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- (𝑆1,𝑆2)𝑇 gence of the method was proved by Akyildirim et al. [5]. 𝑡 𝑡 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 𝑆𝑖 1 oneunderlyingassetandsinglestochasticvolatilitymodel.To 𝑡+1 𝑖 √ 𝑖 𝑖 √ 𝑖 𝑖 ln ( )=𝑟𝑓 −𝑞 +𝜆𝑖 ℎ − ℎ + ℎ ] , the best of our knowledge, there are no papers studying the 𝑆𝑖 𝑡 2 𝑡 𝑡 𝑡+1 𝑡 (1) lattice methods for options with many underlying assets and 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 2 multidimensional stochastic volatility models. Indeed there ℎ𝑡+1 =𝛽0 +𝛽1ℎ𝑡 +𝛽2ℎ𝑡(]𝑡+1 −𝑐) , are many papers in developing lattice methods for pricing 𝑖 options with many underlying assets, for example, Boyle [6], with 𝑖=1,2,where𝑆𝑡 is the price of the 𝑖th asset correspond- 𝑖 Boyle et al. [7], Chen et al. [8], Gamba and Trigeorgis [9], and ingtothestandardBrownmotion,𝑞 isthedividendratefor 𝑖 𝑖 Moon et al. [10].Howeveritisnotseenforlatticemethodsfor the 𝑖th asset, ℎ𝑡 is the volatility of the 𝑖th asset price, ]𝑡+1, multidimensional stochastic volatility models. conditional on information at time 𝑡,isastandardnormal 2 Discrete Dynamics in Nature and Society

𝑖 random variable, 𝑟𝑓 is the riskless rate of return over the is determined by a spacing parameter 𝛾𝑛 for the logarithmic period, and 𝜆𝑖 is the unit risk premium for the 𝑖th 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 𝑖 𝑖 𝛾 𝑖 𝛾 = . 𝑆 1 𝑛 (5) ( 𝑡+1 )=𝑟 −𝑞𝑖 − ℎ𝑖 + √ℎ𝑖 v𝑖 , √𝑛 ln 𝑖 𝑓 𝑡 𝑡 𝑡+1 𝑆𝑡 2 (2) The size of these 2𝑛+1jumps is restricted to integer multiples 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 ∗,𝑖 2 𝑖 of 𝛾𝑛.Anotherimportantissueistoensurevalidprobability ℎ𝑡+1 =𝛽0 +𝛽1ℎ𝑡 +𝛽2ℎ𝑡(v𝑡+1 −𝑐 ) , values over the grid of 2𝑛 + 1 prices; the size of these jumps 𝑖=1,2 v𝑖 𝑡 needstobeadjustedaccordingly.Thisisefficientlyhandled with ,where 𝑡+1, conditional on time information, is 𝜂𝑖 a standard normal random variable with respect to the risk- with the inclusion of a jump parameter ,whichisaninteger 𝑖 𝑖 𝑖 ∗,𝑖 that depends on the level of