Research Article Lattice Methods for Pricing American Strangles …downloads.hindawi.com/journals/ddns/2014/165259.pdf · 2019. 7. 31. · valuable to price American style options

Research ArticleLattice Methods for Pricing American Strangles withTwo-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, China2 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 withone-dimensional stochastic volatility models to the two-dimensional cases with strangle payoff.This proposedmethod is comparedwith the least square Monte-Carlo method via numerical examples.

1. Introduction

Calculating American style options under geometric Brown-ianmotion is far from the realistic financial market. It is morevaluable to price American style options under stochasticmodels. In general the valuation of American options withstochastic volatility models has no closed-form solutionexcept very few cases (see, e.g., Heston [1]).Therefore numer-ical methods or simulation methods are developed to pricefinancial derivatives with stochastic volatility, among whichthe lattice methods receive much more attention. RitchkenandTrevor [2] proposed an efficient latticemethod for pricingAmerican options under GARCHmodels. Later the idea wasfurther developed and applied by several papers, for example,Cakici and Topyan [3] and Wu [4], and recently the conver-gence of the method was proved by Akyildirim et al. [5].

All the abovementioned references focused on the devel-opment of lattice methods for pricing American options withone underlying asset and single stochastic volatilitymodel. Tothe best of our knowledge, there are no papers studying thelattice methods for options with many underlying assets andmultidimensional stochastic volatility models. Indeed thereare many papers in developing lattice methods for pricingoptions with many underlying assets, for example, Boyle [6],Boyle et al. [7], Chen et al. [8], Gamba and Trigeorgis [9], andMoon et al. [10]. However it is not seen for latticemethods formultidimensional stochastic volatility models.

In this paper we give an attempt to this challenging topicby studying an American style option with strangle payoff,which was previously investigated by Chiarella and Ziogas[11] and Moraux [12] for single asset and constant volatility.We develop the lattice methods of Ritchken and Trevor [2] tothe American strangle options with many underlying assetsand multidimensional stochastic volatility GARCH models.We compare the lattice methods with the least squareMonte-Carlo methods via several numerical examples.

2. Two-Dimensional Stochastic VolatilityModels of American Strangles

Assume that the prices of two-dimensional assets St =(𝑆

1

𝑡, 𝑆

2

𝑡)

𝑇 follow a two-dimensional GARCH model (see, e.g.,Duan [13] for more explanation of the one-dimensionalGARCH model). Consider

ln(𝑆

𝑖

𝑡+1

𝑆

𝑖

𝑡

) = 𝑟𝑓− 𝑞

𝑖+ 𝜆𝑖√ℎ

𝑖

𝑡−

1

2

ℎ

𝑖

𝑡+√ℎ

𝑖

𝑡]𝑖𝑡+1

,

ℎ

𝑖

𝑡+1= 𝛽

𝑖

0+ 𝛽

𝑖

1ℎ

𝑖

𝑡+ 𝛽

𝑖

2ℎ

𝑖

𝑡(]𝑖𝑡+1

− 𝑐

𝑖)

2

,

(1)

with 𝑖 = 1, 2, where 𝑆𝑖𝑡is the price of the 𝑖th asset correspond-

ing to the standard Brown motion, 𝑞𝑖 is the dividend rate forthe 𝑖th asset, ℎ𝑖

𝑡is the volatility of the 𝑖th asset price, ]𝑖

𝑡+1,

conditional on information at time 𝑡, is a standard normal

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 165259, 6 pageshttp://dx.doi.org/10.1155/2014/165259

2 Discrete Dynamics in Nature and Society

random variable, 𝑟𝑓is the riskless rate of return over the

period, and 𝜆𝑖is the unit risk premium for the 𝑖th asset.

