE is applied to better analyze the accuracy vi of numerical results depending on explicit solutions. Hopefully and believably, the statements in this paper would provide guiding significance. vii Chapter 1 Introduction 1.1 European options A European option is a kind of security that gives its holders the right to sell or buy the underlying asset (stock, index, equity) only on the maturity date with a certain price. In reality, lots of index and equity options in the U.S. belong to European styles, see Björk (2004) for details. Consider the Black-Scholes model consisting of one risk-free asset B and one risk asset S. The P-dynamics (P is a probability measure) are given by:      dB(t) =rB(t)dt dS(t) =aS(t)dt +σS(t)dW ∗ (t) (1.1) where r: risk-free interest rate σ: volatility a: local mean rate They are all deterministic constants. W ∗ is a Brownian motion on the probability space (Ω,F,P). 1 Let S be a price process of a stock, a simple contingent claimX with expiration date T is a stochastic variable which satisfies: X∈F T , X = Φ(S T ) where Φ is the contract function. European options consist of European call and put options. An European put option is a simple contingent claim with the contract function Φ(x) = (x−K) + =max[x−K, 0] We can directly see the contract in Figure 1.1 (Strike price K=5). Similarly, an European put option can also be seen as a simple contingent claim with the contract function Φ(x) = (K−x) + =max[K−x, 0] We can directly see the contract in Figure 1.2 (Strike price K=5). Let Π(t;X ) be the price process of the contingent claimX, then for European put options, Π(T ;X ) = Φ(S T ) =max[K−x, 0] More generally, Π(T ;X ) =X 2 Figure 1.1: European call option Now we define another probability measure Q such that dS t =rS t dt +σS(t)dW t were W is a Q-Brownian motion. In the new model, the volatility σ remains the same and the local mean rate a disappears. 3 Figure 1.2: European put option Assume the market is free of arbitrage, then under Q-measure, we have the Risk-Neutral valuation: F (t,s) =e −r(T−t) E t,s [X ] =e −r(T−t) E t,s [Φ(S T )] where Π(t,X ) =F (t,S t ) and S t =s Actually, we can obtain the explicit solution ofS T according to the following SDE:      dS u =rS u du +σS u dW u S t =s (1.2) 4 Thus, S T =se (r− 1 2 σ 2 )(T−t)+σ(W T −Wt) were z = (r− 1 2 σ 2 )(T−t) +σ(W T −W t ) has normal distribution with Mean = (r− 1 2 σ 2 )(T−t), StandardDeviation =σ √ T−t BasedontheRisk-NeutralvaluationandtheexactformofS T , thereisafamous result called Black-Scholes formula for an European call, see Davis (2010): C K (t,s) =F (t,s) =sN[d1]−e −r(T−t) KN[d2] where d1 = ln( s K ) + (r + 1 2 σ 2 ) σ √ T−t d2 =d1−σ √ T−t N(x) is the probability density function of the standard normal distribution. On the other hand, for a European put option, the priceP K (t,s) could be directly calculated as the Put-Call parity: P K (t,s) =C K (t,s) +Ke −r(T−t) −s where both the Call and Put has the same expiration date T and strike price K. Since one can compute C K (t,s) by Black-Scholes formula, one can compute P K (t,s) as well by Put-Call parity. 5 1.2 American options American-style options are different from European-style options in the sense that the holders have the right to exercise (sell or buy) the underlying asset on any date prior to (and including) the maturity date. Assume 0 = t = t 0 < t 1 < ... < t N = T be any partition of the time interval [0,T ], where T is the date of expiration. American call options give the owner the right to buy a share of underlying asset at any time stept i , wherei = 0, 1, 2,...,N. American put options can be defined naturally, see Björk (2004) for details. In modern financial market, numerous stock, equity and index options in the U.S. have both European-feature and American-feature, however, some only belong to one style. There is no doubt that valuating American options have sparked our attentions, especially to high dimensional cases. Similar to European options, American options could be viewed as contingent claims that are totally defined by the underlying asset. Recall that the price process satisfies the terminal conditions: Π(T ;X ) =X = Φ(S T ) and the Risk-Neutral valuation