the variance as follows: neutralized measure, the parameters 𝛽0, 𝛽1, 𝛽2, 𝑐 in the model can be obtained by regression on the financial market, √ℎ𝑖 𝑖 1 2 𝑖 𝑎,𝑡 𝑖 and ℎ0 is the initial variance of asset 𝑖.Let𝑓(St)=max(𝑆𝑡 ,𝑆𝑡 ) 𝜂 −1< ≤𝜂. (6) 𝑖 be a single-valued function of St. In this paper, we consider 𝛾 a two-dimensional assets American strangle option whose payoff at maturity 𝑇 is defined by Consequently, the resulting two-asset GARCH model is 𝑖 𝑖 𝑖 𝑖 + + 𝑦𝑎,𝑡+1 =𝑦𝑎,𝑡 +𝑗𝜂𝛾𝑛, max ([𝐾1 −𝑓(ST)] ,[𝑓(ST)−𝐾2] ), (3) (7) 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 ∗,𝑖 2 ℎ𝑎,𝑡+1 =𝛽0 +𝛽1ℎ𝑎,𝑡 +𝛽2ℎ𝑎,𝑡(𝜀𝑎,𝑡+1 −𝑐 ) , in which 𝐾1 and 𝐾2, the strikes for American strangle’s call and put parts, satisfy 𝐾1 <𝐾2. for 𝑖=1,2,where 𝑗𝜂𝑖𝛾𝑖 −(𝑟 −𝑞𝑖 − (1/2) ℎ𝑖 ) 3. Lattice Algorithms 𝜀𝑖 = 𝑛 𝑓 𝑎,𝑡 𝑎,𝑡+1 (8) √ℎ𝑖 Ritchken and Trevor [2] investigated the stochastic lattice 𝑎,𝑡 methods for one-dimensional GARCH model. This paper 𝑖 √ 𝑖 intends to extend the methods to two-dimensional GARCH and 𝑗= 0,±1,±2,...,±𝑛, 𝛾 = ℎ0, 𝑖=1,2.Theprobability model. The aim of this paper is to design an algorithm 𝑖 𝑖 𝑖 distribution for 𝑦 , conditional on 𝑦 and ℎ ,isthen that avoids an exponentially exploding number of states. 𝑎,𝑡+1 𝑎,𝑡 𝑎,𝑡 given by Toward this goal, we begin by approximating the sequence of single period log normal random variables in (2)bya 𝑖 𝑖 𝑖 𝑖 𝑖 Prob (𝑦𝑎,𝑡+1 =𝑦𝑎,𝑡 +𝑗𝜂𝛾𝑛)=𝑃 (𝑗), 𝑗 =0,±1,±2,...,±𝑛, sequence of discrete random variables. In particular, assume 𝑖 𝑖 (9) the information set at date 𝑡 is (𝑆𝑡,ℎ𝑡), 𝑖 = 1,,andlet 2 𝑦𝑖 = (𝑆𝑖 ), 𝑖 = 1, 2 𝑡 𝑦𝑖 ,𝑖= 𝑡 ln 𝑡 .Then,viewedfromdate , 𝑡+1 where 1, 2, are normal random variables with conditional moments. 𝑛 𝑗 𝑗 𝑗 Consider 𝑃𝑖 (𝑗) = ∑ ( )(𝑝𝑖 ) 𝑢 (𝑝𝑖 ) 𝑚 (𝑝𝑖 ) 𝑑 𝑗 𝑗 𝑗 𝑢 𝑚 𝑑 (10) 𝑗 ,𝑗 ,𝑗 𝑢 𝑚 𝑑 1 𝑢 𝑚 𝑑 𝐸 [𝑦𝑖 ]=𝑦𝑖 +𝑟 −𝑞𝑖 − ℎ𝑖 , 𝑡 𝑡+1 𝑡 𝑓 2 𝑡 𝑗 ,𝑗 ,𝑗 ≥0 𝑛=𝑗+𝑗 +𝑗 (4) with 𝑢 𝑚 𝑑 such that 𝑢 𝑚 𝑑 and 𝑖 𝑖 𝑗=𝑗𝑢 −𝑗𝑑.Usethesamelatticetreeforassets𝑆1 and Var 𝑡 [𝑦𝑡+1]=ℎ𝑡. 