Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 165259, 6 pages http://dx.doi.org/10.1155/2014/165259 Research Article Lattice Methods for Pricing American Strangles with Two-Dimensional Stochastic Volatility Models Xuemei Gao,1 Dongya Deng,2 and Yue Shan2 1 School of Economic Mathematics and School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, China 2 School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, China Correspondence should be addressed to Dongya Deng; [email protected] Received 22 January 2014; Accepted 5 April 2014; Published 28 April 2014 Academic Editor: Chuangxia Huang Copyright © 2014 Xuemei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this paper is to extend the lattice method proposed by Ritchken and Trevor (1999) for pricing American options with one-dimensional stochastic volatility models to the two-dimensional cases with strangle payoff. This proposed method is compared with the least square Monte-Carlo method via numerical examples. 1. Introduction In this paper we give an attempt to this challenging topic by studying an American style option with strangle payoff, Calculating American style options under geometric Brown- which was previously investigated by Chiarella and Ziogas ian motion is far from the realistic financial market. It is more [11]andMoraux[12] for single asset and constant volatility. valuable to price American style options under stochastic We develop the lattice methods of Ritchken and Trevor [2]to models. In general the valuation of American options with the American strangle options with many underlying assets stochastic volatility models has no closed-form solution and multidimensional stochastic volatility GARCH models. except very few cases (see, e.g., Heston [1]). Therefore numer- We compare the lattice methods with the least square Monte- ical methods or simulation methods are developed to price Carlo methods via several numerical examples. financial derivatives with stochastic volatility, among which the lattice methods receive much more attention. Ritchken and Trevor [2] proposed an efficient lattice method for pricing 2. Two-Dimensional Stochastic Volatility American options under GARCH models. Later the idea was Models of American Strangles further developed and applied by several papers, for example, Assume that the prices of two-dimensional assets St = Cakici and Topyan [3]andWu[4], and recently the conver- (1,2) gence of the method was proved by Akyildirim et al. [5]. follow a two-dimensional GARCH model (see, e.g., Duan [13] for more explanation of the one-dimensional All the abovementioned references focused on the devel- GARCH model). Consider opment of lattice methods for pricing American options with 1 oneunderlyingassetandsinglestochasticvolatilitymodel.To +1 √ √ ln ( )= − + ℎ − ℎ + ℎ ] , the best of our knowledge, there are no papers studying the 2 +1 (1) lattice methods for options with many underlying assets and 2 multidimensional stochastic volatility models. Indeed there ℎ+1 =0 +1ℎ +2ℎ(]+1 −) , are many papers in developing lattice methods for pricing options with many underlying assets, for example, Boyle [6], with =1,2,where is the price of the th asset correspond- Boyle et al. [7], Chen et al. [8], Gamba and Trigeorgis [9], and ingtothestandardBrownmotion, isthedividendratefor Moon et al. [10].Howeveritisnotseenforlatticemethodsfor the th asset, ℎ is the volatility of the th asset price, ]+1, multidimensional stochastic volatility models. conditional on information at time ,isastandardnormal 2 Discrete Dynamics in Nature and Society random variable, is the riskless rate of return over the is determined by a spacing parameter for the logarithmic period, and is the unit risk premium for the th asset. returns in such a way that all the approximating logarithmic Under the local risk-neutralized measure, the processes (1) prices are separated by are written as = . 1 (5) ( +1 )= − − ℎ + √ℎ v , √ ln +1 2 (2) The size of these 2+1jumps is restricted to integer multiples ∗, 2 of .Anotherimportantissueistoensurevalidprobability ℎ+1 =0 +1ℎ +2ℎ(v+1 − ) , values over the grid of 2 + 1 prices; the size of these jumps =1,2 v needstobeadjustedaccordingly.Thisisefficientlyhandled with ,where +1, conditional on time information, is a standard normal random variable with respect to the risk- with the inclusion of a jump parameter ,whichisaninteger ∗, that depends on the level of the variance as follows: neutralized measure, the parameters 0, 1, 2, in the model can be obtained by regression on the financial market, √ℎ 1 2 , and ℎ0 is the initial variance of asset .Let(St)=max( , ) −1< ≤. (6) be a single-valued function of St. In this paper, we consider a two-dimensional assets American strangle option whose payoff at maturity is defined by Consequently, the