Under the local risk-neutralized measure, the processes (1)are written as

ln(𝑆

𝑖

𝑡+1

𝑆

𝑖

𝑡

) = 𝑟𝑓− 𝑞

𝑖−

1

2

ℎ

𝑖

𝑡+√ℎ

𝑖

𝑡v𝑖

𝑡+1,

ℎ

𝑖

𝑡+1= 𝛽

𝑖

0+ 𝛽

𝑖

1ℎ

𝑖

𝑡+ 𝛽

𝑖

2ℎ

𝑖

𝑡(v𝑖

𝑡+1− 𝑐

∗,𝑖)

2

,

(2)

with 𝑖 = 1, 2, where v𝑖𝑡+1

, conditional on time 𝑡 information, isa standard normal random variable with respect to the risk-neutralized measure, the parameters 𝛽𝑖

0, 𝛽𝑖1, 𝛽𝑖2, 𝑐∗,𝑖 in the

model can be obtained by regression on the financial market,and ℎ𝑖0is the initial variance of asset 𝑖. Let𝑓(St) = max(𝑆1𝑡 , 𝑆

2

𝑡)

be a single-valued function of St. In this paper, we considera two-dimensional assets American strangle option whosepayoff at maturity 𝑇 is defined by

max ([𝐾1− 𝑓 (ST)]

+

, [𝑓 (ST) − 𝐾2]+

) , (3)

in which 𝐾1and 𝐾

2, the strikes for American strangle’s call

and put parts, satisfy𝐾1< 𝐾2.

3. Lattice Algorithms

Ritchken and Trevor [2] investigated the stochastic latticemethods for one-dimensional GARCH model. This paperintends to extend the methods to two-dimensional GARCHmodel. The aim of this paper is to design an algorithmthat avoids an exponentially exploding number of states.Toward this goal, we begin by approximating the sequenceof single period log normal random variables in (2) by asequence of discrete random variables. In particular, assumethe information set at date 𝑡 is (𝑆𝑖

𝑡, ℎ

𝑖

𝑡), 𝑖 = 1, 2, and let

𝑦

𝑖

𝑡= ln(𝑆𝑖

𝑡), 𝑖 = 1, 2. Then, viewed from date 𝑡, 𝑦𝑖

𝑡+1, 𝑖 =

1, 2, are normal random variables with conditional moments.Consider

𝐸𝑡[𝑦

𝑖

𝑡+1] = 𝑦

𝑖

𝑡+ 𝑟𝑓− 𝑞

𝑖−

1

2

ℎ

𝑖

𝑡,

Var𝑡[𝑦

𝑖

𝑡+1] = ℎ

𝑖

𝑡.

(4)

We establish two discrete state Markov chains’ approxima-tion, (𝑦𝑖

𝑎,𝑡, ℎ

𝑖

𝑎,𝑡), 𝑖 = 1, 2, for the dynamics of the discrete

time state variables that converge to the continuous state(𝑦

𝑖

𝑡, ℎ

𝑖

𝑡), 𝑖 = 1, 2. In particular, we approximate the sequence

of conditional normal random variables by a sequence ofdiscrete random variables. Given this period’s logarithmicprice and conditional variance, the conditional normal distri-bution of the next period’s logarithmic price is approximatedby a discrete random variable that takes on 2𝑛 + 1 valuesfor each asset. The lattice we construct has the propertythat the conditional means and variances of one periodreturns match the true means and variances given in (4),and the approximating sequence of discrete random variablesconverges to the true sequence of normal random variables.For each asset, the gap between adjacent logarithmic prices

is determined by a spacing parameter 𝛾𝑖𝑛for the logarithmic

returns in such a way that all the approximating logarithmicprices are separated by

𝛾

𝑖

𝑛=

𝛾

𝑖

√𝑛

. (5)

The size of these 2𝑛+1 jumps is restricted to integer multiplesof 𝛾𝑖𝑛. Another important issue is to ensure valid probability

values over the grid of 2𝑛 + 1 prices; the size of these jumpsneeds to be adjusted accordingly. This is efficiently handledwith the inclusion of a jump parameter 𝜂𝑖, which is an integerthat depends on the level of the variance as follows:

𝜂

𝑖− 1 <

√ℎ

𝑖

𝑎,𝑡

𝛾

𝑖≤ 𝜂

𝑖.