still holds, F (t,s) =e −r(T−t) E t,s [X ] =e −r(T−t) E t,s [Φ(S T )] 6 where Π(t,X ) =F (t,S t ) and S t =s In sprite of Black-Scholes formula, one could solve the price of American put options through the Black-Scholes PDE:      α t F +rxα x F + 1 2 σ 2 x 2 α xx F−rF = 0 F (T,x) = Φ(x) = (K−x) + (1.3) It is feasible to apply many methods to compute the solution F (t,x), such as Fourier transform approach Chiarella & Ziogas (2006), or finite difference techniques Wu & Kwok (1997). But lots of methods fade when the dimensions are higher than three. That’s why we need Least-Squares Monte Carlo (LSM) which is insensitive of dimensions. This popular method is going to be demonstrated in the next subsection. 7 1.3 Least-Squares Monte Carlo (LSM) To briefly review the LSM approach, it focus on the approximating of the conditional expectation function by least squares regressions. We simulate desired number of sample paths at each time step. Further, the holders compare the expected payoff from continuation with the payoff from immediate exercise, then a decision is made at each possible exercise step. Specifically, if the continued pay-off is higher, the holders may keep the option alive, otherwise, the option will be exercised immediately. Least-Squares Monte Carlo (LSM) is a backward method. Applying the LSM algorithm starting from the maturity date, the estimated value of the option at time zero following each path is obtained. In the end, the averaging value among all the paths would produce the option price. In fact, the expected pay-off for continuation is conditional on the information at the current time step. Longstaff & Schwartz (2001) estimated the conditional expectations through least-squares regression of discounted optimal payoffs from the next step, on a set of basis functions towards underlying assets prices. The details of the algorithm can be found in Longstaff & Schwartz (2001). For its convergence, see Stentoft (2004). The original method has been improved by Areal et al. (2008). 8 Chapter 2 Pricing American options using LSM 2.1 Pricing American put options on one share of stock Firstly, all examples in Page 115-120, 127 from Longstaff & Schwartz (2001) have been simulated using the Matlab code in this paper, see Appendix MyAmericanLSM_test_ex1 and MyAmericanLSM_test_ex2. One can check that the numerical results are consistent. Nowadays, the simulation technique is so popular that it has drawn great attentions, one can see Coskan (2008), Larsson (2007), etc. There are several sets of basis functions in this paper: 1) 1,x,x 2 ,x 3 ,....,x n 2) Hermite polynomials: H(1),H(2),.....,H(n) 3) Laguerre polynomials 4) Chebyshev polynomials 5) Legendre polynomials Remark: In the following contents, the order n stands for the highest degree in 9 Figure 2.1: Pricing American put options on one share of stock the set of basis functions used in Least-Squares Monte Carlo (LSM). For parameters t = 0, T = 1, r = 0.06, v = 0.2, S0 = 40, K = 40, N exercise = 50, N simu = 100000, the numerical results are shown in Figure 2.1. 10 In order to avoid rank deficiency issues, I normalized the stock price and used QR decomposition. For each type of basis functions, I computed the value of American put option by LSM, as well as the residuals generated in regressions. The numerical method is presented in Brandimarte (2013). Using each set of basis functions, I found that the first three orders were sufficient to generate a stable value of American put option. This conclusion is consistent with many papers including Longstaff & Schwartz (2001), in which the authors only used 1,x,x 2 . The change of value was negligible with an increasing number of orders in each set. It is naturally to accept because there is only one asset in market. However, the most challenging issue is the high dimensions. This paper will indicates this issue later. 