resulting two-asset GARCH model is + + ,+1 =, +, max ([1 −(ST)] ,[(ST)−2] ), (3) (7) ∗, 2 ℎ,+1 =0 +1ℎ, +2ℎ,(,+1 − ) , in which 1 and 2, the strikes for American strangle’s call and put parts, satisfy 1 <2. for =1,2,where −( − − (1/2) ℎ ) 3. Lattice Algorithms = , ,+1 (8) √ℎ Ritchken and Trevor [2] investigated the stochastic lattice , methods for one-dimensional GARCH model. This paper √ intends to extend the methods to two-dimensional GARCH and = 0,±1,±2,...,±, = ℎ0, =1,2.Theprobability model. The aim of this paper is to design an algorithm distribution for , conditional on and ℎ ,isthen that avoids an exponentially exploding number of states. ,+1 , , given by Toward this goal, we begin by approximating the sequence of single period log normal random variables in (2)bya Prob (,+1 =, +)= (), =0,±1,±2,...,±, sequence of discrete random variables. In particular, assume (9) the information set at date is (,ℎ), = 1,,andlet 2 = ( ), = 1, 2 ,= ln .Then,viewedfromdate , +1 where 1, 2, are normal random variables with conditional moments. Consider () = ∑ ( )( ) ( ) ( ) (10) , , 1 [ ]= + − − ℎ , +1 2 , , ≥0 =+ + (4) with such that and = −.Usethesamelatticetreeforassets1 and Var [+1]=ℎ. 2 independently and assume each asset node has three possible paths to the next node: up, middle, and down. We establish two discrete state Markov chains’ approxima- 9 ( ,ℎ ), = 1, 2 Then there are possible combinations. The order of cal- tion, , , ,forthedynamicsofthediscrete (up,up) (up,middle) (up,down) (middle,up) time state variables that converge to the continuous state culation is 1 2 , 1 2 , 1 2 , 1 2 , (middle,middle), (middle,down), (down,up), (down,middle), (,ℎ), = 1,. 2 In particular, we approximate the sequence 1 2 1 2 1 2 1 2 (down,down) of conditional normal random variables by a sequence of and 1 2 ,whichisillustratedbyFigure 1. The possibilities for the nine combinations are discrete random variables. Given this period’s logarithmic 1 2 1 2 1 2 1 2 (1) (1), (1) (0), (1) (−1), (0) (1), price and conditional variance, the conditional normal distri- 1 2 1 2 1 2 1 2 (0) (0), (0) (−1), (−1) (1), (−1) (0),and bution of the next period’s logarithmic price is approximated 1 2 by a discrete random variable that takes on 2 + 1 values (−1) (−1). Then, the volatility pattern by restricting for each asset. The lattice we construct has the property the storage of conditional variance to the minimum and that the conditional means and variances of one period maximum values at each node under the forward-building returnsmatchthetruemeansandvariancesgivenin(4), process needs to be constructed. At each node for each and the approximating sequence of discrete random variables asset, the option prices over a grid of points are evaluated, converges to the true sequence of normal random variables. covering the state space of the variances from the minimum ,max ,min For each asset, the gap between adjacent logarithmic prices to the maximum for each asset. Let ℎ, () and ℎ, () Discrete Dynamics in Nature and Society 3 Table 1: Full volatility information at node (, ). According to Wu [4], we have ℎ1 (3, ) ℎ2 (3, ) , , , ℎ1 (3, ) ℎ2 (2, ) ℎ, − − (1/2) ℎ, , , , = + ℎ1 (3, ) ℎ2 (1, ) 2 2 , , , 2( ) ℎ1 (2, ) ℎ2 (3, ) , , , 2 1 2 ℎ (2, ), ℎ (2, ) ( − − (1/2) ℎ,) , , + , ℎ1 (2, ) ℎ2 (1, ) 2 , , , 2( ) 1 2 ℎ, (1, ), ℎ, (3, ) 1 2 2 ℎ (1, ), ℎ (2, ) ( − − (1/2) ℎ ) , , ℎ, , ℎ1 (1, ) ℎ2 (1, ) =1− − , (13) , , , 2 2 ( ) ( ) − − (1/2) ℎ ℎ, , 1, 2, = − S up ,S up 2 2 1 1 2( ) 1,up 2,middle S1 ,S1 2 ( − − (1/2) ℎ ) 1,up 2,down , S1 ,S1 + , 2 2( ) where =1,2represent the asset, and ℎ, is the approx- imation volatility of asset at time ,and is the jump S1,middle ,S2,up 1 1 parameter of asset .CakiciandTopyan[3]modifiedthe S1,S2 1,middle 2,middle 0 0 S1 ,S1 forward-building process and used interpolated variances only during the backward recursion to make the algorithm S1,middle ,S2,down 1 1 more efficient. They adopted only real node maximum and minimum variances, not the interpolated ones that fell betweenthemaximumandminimumvariances.Itisintuitive to use interpolation for points in the backward procedure. S1,down ,S2,up At the terminal time ,thetwo-assetAmericanstrangle 1 1 option’s cash flow is 1,down 2,middle S1 ,S1 + + S1,down ,S2,down 1 2 1 2 1 1 max {[1 − max ( , )] ,[max ( , )−2] }. (14) (, ) (, ) Figure 1: Two-asset GARCH tree. Let , be the th option price at the node , for = 1,2,...,, and the variance is ℎ,(, ), =. 1,2 Note that the boundary condition for a two-asset American strangle option with strike which expires in period is represent the maximum and minimum variances that can be attained at node for asset .Optionpricesatthisnodeare (,) 1 = (,) 2 =⋅⋅⋅= (,) computed for levels of variance ranging from the lowest to , , , ℎ (, ) + the highest at equidistant intervals.