𝑆2 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 (𝑦𝑖 ,ℎ𝑖 ), 𝑖 = 1, 2 Then there are possible combinations. The order of cal- tion, 𝑎,𝑡 𝑎,𝑡 ,forthedynamicsofthediscrete (𝑆up,𝑆up) (𝑆up,𝑆middle) (𝑆up,𝑆down) (𝑆middle,𝑆up) time state variables that converge to the continuous state culation is 1 2 , 1 2 , 1 2 , 1 2 , 𝑖 𝑖 (𝑆middle,𝑆middle), (𝑆middle,𝑆down), (𝑆down,𝑆up), (𝑆down,𝑆middle), (𝑦𝑡,ℎ𝑡), 𝑖 = 1,. 2 In particular, we approximate the sequence 1 2 1 2 1 2 1 2 (𝑆down,𝑆down) 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 𝑃 (1)𝑃 (1), 𝑃 (1)𝑃 (0), 𝑃 (1)𝑃 (−1), 𝑃 (0)𝑃 (1), price and conditional variance, the conditional normal distri- 1 2 1 2 1 2 1 2 𝑃 (0)𝑃 (0), 𝑃 (0)𝑃 (−1), 𝑃 (−1)𝑃 (1), 𝑃 (−1)𝑃 (0),and bution of the next period’s logarithmic price is approximated 1 2 by a discrete random variable that takes on 2𝑛 + 1 values 𝑃 (−1)𝑃 (−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 𝐾 points are evaluated, converges to the true sequence of normal random variables. covering the state space of the variances from the minimum 𝑖,max 𝑖,min For each asset, the gap between adjacent logarithmic prices to the maximum for each asset. Let ℎ𝑎,𝑡 (𝑚) and ℎ𝑎,𝑡 (𝑚) Discrete Dynamics in Nature and Society 3

Table 1: Full volatility information at node (𝑡, 𝑚). According to Wu [4], we have ℎ1 (3, 𝑚) ℎ2 (3, 𝑚) 𝑎,𝑡 , 𝑎,𝑡 𝑖 𝑖 𝑖 ℎ1 (3, 𝑚) ℎ2 (2, 𝑚) ℎ𝑎,𝑡 𝑟𝑓 −𝑞 − (1/2) ℎ𝑎,𝑡 𝑎,𝑡 , 𝑎,𝑡 𝑝𝑖 = + ℎ1 (3, 𝑚) ℎ2 (1, 𝑚) 𝑢 𝑖 𝑖 2 2𝑛𝜂𝑖𝛾𝑖 𝑎,𝑡 , 𝑎,𝑡 2𝑛(𝜂 𝛾𝑛) 𝑛 ℎ1 (2, 𝑚) ℎ2 (3, 𝑚) 𝑎,𝑡 , 𝑎,𝑡 2 1 2 𝑖 𝑖 ℎ (2, 𝑚), ℎ (2, 𝑚) (𝑟𝑓 −𝑞 − (1/2) ℎ𝑎,𝑡) 𝑎,𝑡 𝑎,𝑡 + , ℎ1 (2, 𝑚) ℎ2 (1, 𝑚) 2 𝑎,𝑡 , 𝑎,𝑡 2(𝑛𝜂𝑖𝛾𝑖 ) 1 2 𝑛 ℎ𝑎,𝑡 (1, 𝑚), ℎ𝑎,𝑡 (3, 𝑚) 1 2 2 ℎ (1, 𝑚), ℎ (2, 𝑚) 𝑖 (𝑟 −𝑞𝑖 − (1/2) ℎ𝑖 ) 𝑎,𝑡 𝑎,𝑡 𝑖 ℎ𝑎,𝑡 𝑓 𝑎,𝑡 ℎ1 (1, 𝑚) ℎ2 (1, 𝑚) 𝑝 =1− − , (13) 𝑎,𝑡 , 𝑎,𝑡 𝑚 𝑖 𝑖 2 𝑖 𝑖 2 𝑛(𝜂 𝛾𝑛) (𝑛𝜂 𝛾𝑛) 𝑖 𝑟 −𝑞𝑖 − (1/2) ℎ𝑖 𝑖 ℎ𝑎,𝑡 𝑓 𝑎,𝑡 1, 2, 𝑝 = − S up ,S up 𝑑 𝑖 𝑖 2 2𝑛𝜂𝑖𝛾𝑖 1 1 2𝑛(𝜂 𝛾𝑛) 𝑛 1,up 2,middle S1 ,S1 2 (𝑟 −𝑞𝑖 − (1/2) ℎ𝑖 ) 1,up 2,down 𝑓 𝑎,𝑡 S1 ,S1 + , 𝑖 𝑖 2 2(𝑛𝜂 𝛾𝑛)