11 2.2 Pricing American put options in multidi- mensional models There are many ways to manipulate N stocks, such as taking the maximum, weighted sum, geometric mean and so on. In this paper, I used the last measure- ment. As for the error analysis, one may concentrate on the basis functions used in linear regressions for approximating conditional expectations. However, we need to know the exact solutions. For low dimensional cases, we are able to calculate the explicit solution through finite difference techniques. For example, in 2-D, we could use Broadie & Glasserman (1997), Broadie & Glasserman (1997). For stochastic mesh method, see Broadie & Glasserman (1998). However, for high-dimensional cases, such as 5-D or 10-D, we are not able to know the explicit solutions, neither can we compute the absolute and relative errors. In this paper, I desired to used two measures of the "best" choice of basis functions. The first one is the mean of option price, and the other one is the standard error (se). We note that for normally distributed variables, the 95% confident interval could be calculated as (mean−se× 1.95, mean +se× 1.95). In practise, a trade-off exists, since we expect higher mean and narrower interval (lower standard error). This problem would be explained based on numerical experiments later. 12 2.2.1 High-dimensional underlying stocks Nowweassumetherearenriskyassets(stocks)inmarketwhosepriceprocesses are S 1 (t),S 2 (t),...,S n (t). Let S(t) denote the entire price in a form of a column vector S(t) =        S 1 (t) . . . S n (t)        (2.1) We further assume W ∗ 1 ,...,W ∗ n are n independent Wiener process on the probabil- ity space (Ω,F,P). The P-dynamics of the assets are given by        dB(t) =rB(t)dt dS i (t) =a i S i (t)dt +S i (t) n P k=1 σ ik dW ∗ k (t), i = 1, 2,...,n (2.2) where r: risk-free interest rate σ = (σ ik ): volatility matrix a: local mean rate of return of S We are willing to represent the dynamics in the matrix form. Let W ∗ (t) denote the n Wiener processes such that W ∗ (t) =        W ∗ 1 (t) . . . W ∗ n (t)        (2.3) 13 and let a denote the column vector a =        a 1 . . . a n        (2.4) Let D[s] denote the diagonal matrix for a vector s = (s 1 ,s 2 ,...,s n ) such that D[s] =            s 1 0 ··· 0 0 s 2 ··· 0 . . . . . . . . . . . . 0 0 ··· s n            (2.5) Finally, we have the P-dynamics given by Björk (2004) dS(t) =D[S(t)]adt +D[S(t)]σdW ∗ (t) Furthermore, we assume as usual, T is the expiration date, and the contingent claimX = Φ(S(T )) has the price process Π(t;X ) = F (t,S(t)). Recall the Risk- Neural valuation, we have Q-dynamics dS(t) =D[S(t)]r1 n dt +D[S(t)]σdW (t) where 1 n =        1 . . . 1        (2.6) 14 is an n dimensional column vector, W 1 ,...,W n are n independent Q-Wiener processes. The price function F (t,s) could be represented as F (t,s) =e −r(T−t) E t,s [Φ(S(T ))] and F (t,x) is the solution of the following PDE      α t F + (r1 n , ΔF ) + 1 2 trace{σ 0 D[x]α xx FD[x]σ}−rF = 0 F (T,x) = Φ(x) (2.7) where ΔF =        dF dx 1 . . . dF dxn        (2.8) α xx F =            d 2 F dx 1 dx 1 ··· d 2 F dx 1 dxn d 2 F dx 2 dx 1 ··· d 2 F dx 2 dxn . . . . . . . . . d 2 F dxndx 1 ··· d 2 F dxndxn            (2.9) The former statements are under the assumption that W 1 ,...,W n are indepen- dent, i.e. d<W i ,W k > t =      dt, for i =k 0, for i6=k (2.10) 15 However, it is possible that they are not independent, a general representation is given by d<W i ,W k > t =      dt, for i =k ρ ik dt, for i6=k (2.11) where ρ ik is the correlation coefficients. This paper will use the general formula in pricing high-dimensional options, the detailed indications will show later. 16 Figure 2.2: Standard error of pricing 5-D American put options 2.2.2 Pricing American put options with 5-dimensional stocks For 5-dimensional American options, I simulated using a set of Hermite poly- nomials with the highest degree from 3 to 15. The numerical results are presented in Figure 2.2 and Figure 2.3. 17 Figure 2.3: Mean of pricing 5-D American put options 18 Figure 2.4: Standard error of pricing 10-D American put options 2.2.3 Pricing American put options with 10-dimensional stocks For 10-dimensional cases, I used Hermite, Laguerre and Chebyshev to generate regression matrices with the highest degree from 10 to 20. It was interesting that the numerical results (in Figure 2.4, Figure 2.5) for these three choices of basis functions were the same! 19 Figure 2.5: Mean of pricing 10-D American put options 20 2.2.4 Conclusion To find the ”best” choice of basis functions, there was a trade-off between the higher price and the narrower confident interval (i.e. the lower standard error). It might be reasonable to focus on the later measure if we were supposed to compute the most stable solution. However, if we were willing to see the highest price possible, then the previous measure would be more important. 