(6)

Consequently, the resulting two-asset GARCHmodel is

𝑦

𝑖

𝑎,𝑡+1= 𝑦

𝑖

𝑎,𝑡+ 𝑗𝜂

𝑖𝛾

𝑖

𝑛,

ℎ

𝑖

𝑎,𝑡+1= 𝛽

𝑖

0+ 𝛽

𝑖

1ℎ

𝑖

𝑎,𝑡+ 𝛽

𝑖

2ℎ

𝑖

𝑎,𝑡(𝜀

𝑖

𝑎,𝑡+1− 𝑐

∗,𝑖)

2

,

(7)

for 𝑖 = 1, 2, where

𝜀

𝑖

𝑎,𝑡+1=

𝑗𝜂

𝑖𝛾

𝑖

𝑛− (𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡)

√ℎ

𝑖

𝑎,𝑡

(8)

and 𝑗 = 0, ±1, ±2, . . . , ±𝑛, 𝛾𝑖 = √ℎ𝑖0, 𝑖 = 1, 2. The probability

distribution for 𝑦𝑖𝑎,𝑡+1

, conditional on 𝑦𝑖𝑎,𝑡

and ℎ𝑖𝑎,𝑡, is then

given by

Prob (𝑦𝑖𝑎,𝑡+1

= 𝑦

𝑖

𝑎,𝑡+ 𝑗𝜂

𝑖𝛾

𝑖

𝑛) = 𝑃

𝑖(𝑗) , 𝑗 = 0, ±1, ±2, . . . , ±𝑛,

(9)

where

𝑃

𝑖(𝑗) = ∑

𝑗𝑢,𝑗𝑚,𝑗𝑑

(

𝑛

𝑗𝑢𝑗𝑚𝑗𝑑

) (𝑝

𝑖

𝑢)

𝑗𝑢

(𝑝

𝑖

𝑚)

𝑗𝑚

(𝑝

𝑖

𝑑)

𝑗𝑑

(10)

with 𝑗𝑢, 𝑗𝑚, 𝑗𝑑

≥ 0 such that 𝑛 = 𝑗𝑢

+ 𝑗𝑚

+ 𝑗𝑑and

𝑗 = 𝑗𝑢

− 𝑗𝑑. Use the same lattice tree for assets 𝑆

1and

𝑆2independently and assume each asset node has three

possible paths to the next node: up, middle, and down.Then there are 9 possible combinations. The order of cal-culation is (𝑆up

1, 𝑆

up2), (𝑆up1, 𝑆

middle2

), (𝑆up1, 𝑆

down2

), (𝑆middle1

, 𝑆

up2),

(𝑆

middle1

, 𝑆

middle2

), (𝑆middle1

, 𝑆

down2

), (𝑆down1

, 𝑆

up2), (𝑆down1

, 𝑆

middle2

),and (𝑆down

1, 𝑆

down2

), which is illustrated by Figure 1.The possibilities for the nine combinations are

𝑃

1(1)𝑃

2(1), 𝑃1(1)𝑃2(0), 𝑃1(1)𝑃2(−1), 𝑃1(0)𝑃2(1),

𝑃

1(0)𝑃

2(0), 𝑃1(0)𝑃2(−1), 𝑃1(−1)𝑃2(1), 𝑃1(−1)𝑃2(0), and

𝑃

1(−1)𝑃

2(−1). Then, the volatility pattern by restricting

the storage of conditional variance to the minimum andmaximum values at each node under the forward-buildingprocess needs to be constructed. At each node for eachasset, the option prices over a grid of 𝐾 points are evaluated,covering the state space of the variances from the minimumto the maximum for each asset. Let ℎ𝑖,max

𝑎,𝑡(𝑚) and ℎ𝑖,min

𝑎,𝑡(𝑚)

Discrete Dynamics in Nature and Society 3

Table 1: Full volatility information at node (𝑡,𝑚).