𝑖 where 𝑖=1,2represent the 𝑖 asset, and ℎ𝑎,𝑡 is the approx- 𝑖 imation volatility of asset 𝑖 at time 𝑡,and𝜂 is the jump S1,middle ,S2,up 1 1 parameter of asset 𝑖.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 𝐾 points in the backward procedure. 𝑇 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 {[𝐾1 − max (𝑆𝑡 ,𝑆𝑡 )] ,[max (𝑆𝑡 ,𝑆𝑡 )−𝐾2] }. (14)

𝐶 (𝑚, 𝑘) 𝑘 (𝑚, 𝑘) Figure 1: Two-asset GARCH tree. Let 𝑎,𝑡 be the th option price at the node , 𝑖 for 𝑘 = 1,2,...,𝐾, and the variance is ℎ𝑎,𝑡(𝑚, 𝑘), 𝑖 =. 1,2 Note that the boundary condition for a two-asset American strangle option with strike 𝑋 which expires in period 𝑇 is represent the maximum and minimum variances that can be attained at node 𝑚 for asset 𝑖.Optionpricesatthisnodeare 𝐶 (𝑚,) 1 =𝐶 (𝑚,) 2 =⋅⋅⋅=𝐶 (𝑚, 𝐾) computed for 𝐾 levels of variance ranging from the lowest to 𝑎,𝑇 𝑎,𝑇 𝑎,𝑇 𝑖 ℎ (𝑘, 𝑚) + the highest at equidistant intervals. In particular, 𝑎,𝑡 = {[𝐾 − (𝑆1 ,𝑆2 )] , representing the 𝑘th level of the variance at node (𝑡, 𝑚) with max 1 max 𝑇 𝑇 (15) 𝑘=1,...,𝐾 is defined by an interpolation as follows: 1 2 + [max (𝑆𝑇,𝑆𝑇)−𝐾2] }. 𝑖 𝑖,min 𝑖 ℎ𝑎,𝑡 (𝑘,) 𝑚 =ℎ𝑎,𝑡 (𝑚) +𝑙𝑡 (𝑚)(𝑘−1) , 𝑘=1,...,𝐾, (11) We apply backward recursion to establish the option price at date 0. Consider a node (𝑚, 𝑘) at time 𝑡.Thenwecomputethe 𝑖 option price 𝐶𝑎,𝑡(𝑚, 𝑘) corresponding to variance ℎ𝑎,𝑡(𝑚, 𝑘) where 𝑖 at the node. Given the variance ℎ𝑎,𝑡(𝑚, 𝑘),wecomputethe 𝜂𝑖 ℎ𝑖,max (𝑚) −ℎ𝑖,min (𝑚) appropriate jump parameter, for each asset, by (6). The 𝑙𝑖 𝑚 = 𝑎,𝑡 𝑎,𝑡 . successive nodes for this variance combination are ((𝑡+1, 𝑚+ 𝑡 ( ) (12) 1 1 2 2 1 𝐾−1 𝑗 𝜂 ), (𝑡 + 1, 𝑚 +𝑗 𝜂 )),where𝑗 = 0,±1,±2,...,±𝑛 and 2 𝑗 = 0,±1,±2,...,±𝑛.Equation(11)isusedtocomputethe For 𝐾=3, the full volatility information at node (𝑡, 𝑚) is period (𝑡+1) variance for each of these nodes. Specifically, for described by Table 1. the transition from the 𝑘th variance element of node (𝑡, 𝑚) to 4 Discrete Dynamics in Nature and Society