21 Chapter 3 Pricing 10-D European options using LSM 3.1 LSM combined with Black-Scholes PDE 3.1.1 Flexible PDE method to obtain the explicit solution LetY t(i) denote the price of an European put option at time t(i), based on the risk-neutral valuation, Y t(i) =E t(i) [e (−rΔt) Y t(i+1) ] where Δt = T N excercise Let Y t =u(t,S t ), then the price function u(t,x) is the solution of the PDE (in 1-D):      α t u +rxα x u + 1 2 σ 2 x 2 α xx u−ru = 0 u(T,x) = (K−x) + (3.1) In order to compare the numerical results by using Least-Squares Monte Carlo (LSM) with the explicit solution, one can firstly choose a smooth function u(t,x), then a function f(t,x) would be generated based on the following PDE (1-D):      α t u +rxα x u + 1 2 σ 2 x 2 α xx u−ru =f(t,x) u(T,x) =g(x) (3.2) 22 As for high-dimensional cases, assume      dS i t =S i t [rdt +σ i dw i t ], i = 1, 2,...,n d(w i t ,w j t ) =ρ ij dt, i,j = 1, 2,...,n (3.3) then the general PDE becomes:      α t u +r(x, Δu) + 1 2 trace{σD[x]D 2 [u]D[x]}−ru =f(t,x) u(T,x) =g(x) (3.4) where (, ) is the inner product, x = (x1,x2,x3,...,xn) T is an n dimensional column vector, Δu = (α x1 u,α x2 u,...,α xn u) T , D[x] = diag(x1,x2,...,xn), (D 2 [u]) ij = α 2 u αxiαxj , (σ) ij =ρ ij σ i σ j After generating the function f(t,x), we can apply Least-Squares Monte Carlo (LSM) to estimate the conditional expectation: Y t(i) =E t(i) [e −rΔt Y t(i+1) − Δtf(t(i),S t(i) )] (3.5) Since European options can only be exercised at the maturity date, it is not required to compare payoffs from continuation and payoffs from immediate exercise at each time step. Thus the LSM algorithm could be simplified. Following the backward algorithm, one can compute the price of European option Y t(0) based on different choices of basis functions. Finally, we are able to analyze the effect of basis functions depending on the "explicit" solution u(t(0),S0), that is the magic of combined mathematical tools. 23 3.1.2 Numerical experiments In this section, the parameters are set as follows: S(0) = 1, t(0) = 0, T = 1, K = 5, r = 0.06, σ i = 0.2, ρ ij = 0.25, N exercise = 50, N simu = 1000. Besides, the price function is chosen as u(t,x) = (1 +t)x 1 x 2 ...x n , thus, the exact solution should be u(0,S(0)) = 1. First of all, to test the validness of the Matlab code for this section, we can estimate the price of European options by Y t(i) =E t(i) [e −rΔt Y t(i+1) ] As a result, Y t(0) = 0.8788 and the absolute error is 0.0002 which is sufficiently small. On the other hand, one can calculate directly by Y t(0) =e −rT E t(0) [e −rT max(K− (S 1 T S 2 T ...S 10 T ) 1 10 , 0)] then Y t(0) = 0.8786 which is almost the same with the result by LSM. Moreover, in terms of the Equation(3.5), this paper used three choices of basis functions : 1) Hermite/chebyshelv/Laguerre polynomials: p(1),p(2),...,p(n) 2) 1,x,x 2 ,x 3 ,...,x n 3) T 1 =α t u, T 2 =r(x, Δu), T 3 = 1 2 trace{σD[x]D 2 [u]D[x]}, T 4 =ru 24 Remark: Through numerical computations, it was found that T 1 = c1× T 3 = c2× T 4, where c1 and c2 are constants. Thus, in order to avoid singular regression matrices, I only chose T 1 and T 2 as the third set of basis functions. The numerical results are shown in Figure 3.1, Figure 3.2, Figure 3.3. These figures indicate that with the growth of the highest degree of basis functions, the price of option firstly keeps stable during n=3 to 7, and then jumps to a high level which is quite close to 1. At the same time, the absolute error stays high at the beginning, and then sharply falls down after n=8. Intuitively, one can understand the indications. Recall that f(t,x) is generated by derivatives of u(t,x), and u(t,x) = (1 +t)x 1 x 2 ...x n so x 10 and x 9 are involved in the expression of f(t,x). Thus, n=9, 10 of basis functions are able to produce better regressions. In fact, when I just used the derivatives of u(t,x) as the basis functions, the option price is much more close to u(0,S(0)) = 1. 