ℎ

1

𝑎,𝑡(3, 𝑚), ℎ2

𝑎,𝑡(3, 𝑚)

ℎ

1

𝑎,𝑡(3, 𝑚), ℎ2

𝑎,𝑡(2, 𝑚)

ℎ

1

𝑎,𝑡(3, 𝑚), ℎ2

𝑎,𝑡(1, 𝑚)

ℎ

1

𝑎,𝑡(2, 𝑚), ℎ2

𝑎,𝑡(3, 𝑚)

ℎ

1

𝑎,𝑡(2, 𝑚), ℎ2

𝑎,𝑡(2, 𝑚)

ℎ

1

𝑎,𝑡(2, 𝑚), ℎ2

𝑎,𝑡(1, 𝑚)

ℎ

1

𝑎,𝑡(1, 𝑚), ℎ2

𝑎,𝑡(3, 𝑚)

ℎ

1

𝑎,𝑡(1, 𝑚), ℎ2

𝑎,𝑡(2, 𝑚)

ℎ

1

𝑎,𝑡(1, 𝑚), ℎ2

𝑎,𝑡(1, 𝑚)

S1

0, S

2

0

S1,up1

, S2,up1

S1,up1

, S2,middle1

S1,up1

, S2,down1

S1,middle1

, S2,up1

S1,middle1

, S2,middle1

S1,middle1

, S2,down1

S1,down1

, S2,up1

S1,down1

, S2,middle1

S1,down1

, S2,down1

Figure 1: Two-asset GARCH tree.

represent the maximum and minimum variances that can beattained at node 𝑚 for asset 𝑖. Option prices at this node arecomputed for𝐾 levels of variance ranging from the lowest tothe highest at equidistant intervals. In particular, ℎ𝑖

𝑎,𝑡(𝑘,𝑚)

representing the 𝑘th level of the variance at node (𝑡, 𝑚) with𝑘 = 1, . . . , 𝐾 is defined by an interpolation as follows:

ℎ

𝑖

𝑎,𝑡(𝑘,𝑚) = ℎ

𝑖,min𝑎,𝑡

(𝑚) + 𝑙

𝑖

𝑡(𝑚) (𝑘 − 1) , 𝑘 = 1, . . . , 𝐾,

(11)

where

𝑙

𝑖

𝑡(𝑚) =

ℎ

𝑖,max𝑎,𝑡

(𝑚) − ℎ

𝑖,min𝑎,𝑡

(𝑚)

𝐾 − 1

.(12)

For 𝐾 = 3, the full volatility information at node (𝑡, 𝑚) isdescribed by Table 1.

According to Wu [4], we have

𝑝

𝑖

𝑢=

ℎ

𝑖

𝑎,𝑡

2𝑛(𝜂

𝑖𝛾

𝑖

𝑛)

2+

𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡

2𝑛𝜂

𝑖𝛾

𝑖

𝑛

+

(𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡)

2

2(𝑛𝜂

𝑖𝛾

𝑖

𝑛)

2,

𝑝

𝑖

𝑚= 1 −

ℎ

𝑖

𝑎,𝑡

𝑛(𝜂

𝑖𝛾

𝑖

𝑛)

2−

(𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡)

2

(𝑛𝜂

𝑖𝛾

𝑖

𝑛)

2,

𝑝

𝑖

𝑑=

ℎ

𝑖

𝑎,𝑡

2𝑛(𝜂

𝑖𝛾

𝑖

𝑛)

2−

𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡

2𝑛𝜂

𝑖𝛾

𝑖

𝑛

+

(𝑟𝑓− 𝑞

𝑖− (1/2) ℎ

𝑖

𝑎,𝑡)

2

2(𝑛𝜂

𝑖𝛾

𝑖

𝑛)