1 1 2 2 node ((𝑡+1,𝑚+𝑗 𝜂 ), (𝑡 + 1, 𝑚 +𝑗 𝜂 )),theperiod(𝑡 + 1) Table 2: Numerical results for Example 1. variance for each asset is given by 𝑇𝑛 Option prices with lattice Option prices and intervals 𝑖,next 1,2 𝑖 𝑖 𝑖 ℎ𝑎,𝑡+1 (𝑗 )=𝛽0 +𝛽1ℎ𝑎,𝑡 (𝑚, 𝑘) with LSM 1 0.014987254820724 𝑖 𝑖 𝑖 2 (𝑗𝜂 𝛾 −𝑟 +ℎ (𝑚, 𝑘)) 2 0.030190542190626 𝑖 𝑖 [ 𝑛 𝑓 𝑎,𝑡 ] 0.029606167588959 +𝛽2ℎ𝑎,𝑡 (𝑚, 𝑘) , √ℎ𝑖 (𝑚, 𝑘) −𝑐𝑖,∗ 5 4 0.030393224336490 [0.02804 0.03117] [ 𝑎,𝑡 ] 7 0.030981333835160 (16) 10 0.031863879158063 1,2 1 2 where 𝑗 represents the combination of 𝑗 and 𝑗 .Linear 1 0.068487810335143 interpolation of the two stored option prices corresponding 2, 2 2 0.081254039347033 0.081499912354419 to the two stored variance entries closest to ℎ next(𝑗 ) is used 𝑎,𝑡+1 [0.07999 0.08500] to obtain the option price corresponding to a variance of 7 4 0.083605936631716 2,next 2 1,next 1 7 0.081680163494625 ℎ𝑎,𝑡+1 (𝑗 ) when ℎ𝑎,𝑡+1 (𝑗 ) is already chosen. Let 𝐿 be an integer smaller than 𝐾 defined via 10 0.080301350887482 1 0.178948896057298 ℎ2 (𝑚 + 𝑗2𝜂2,𝐿)<ℎ2,next (𝑗2)<ℎ2 (𝑚 +2 𝑗 𝜂2,𝐿+1). 𝑎,𝑡+1 𝑎,𝑡+1 𝑎,𝑡+1 2 0.192329465764766 (17) 0.193857807991106 10 4 0.195314383716280 [0.18842 0.19929] The interpolated option price is 7 0.199868029599249 10 0.193058780881166 interp 1,2 1,2 𝐶 (𝑚) =𝑞(𝑗)𝐶𝑎,𝑡+1 (𝑚 + 𝑗 𝜂 ,𝐿)

+(1−𝑞(𝑗))𝐶 (𝑚 + 𝑗1,2𝜂1,2,𝐿+1), 𝑎,𝑡+1 4. Numerical Examples (18) In this section, several examples are implemented using the where latticemethodinthispaperandleastsquareMonte-Carlo 2 2 2 2,next 2 method (LSM) developed by Longstaff and Schwartz [14]. ℎ𝑎,𝑡+1 (𝑚 + 𝑗 𝜂 ,𝐿+1)−ℎ𝑎,𝑡+1 (𝑗 ) 𝑞 (𝑗) = . In Examples 1, 2,and3,wefocusonthesingleasset ℎ2 (𝑚 +2 𝑗 𝜂2,𝐿+1)−ℎ2 (𝑚 + 𝑗2𝜂2,𝐿) 𝑎,𝑡+1 𝑎,𝑡+1 American strangle options under GARCH model where the (19) convergence with respect to 𝑛 and 𝐾 are studied, respectively, in the first two examples, and the optimal exercise boundaries In this way an option price is identified for each of the are drawn for the third example. In Examples 4 and 5, (2𝑛 + 1)(2𝑛 +1) jumps from node (𝑡, 𝑚) with variance 1,next 1 2,next 2 we compute the two-dimensional assets American strangle combination (ℎ𝑎,𝑡+1 (𝑗 ), ℎ𝑎,𝑡+1 (𝑗 )). In each case, either node 1,2 1,2 options. (𝑡+1,𝑚+𝑗 𝜂 ) contains a variance entry (and hence option