25 Figure 3.1: Numerical results by using different basis functions Figure 3.2: Option price by using different basis functions 26 Figure 3.3: Standard error by using different basis functions 27 3.2 LSM combined with modified price PDE 3.2.1 Modified PDE method to obtain the explicit solution Followingtheformermethod, thereisamodifiedPDEtogeneratef 0 (t,x)based on u(t,x) and f(t,x,u), the two functions are very smooth:      α t u + 1 2 trace{D 2 [u]} +f(t,x,u) =f 0 (t,x) u(T,x) =g(x) (3.6) We are willing to take advantage of Least-Squares Monte Carlo (LSM) to estimate Y t(i) =E t(i) [Y t(i+1) + Δtf(t(i),x t(i) ,Y t(i+1) )]− Δtf 0 (t(i),x t(i) ) where Y t(i) =u(t(i),S t(i) ). Ideally, Y t(0) ≈u(t(0),S t(0) ). 28 3.2.2 Numerical experiments In this paper, the flexible functions f and u are chosen as u(t,x) = (1 +t)x 1 x 2 ...x 10 f(t,x,u) = q |t∗ (x 1 x 2 ...x 10 ) 1 10 ∗u| The choices of basis functions are as follows: 1) Hermite/Chbyshelv/Laguerre polynomials: p(1), p(2), ... ,p(n) 2) 1,x,x 2 ,...,x n 3) α t u, f(t,x, 1) Remark: α xx u = 0, so it is not included in 3). The numerical results are presented in Figure 3.4 and Figure 3.5. In Figure 3.5, the comparison with the first method demonstrates that when construction functions are not able to be expressed by a linear combinations of polynomials, the estimated price by LSM has considerable difference from the exact solution. However, the first method could provide relative good values although using a set of polynomials 1) and 2) as the basis functions. 29 Figure 3.4: Numerical results by using different basis functions Figure 3.5: Option price by using different basis functions 30 Chapter 4 Conclusions This paper studied the effect of basis functions for pricing European and American options by using Least-Squares Monte Carlo (LSM). Since LSM focus on linear regressions to estimate the conditional expectations, then the choices of basis functions play an important role to compute more explicit solutions. For a market consisting of one underlying stock, one can just use 1,x,x 2 as the basis functions or a set of Hermit/Chebyshev/Laguerre polynomials with the highest degree three, they can already produce good numerical values. However, the more difficult and challenging problems exist in high-dimensional cases. Although the Least-Squares Monte Carlo (LSM) has the advantage of dimensional insensitivity, its hard to choose the basis functions when there are lots of assets in market. Additionally, we don’t know the explicit solutions, thus it is undirect to analyze the numerical results. In this paper, two measurements are applied: the mean of price and the standard error. In practise, there is a trade-off between the two measurements. In addition, for high-dimensional European options, one can not compute the exact price according to the Black-Scholes formula, neither by Black-Scholes PDE since the finite difference technique could only work for low dimensional cases. In this paper, two modified PDE formulas of the price function are introduced 31 so that we are able to grasp the explicit solutions and better study the effect of basis functions. As a result, we can use a set of Hermit/Chebyshev/Laguerre polynomials to compute good solutions for the first modified formula, however, for the second PDE formula, we are not able to obtain ideal solutions by using theses polynomials unless taking advantages of the derivatives of the construction functions. 