2,

(13)

where 𝑖 = 1, 2 represent the 𝑖 asset, and ℎ𝑖𝑎,𝑡

is the approx-imation volatility of asset 𝑖 at time 𝑡, and 𝜂𝑖 is the jumpparameter of asset 𝑖. Cakici and Topyan [3] modified theforward-building process and used interpolated variancesonly during the backward recursion to make the algorithmmore efficient. They adopted only real node maximum andminimum variances, not the interpolated ones that fellbetween themaximumandminimumvariances. It is intuitiveto use interpolation for 𝐾 points in the backward procedure.At the terminal time 𝑇, the two-asset American strangleoption’s cash flow is

max {[𝐾1−max (𝑆1

𝑡, 𝑆

2

𝑡)]

+

, [max (𝑆1𝑡, 𝑆

2

𝑡) − 𝐾2]

+

} . (14)

Let 𝐶𝑎,𝑡(𝑚, 𝑘) be the 𝑘th option price at the node (𝑚, 𝑘),

for 𝑘 = 1, 2, . . . , 𝐾, and the variance is ℎ𝑖𝑎,𝑡(𝑚, 𝑘), 𝑖 = 1, 2.

Note that the boundary condition for a two-asset Americanstrangle option with strike𝑋 which expires in period 𝑇 is

𝐶𝑎,𝑇

(𝑚, 1) = 𝐶𝑎,𝑇

(𝑚, 2) = ⋅ ⋅ ⋅ = 𝐶𝑎,𝑇

(𝑚,𝐾)

= max {[𝐾1−max (𝑆1

𝑇, 𝑆

2

𝑇)]

+

,

[max (𝑆1𝑇, 𝑆

2

𝑇) − 𝐾2]

+

} .

(15)

We apply backward recursion to establish the option price atdate 0. Consider a node (𝑚, 𝑘) at time 𝑡.Thenwe compute theoption price 𝐶

𝑎,𝑡(𝑚, 𝑘) corresponding to variance ℎ𝑖

𝑎,𝑡(𝑚, 𝑘)

at the node. Given the variance ℎ𝑖𝑎,𝑡(𝑚, 𝑘), we compute the

appropriate jump parameter, 𝜂𝑖 for each asset, by (6). Thesuccessive nodes for this variance combination are ((𝑡+1,𝑚+𝑗

1𝜂

1), (𝑡 + 1,𝑚 + 𝑗

2𝜂

2)), where 𝑗1 = 0, ±1, ±2, . . . , ±𝑛 and

𝑗

2= 0, ±1, ±2, . . . , ±𝑛. Equation (11) is used to compute the

period (𝑡+1) variance for each of these nodes. Specifically, forthe transition from the 𝑘th variance element of node (𝑡, 𝑚) to


node ((𝑡 + 1,𝑚 + 𝑗1𝜂1), (𝑡 + 1,𝑚 + 𝑗2𝜂2)), the period (𝑡 + 1)variance for each asset is given by

ℎ

𝑖,next𝑎,𝑡+1

(𝑗

1,2) = 𝛽

𝑖

0+ 𝛽

𝑖

1ℎ

𝑖


+ 𝛽

𝑖

2ℎ

𝑖


[

[

(𝑗𝜂

𝑖𝛾

𝑖

𝑛− 𝑟𝑓+ ℎ

𝑖

𝑎,𝑡(𝑚, 𝑘))

√ℎ

𝑖

𝑎,𝑡(𝑚, 𝑘) − 𝑐

𝑖,∗

]

]

2

,

(16)

where 𝑗1,2 represents the combination of 𝑗1 and 𝑗2. Linearinterpolation of the two stored option prices correspondingto the two stored variance entries closest to ℎ2,next

𝑎,𝑡+1(𝑗

2) is used

to obtain the option price corresponding to a variance ofℎ

2,next𝑎,𝑡+1

(𝑗

2) when ℎ1,next

𝑎,𝑡+1(𝑗

1) is already chosen. Let 𝐿 be an

integer smaller than𝐾 defined via

ℎ

2

𝑎,𝑡+1(𝑚 + 𝑗

2𝜂

2, 𝐿) < ℎ

2,next𝑎,𝑡+1

(𝑗

2) < ℎ

2

𝑎,𝑡+1(𝑚 + 𝑗

2𝜂

2, 𝐿 + 1) .