In Tables 2 and 3, the prices of the options using LSM 1,next 1 2,next 1 value) that matches (ℎ𝑎,𝑡+1 (𝑗 ), ℎ𝑎,𝑡+1 (𝑗 )),ortherelevant with 5,000 paths are calculated and the intervals that the information is interpolated from the closest two entries. We true prices fall into are provided. From the comparisons we use the following formula to compute the unexercised option confirm that the lattice methods developed in this paper are go value 𝐶𝑎,𝑡(𝑚, 𝑘): correct and reliable. Furthermore from Table 2 we observe 𝑛 𝑛 𝑛 that the lattice method converges as goes larger and from −𝑟 1 1 2 2 go 𝑓 interp Table 3 the lattice method converges as 𝐾 goes larger. Figure 2 𝐶𝑎,𝑡 (𝑚, 𝑘) =𝑒 ∑ 𝑃 (𝑗 ) ∑ 𝑃 (𝑗 )𝐶 (𝑚) . 𝑗1=−𝑛 𝑗2=−𝑛 shows exercise and holding regions: the middle part is the (20) holding region and the top and bottom parts are the exercise regions. 𝑎,stop Denote the exercised value of the claim by 𝐶𝑡 (𝑚, 𝑘).For Example 1. Consider single asset GARCH model with a two-asset American strangle option with strikes 𝐾1 and 𝐾2, 𝑟 =5 𝑞=10 𝛽 = 6.575 × 10−6 𝛽 =0.9 𝐾1 <𝐾2, parameters 𝑓 %, %, 0 , 1 , 𝛽 = 0.04 𝑆 = 100 ℎ = 0.0001096 𝐾 = 105 𝐾 =95 + 2 , 0 , 0 , 1 , 2 , stop 1 2 𝛾=ℎ 𝑐∗ =0 𝐾=20 𝐶𝑎,𝑡 (𝑚, 𝑘) = max {[𝐾1 − max (𝑆𝑡 ,𝑆𝑡 )] , 0,and . Fixing , we investigate the 𝑛 (21) convergence behavior as increases. 1 2 + [max (𝑆 ,𝑆 )−𝐾2] }. 𝑡 𝑡 Example 2. Consider single asset GARCH model with the 𝑛=5 Thevalueoftheclaimatthe𝑘th entry of node (𝑡, 𝑚) is then same parameters as Example 1. In this example, and the sensitivity to the volatility space parameter, 𝐾, is explored. go stop 𝐶𝑎,𝑡 (𝑚, 𝑘) = max {𝐶𝑎,𝑡 (𝑚, 𝑘) ,𝐶𝑎,𝑡 (𝑚, 𝑘)}. (22) Example 3. Consider single asset GARCH model with the The final option price, obtained by backward recursion ofthis same parameters as Example 1. Draw the figure of the optimal procedure, is given by 𝐶𝑎,0(0, 1). exercise boundaries for American strangle. Discrete Dynamics in Nature and Society 5

Table 3: Numerical results for Example 2. Table 4: Numerical results for Example 4.