32 Reference List Areal N, Rodrigues A, Armada MR (2008) On improving the least squares monte carlo option valuation method. Review of Derivatives Research 11:119–151. Björk T (2004) Arbitrage theory in continuous time Oxford university press. Brandimarte P (2013) Numerical methods in finance and economics: a MATLAB- based introduction John Wiley & Sons. BroadieM,GlassermanP(1997) Pricingamerican-stylesecuritiesusingsimulation. Journal of Economic Dynamics and Control 21:1323–1352. Broadie MN, Glasserman P (1998) A stochastic mesh method for pricing high- dimensional american options . Chiarella C, Ziogas A (2006) Pricing american options under stochastic volatil- ity. School of Finance and Economics, University of Technology, Sydney, AUS, forthcoming . Coskan C (2008) Pricing American Options by Simulation Ph.D. diss., Bogaziçi University. Davis MH (2010) Black–scholes formula. Encyclopedia of Quantitative Finance . Larsson K (2007) Pricing american options using simulation . Longstaff FA, Schwartz ES (2001) Valuing american options by simulation: A simple least-squares approach. Review of Financial studies 14:113–147. Stentoft L (2004) Convergence of the least squares monte carlo approach to amer- ican option valuation. Management Science 50:1193–1203. Wu L, Kwok YK (1997) A front-fixing finite difference method for the valuation of american options. Journal of Financial Engineering 6:83–97. 33 Appendix A Pricing American options on one share of stock by LSM A.1 MyAmericanLSM_test_ex1 %% Valuating American put option on one share of stock %% in order to test my coding %% using examples at Page 115−120 of Longstaff and Schwart ’ s paper %% numerical results are the same as that in the paper %%name: Hao Wu %%date : Feb.2014 clear all ; T=3;%maturity date N=3; %number of time steps M=8; %number of simulated paths K=1.1;%strike price r=0.06;%risk−free interest rate 34 dt=T/N;%length of the small time interval S0=1; %Initial stock price S=ones(M,N+1);%Stock Price Matrix %Stock price paths in the paper S=[1 1.09 1.08 1.34; 1 1.16 1.26 1.54; 1 1.22 1.07 1.03; 1 0.93 0.97 0.92; 1 1.11 1.56 1.52; 1 0.76 0.77 0.9; 1 0.92 0.84 1.01; 1 0.88 1.22 1.34; ]; CF=zeros(M,N+1);%cash flow matrix CF(: ,N+1)=max((K−S(: ,N+1)) ,0) ;%if exercise at maturity date for i=N:−1:2 Idx=find(S(: , i )<K) ; %Find paths which are in the money X=S(Idx , i ) ; %Regression matrices used in the paper Reg=[ones(size(X)) X X.^2]; 35 Y=CF(Idx , i+1)∗exp(−r∗dt) ;%discounted value from continuation E=max((K−S(Idx , i )) ,0) ;%values from immediate exercise %Linear regression coeff=Reg\Y; C=Reg∗coeff ; %This is improved by QR decomposition later %compare values from immediate exercise and continuation(regression) %make a decision Jdx=E > C; nIdx=setdiff ((1:M) ’ ,Idx(Jdx)) ; CF(Idx(Jdx) , i )=E(Jdx) ; %Since we need to discount all following cash flows %To make our calculation easier CF(nIdx , i )=exp(−r∗dt)∗CF(nIdx , i+1); end value=mean(CF(: ,2) )∗exp(−r∗dt) 36 A.2 MyAmericanLSM_test_ex2 %%Valuating American Put Option on a share of stock %%in order to test my coding %%using examples on page 127 of Longstaff and Schwart ’s paper %%numerical results are the same as that in the table on Page 127 of this paper %%name: Hao Wu %%date : Feb.2014 clear all ; T=1;%maturity date N=50; %number of time steps M=100000; %number of simulated paths dt=T/N;%length of the small time interval K=40;%strike price r=0.06;%risk−free interest rate sigma=0.2;%volatility S0=40; %Initial stock price S=ones(M,N+1);%Stock Price Matrix %Generate Stock Price Matrix R = exp((r−sigma^2/2)∗dt+sigma∗sqrt(dt)∗randn(N,M)) ; 37 SS = cumprod([S0∗ones(1 ,M) ; R]) ; S=SS ’; CF=zeros(M,N+1);%cash flow matrix CF(: ,N+1)=max((K−S(: ,N+1)) ,0) ;%exercise at maturity data for i=N:−1:2 Idx=find(S(: , i )<K) ; %Find paths which are in the money at time i X=S(Idx , i ) ; %Regression matrices used in the paper Reg=[ones(size(X)) X X.^2]; Y=CF(Idx , i+1)∗exp(−r∗dt) ;%discounted value from continuation E=max((K−S(Idx , i )) ,0) ;%values from immediate exercise %Linear regression coeff=Reg\Y; C=Reg∗coeff ; % This is improved by QR decomposition later 38 %compare values from immediate exercise and continuation(regression) %make a decision Jdx=E > C; nIdx=setdiff ((1:M) ’ ,Idx(Jdx)) ; CF(Idx(Jdx) , i )=E(Jdx) ; %Since we need to discount all following cash flows %To make our calculation easier CF(nIdx , i )=exp(−r∗dt)∗CF(nIdx , i+1); end value=mean(CF(: ,2) )∗exp(−r∗dt) 39 Appendix B Pricing 10-D European options by LSM combined with PDE % High dimensional European put options using LSM %Geometric average %Name: Hao Wu %Date:3−19−2014 %Topic: choice of basis functions clear all ; N_dim = 10; % Dimensions of assets (stocks) N_simu = 10000; % number of simulated paths N_exercise = 50; % number of time steps randn( ’ state ’ ,0) ; %parameters setting S0 = 1; % initial price rho=0.25;%correlation coefficient v=0.2; % volatility sigma = rho ∗v.