(17)

The interpolated option price is

𝐶

interp(𝑚) = 𝑞 (𝑗) 𝐶

𝑎,𝑡+1(𝑚 + 𝑗

1,2𝜂

1,2, 𝐿)

+ (1 − 𝑞 (𝑗)) 𝐶𝑎,𝑡+1

(𝑚 + 𝑗

1,2𝜂

1,2, 𝐿 + 1) ,

(18)

where

𝑞 (𝑗) =

ℎ

2

𝑎,𝑡+1(𝑚 + 𝑗

2𝜂

2, 𝐿 + 1) − ℎ

2,next𝑎,𝑡+1

(𝑗

2)

ℎ

2

𝑎,𝑡+1(𝑚 + 𝑗

2𝜂

2, 𝐿 + 1) − ℎ

2

𝑎,𝑡+1(𝑚 + 𝑗

2𝜂

2, 𝐿)

.

(19)

In this way an option price is identified for each of the(2𝑛 + 1)(2𝑛 + 1) jumps from node (𝑡, 𝑚) with variancecombination (ℎ1,next

𝑎,𝑡+1(𝑗

1), ℎ

2,next𝑎,𝑡+1

(𝑗

2)). In each case, either node

(𝑡+1,𝑚+𝑗

1,2𝜂

1,2) contains a variance entry (and hence option

value) that matches (ℎ1,next𝑎,𝑡+1

(𝑗

1), ℎ

2,next𝑎,𝑡+1

(𝑗

1)), or the relevant

information is interpolated from the closest two entries. Weuse the following formula to compute the unexercised optionvalue 𝐶go

𝑎,𝑡(𝑚, 𝑘):

𝐶

go𝑎,𝑡

(𝑚, 𝑘) = 𝑒

−𝑟𝑓

𝑛

∑

𝑗1=−𝑛

𝑃

1(𝑗

1)

𝑛

∑

𝑗2=−𝑛

𝑃

2(𝑗

2) 𝐶

interp(𝑚) .

(20)

Denote the exercised value of the claim by 𝐶𝑎,stop𝑡

(𝑚, 𝑘). Fora two-asset American strangle option with strikes𝐾

1and𝐾

2,

𝐾1< 𝐾2,

𝐶

stop𝑎,𝑡

(𝑚, 𝑘) = max {[𝐾1−max (𝑆1

𝑡, 𝑆

2

𝑡)]

+

,

[max (𝑆1𝑡, 𝑆

2

𝑡) − 𝐾2]

+

} .

(21)

The value of the claim at the 𝑘th entry of node (𝑡, 𝑚) is then

𝐶𝑎,𝑡

(𝑚, 𝑘) = max {𝐶go𝑎,𝑡

(𝑚, 𝑘) , 𝐶

stop𝑎,𝑡

(𝑚, 𝑘)} . (22)

The final option price, obtained by backward recursion of thisprocedure, is given by 𝐶

𝑎,0(0, 1).

Table 2: Numerical results for Example 1.

𝑇 𝑛 Option prices with lattice Option prices and intervalswith LSM1 0.014987254820724

0.029606167588959[0.02804 0.03117]

2 0.0301905421906265 4 0.030393224336490

7 0.03098133383516010 0.0318638791580631 0.068487810335143

0.081499912354419[0.07999 0.08500]

2 0.0812540393470337 4 0.083605936631716

7 0.08168016349462510 0.0803013508874821 0.178948896057298

0.193857807991106[0.18842 0.19929]

2 0.19232946576476610 4 0.195314383716280

7 0.19986802959924910 0.193058780881166

4. Numerical Examples

In this section, several examples are implemented using thelattice method in this paper and least square Monte-Carlomethod (LSM) developed by Longstaff and Schwartz [14].