𝑇𝐾Option prices with lattice Option prices and intervals 𝑇𝑛 Option prices with lattice Option prices and interval with LSM with LSM 2 0.028187456298155 1 0.005974297 4 0.029090524706680 2 0.004223692 0.005065587636226 0.029606167588959 3 5 6 0.029930565933099 3 0.004502351 [0.00261 0.006316] 10 0.030190542190626 [0.02804 0.03117] 4 0.005530355 20 0.029391074539888 5 0.005040507 40 0.029489977773055 1 0.014427788 2 0.197070210662258 2 0.014590056 0.015018277749022 4 4 0.195700639175277 3 0.013984718 [0.01111 0.01892] 0.193857807991106 10 6 0.196465071653130 4 0.017223364 10 0.192329465764766 [0.18842 0.19929] 5 0.015988485 20 0.193696054251151 1 0.022129134 40 0.194423204241922 2 0.030507636 0.034841312405081 5 2 1.182560819901510 3 0.030853276 [0.02844 0.041235] 4 1.197816540629624 4 0.035594538 1.205198709585444 30 6 1.200350772727828 5 0.034694665 10 1.202714267674711 [1.19372 1.22670] 20 1.203407511987706 Table 5: Numerical results for Example 5. 40 1.203545193979154 𝑇𝐾 Option prices with lattice Option prices and intervals 120 with LSM 2 0.005974297 115 4 0.005974297 0.005065587636226 110 3 6 0.005974297 [0.00261 0.006316] 105 8 0.005974297 100 10 0.005974297 2 0.014427788 S 95 4 0.014427788 0.015018277749022 90 4 6 0.014427788 [0.01111 0.01892] 85 8 0.014427788 80 10 0.014427788 75 2 0.02194509 4 0.022129134 0.034841312405081 70 5 6 8 101214161820 6 0.022164226 [0.02844 0.041235] T 8 0.024355984 10 0.024276964 American strangle call side American strangle put side

Figure 2: Optimal exercise boundaries for Example 3. 1 2 1 2 1 ℎ0 =ℎ0 = 0.0001096, 𝐾1 = 105, 𝐾2 =95, 𝛾 =𝛾 =ℎ0,and ∗,1 ∗,2 𝑐 =𝑐 =0. Fixing 𝐾=4, we investigate the convergence behavior as 𝑛 increases. In Examples 4 and 5, we examine the stochastic lattice methods for pricing American strangle options under multi- Example 5. Consider the two-dimensional GARCH model 𝑛=1 asset under stochastic volatility model where the convergence with the same parameters as Example 4. Fixing ,we 𝐾 with 𝑛 and 𝐾 are studied. From the numerics in Tables 4 and study the sensitivity to the volatility space parameter . 5,weconfirmthatthelatticemethodsfortwo-dimensional models are correct and reliable and the convergence of the 5. Conclusions lattice methods with respect to 𝑛 and 𝐾 is observed. In this paper we studied pricing methods for stochastic Example 4. Consider two-asset American strangles with volatility models of the American strangles with single asset 𝑟 =5 𝑞1 =𝑞2 =10 𝛽1 =𝛽2 = 6.575×10−6 param- eters 𝑓 %, %, 0 0 , and multiassets. Both lattice methods and LSM methods are 1 2 1 1 1 2 𝛽1 =𝛽1 = 0.9, 𝛽2 =𝛽2 = 0.04, 𝑆0 =𝑆0 = 100, developed and implemented. To the best of our knowledge, 6 Discrete Dynamics in Nature and Society

there are no results on the lattice methods for multidi- [11] C. Chiarella and A. Ziogas, “Evaluation of American strangles,” mensional stochastic volatility models. We first extended Journal of Economic Dynamics & Control,vol.29,no.1-2,pp. the stochastic lattice methods invented by Ritchken and 31–62, 2005. Trevor [2] which are for one-dimensional GARCH models [12] F. Moraux, “On perpetual American strangles,” Journal of of American call to the multidimensional GARCH models Derivatives,vol.16,no.4,pp.82–97,2009. of American strangles. Numerical examples confirm the [13] J.-C. Duan, “The GARCH option pricing model,” Mathematical correctness and reliability of the lattice methods. Future chal- Finance,vol.5,no.1,pp.13–32,1995. lenging works include the development of the lattice methods [14] F. A. Longstaff and E. S. Schwartz, “Valuing American options for multidimensional volatility models with correlations and by simulation: a simple least-squares approach,” Review of recently developed models (e.g., [15]). One possible solution Financial Studies,vol.14,no.1,pp.113–147,2001. to the case of correlation is to adopt the idea (using moment- [15] C.X.Huang,X.Gong,X.Chen,andF.H.Wen,“Measuringand generating function) in [7]. However it needs to develop new forecasting volatility in Chinese stock market using HAR-CJ- techniques when the stochastic volatility models are involved. Mmodel,”Abstract and Applied Analysis,vol.2013,ArticleID Furthermore, a dimensional-reduction technique should be 143194,13pages,2013. developedtoreducethecomputationalcost.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment The work was supported by the Fundamental Research Funds for the Central Universities (Grant no. JBK130401).

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