^2∗ ones(N_dim,N_dim) ; % variance matrix for i = 1:N_dim sigma(i , i ) = v.^2; 40 end r = 0.06; % risk−free interest rate T = 1; % maturity date t = 0; dt = (T−t)/N_exercise ; % length of the small time interval K = 5; % strike price %discount factor discount=exp(−r∗dt)∗ones(N_simu,1) ; % regression parameters parameter = zeros(4 ,1) ; % Stock Price Stock = Func(N_dim, N_exercise ,S0,r ,sigma ,T,N_simu) ; Stock (: ,1 ,:) = []; %Cash flow CF = max(0 , X− prod(Stock (: , N_exercise ,:) ,3) .^(1/N_dim)) ; %to check the validness of the code original=mean(exp(−r∗T)∗CF) ; % 0.8786 % European options can only be exercised on the maturity date Exercise = N_exercise∗ones(N_simu,1) ; for step = N_exercise−1:−1:1 41 XData = prod(Stock (: , step ,:) ,3) .^(1/N_dim) ; % reg1=zeros(N_simu,1) ; % reg2=zeros(N_simu,1) ; % % for k=1:N_simu % reg1(k)=REG1(dt∗step , Stock(k, step ,:) ); % reg2(k)=REGnl(dt∗step , Stock(k, step ,:) ); % % % end % RegressionMatrix= [reg1 , reg2 ]; %Regression matrix RegressionMatrix = [ ones(length(XData) ,1) , XData, XData.^2 , XData.^3 , XData.^4 , XData.^5 , XData.^6 , XData.^7 , XData .^8 ,XData.^9 , XData.^10]; %basis functions : polynomials %Reg=Reggen(11,XData); %hermit 0.3662 error 0.0110 %chebyshev %Laguerre %RegressionMatrix=Reg; %nonlinear term nlterm=zeros(N_simu,1) ; 42 addterm=zeros(N_simu,1) ; for m=1:N_simu addterm(m)=ADDTERM2(dt∗step , Stock(m, step ,:) ) ; nlterm(m)=nonlinear(dt∗step ,CF(m) ,Stock(m, step ,:) ) ; end YData = CF +dt∗nlterm − dt∗addterm; %without addterm 0.8788 %parameter = RegressionMatrix\ YData;%backslash %To avoid singular regression matrices , using QR decomposition Least Squares [Q, R] = qr(RegressionMatrix ,0) ; parameter=R\(Q’∗YData) ; Continuation = RegressionMatrix∗parameter ; CF=Continuation ; end price= mean(CF.∗ discount) error= 1.96∗std(CF.∗ discount) / sqrt(N_simu) 43

Abstract (if available)

Abstract In modern financial world, it is one of the most challenging problems to valuate American-style options. Finite difference methods could be used only if the dimensions of derivatives are no more than three. In order to overcome the restriction, a simple and powerful approach, known as Least‐Squares Monte Carlo (LSM), appeared in our sight, which was firstly proposed by Longstaff & Schwartz (2001). This approach is really easy to implement, because only simple least‐squares is essentially required. Besides, it could be widely applied to more complex and general options, and LSM has its advantage of dimensional insensitiveness. ❧ Nowadays, the Least‐Squares Monte Carlo (LSM) approach has definitely become a powerful method for pricing options. This backward method considerably supports the growing interest in financial models that involve multiple assets, in these situations, traditional finite difference method fades certainly. However, one would face an important and hard choice in the step of regression by using Least‐Squares Monte Carlo (LSM), which in fact, is the choice of basis functions. ❧ This paper aims at the effect of different basis functions in high‐dimensional cases for pricing American and European options. Particularly, for European options, a flexible method with PDE is applied to better analyze the accuracy of numerical results depending on explicit solutions. Hopefully and believably, the statements in this paper would provide guiding significance.

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Creator Wu, Hao (author)

Core Title Effect of basis functions in least-squares Monte Carlo (LSM) for pricing options

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 04/17/2014

Defense Date 03/20/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag American options,basis functions,European options,least‐squares Monte Carlo (LSM),OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Zhang, Jianfeng (committee chair), Ma, Jin (committee member), Sacker, Robert (committee member)

Creator Email hwu409@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-379998

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