In Examples 1, 2, and 3, we focus on the single assetAmerican strangle options under GARCH model where theconvergence with respect to 𝑛 and𝐾 are studied, respectively,in the first two examples, and the optimal exercise boundariesare drawn for the third example. In Examples 4 and 5,we compute the two-dimensional assets American strangleoptions.

In Tables 2 and 3, the prices of the options using LSMwith 5,000 paths are calculated and the intervals that thetrue prices fall into are provided. From the comparisons weconfirm that the lattice methods developed in this paper arecorrect and reliable. Furthermore from Table 2 we observethat the lattice method converges as 𝑛 goes larger and fromTable 3 the latticemethod converges as𝐾 goes larger. Figure 2shows exercise and holding regions: the middle part is theholding region and the top and bottom parts are the exerciseregions.

Example 1. Consider single asset GARCH model withparameters 𝑟

𝑓= 5%, 𝑞 = 10%, 𝛽

0= 6.575 × 10

−6, 𝛽1= 0.9,

𝛽2= 0.04, 𝑆

0= 100, ℎ

0= 0.0001096, 𝐾

1= 105, 𝐾

2= 95,

𝛾 = ℎ0, and 𝑐∗ = 0. Fixing 𝐾 = 20, we investigate the

convergence behavior as 𝑛 increases.

Example 2. Consider single asset GARCH model with thesame parameters as Example 1. In this example, 𝑛 = 5 and thesensitivity to the volatility space parameter, 𝐾, is explored.

Example 3. Consider single asset GARCH model with thesame parameters as Example 1. Draw the figure of the optimalexercise boundaries for American strangle.

Discrete Dynamics in Nature and Society 5


𝑇 𝐾 Option prices with lattice Option prices and intervalswith LSM

5

2 0.028187456298155

0.029606167588959[0.02804 0.03117]

4 0.0290905247066806 0.02993056593309910 0.03019054219062620 0.02939107453988840 0.029489977773055

10

2 0.197070210662258

0.193857807991106[0.18842 0.19929]

4 0.1957006391752776 0.19646507165313010 0.19232946576476620 0.19369605425115140 0.194423204241922

30

2 1.182560819901510

1.205198709585444[1.19372 1.22670]

4 1.1978165406296246 1.20035077272782810 1.20271426767471120 1.20340751198770640 1.203545193979154

6 8 10 12 14 16 18 20

120

115

110

105

100

95

90

85

80

75

70

T

S

American strangle call sideAmerican strangle put side

Figure 2: Optimal exercise boundaries for Example 3.

In Examples 4 and 5, we examine the stochastic latticemethods for pricing American strangle options under multi-asset under stochastic volatilitymodel where the convergencewith 𝑛 and𝐾 are studied. From the numerics in Tables 4 and5, we confirm that the lattice methods for two-dimensionalmodels are correct and reliable and the convergence of thelattice methods with respect to 𝑛 and𝐾 is observed.

Example 4. Consider two-asset American strangles withparameters 𝑟

𝑓= 5%, 𝑞1 = 𝑞2 = 10%, 𝛽1

0= 𝛽

2

0= 6.575×10

−6,𝛽

1

1= 𝛽

2

1= 0.9, 𝛽1

2= 𝛽

1

2= 0.04, 𝑆1

0= 𝑆

2

0= 100,


𝑇 𝑛 Option prices with lattice Option prices and intervalwith LSM

3

1 0.005974297

0.005065587636226[0.00261 0.006316]

2 0.0042236923 0.0045023514 0.0055303555 0.005040507

4

1 0.014427788

0.015018277749022[0.01111 0.01892]

2 0.0145900563 0.0139847184 0.0172233645 0.015988485

5

1 0.022129134

0.034841312405081[0.02844 0.041235]

2 0.0305076363 0.0308532764 0.0355945385 0.034694665


𝑇 𝐾 Option prices with lattice Option prices and intervalswith LSM

3

2 0.005974297

0.005065587636226[0.00261 0.006316]

4 0.0059742976 0.0059742978 0.00597429710 0.005974297

4

2 0.014427788

0.015018277749022[0.01111 0.01892]

4 0.0144277886 0.0144277888 0.01442778810 0.014427788

5

2 0.02194509

0.034841312405081[0.02844 0.041235]

4 0.0221291346 0.0221642268 0.02435598410 0.024276964

ℎ

1

0= ℎ

2

0= 0.0001096, 𝐾

1= 105, 𝐾

2= 95, 𝛾1 = 𝛾2 = ℎ1

0, and

𝑐

∗,1= 𝑐

∗,2= 0. Fixing 𝐾 = 4, we investigate the convergence

behavior as 𝑛 increases.

Example 5. Consider the two-dimensional GARCH modelwith the same parameters as Example 4. Fixing 𝑛 = 1, westudy the sensitivity to the volatility space parameter 𝐾.

5. Conclusions

In this paper we studied pricing methods for stochasticvolatility models of the American strangles with single assetand multiassets. Both lattice methods and LSM methods aredeveloped and implemented. To the best of our knowledge,


there are no results on the lattice methods for multidi-mensional stochastic volatility models. We first extendedthe stochastic lattice methods invented by Ritchken andTrevor [2] which are for one-dimensional GARCH modelsof American call to the multidimensional GARCH modelsof American strangles. Numerical examples confirm thecorrectness and reliability of the lattice methods. Future chal-lenging works include the development of the latticemethodsfor multidimensional volatility models with correlations andrecently developed models (e.g., [15]). One possible solutionto the case of correlation is to adopt the idea (using moment-generating function) in [7]. However it needs to develop newtechniques when the stochastic volatilitymodels are involved.Furthermore, a dimensional-reduction technique should bedeveloped to reduce the computational cost.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant no. JBK130401).

References

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[2] P. Ritchken and R. Trevor, “Pricing options under generalizedGARCH and stochastic volatility processes,” Journal of Finance,vol. 54, no. 1, pp. 377–402, 1999.

[3] N. Cakici and K. Topyan, “The GARCH option pricing model:a lattice approach,” Quantitative Finance, vol. 2, pp. 432–442,2000.

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[5] E. Akyildirim, Y. Dolinsky, and H. Soner, “Approximating sto-chastic volatility by recombinant trees,” http://arxiv.org/abs/1205.3555v1.

[6] P. Boyle, “A lattice framework for option pricing with two statevariables,” Journal of Financial and Quantitative Analysis, vol.23, pp. 1–2, 1988.

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[10] K.-S. Moon, W.-J. Kim, and H. Kim, “Adaptive lattice methodsfor multi-asset models,” Computers & Mathematics with Appli-cations, vol. 56, no. 2, pp. 352–366, 2008.

[11] C. Chiarella and A. Ziogas, “Evaluation of American strangles,”Journal of Economic Dynamics & Control, vol. 29, no. 1-2, pp.31–62, 2005.

[12] F. Moraux, “On perpetual American strangles,” Journal ofDerivatives, vol. 16, no. 4, pp. 82–97, 2009.

[13] J.-C. Duan, “The GARCH option pricing model,”MathematicalFinance, vol. 5, no. 1, pp. 13–32, 1995.

[14] F. A. Longstaff and E. S. Schwartz, “Valuing American optionsby simulation: a simple least-squares approach,” Review ofFinancial Studies, vol. 14, no. 1, pp. 113–147, 2001.

[15] C. X. Huang, X. Gong, X. Chen, and F. H.Wen, “Measuring andforecasting volatility in Chinese stock market using HAR-CJ-M model,” Abstract and Applied Analysis, vol. 2013, Article ID143194, 13 pages, 2013.

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Research Article Lattice Methods for Pricing American Strangles …downloads.hindawi.com/journals/ddns/2014/165259.pdf · 2019. 7. 31. · valuable to price American style options