!!Muthuraman-American Options Under Stochastic Volatility

7/28/2019 !!Muthuraman-American Options Under Stochastic Volatility

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OPERATIONS RESEARCHVol. 59, No. 4, JulyAugust 2011, pp. 793809

issn 0030-364X eissn 1526-5463 11 5904 0793 http://dx.doi.org/10.1287/opre.1110.0945 2011 INFORMS

American Options Under Stochastic Volatility

Arun ChockalingamSchool of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907, [email protected]

Kumar MuthuramanMcCombs School of Business, The University of Texas at Austin, Austin, Texas 78712, [email protected]

The problem of pricing an American option written on an underlying asset with constant price volatility has been studiedextensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatilitymodels are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature forpricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American optionsunder stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformationprocedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence.Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of optionprices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset and

stochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.Subject classifications: American option; stochastic volatility; free boundary.

Area of review: Financial Engineering.History : Received August 2008; revisions received January 2010, July 2010, August 2010; accepted October 2010.

1. Introduction

Options are contracts that give the holder the right to sell

(put) or buy (call) an underlying asset at a predetermined

strike price. A European option allows the holder to exer-

cise the option only on a predetermined expiration date,

while an American option allows the holder to exercise the

option at any point in time until the expiration date. Optionpricing has always played a prominent role in financial the-

ory as well as real derivative markets. In a celebrated paper,

Black and Scholes (1973) derive a closed-form solution for

the price of a European option by characterizing the price

as the expected payoff under a risk-neutral measure. Their

model assumes that the underlying asset price follows a

geometric Brownian motion with constant volatility.

Even under the constant volatility assumption, that is,

the classical Black-Scholes setting, closed-form solutions

do not exist for American options. Due to the possibility

of early exercise, the American option price depends on

the optimal-exercise policy, which can be represented by

an exercise boundary (also known as the free boundary) onthe price-time space. The exercise boundary partitions the

price-time space into hold and exercise regions.

Most of option pricing literature consider the con-

stant volatility model. Rubinstein (1994), however, provides

empirical evidence, using implied volatilities obtained from

index options on the S&P 500, that suggests that the con-

stant volatility assumption does not hold. Using data for

the OEX contract, Broadie et al. (2000) find that dividends

alone are not accountable for all aspects of option pric-

ing and exercise decisions, and they suggest that stochastic

volatility needs to be included as well. Furthermore, Scott(1987) and the references therein provide ample evidence ofvolatility changing over time. This can also be readily seenfrom the implied volatilities calculated from market prices.Implied volatilities are the volatilities that, when used inthe Black-Scholes formula, provide European option pricesconsistent with option prices observed in the market. When

implied volatilities are plotted against strike prices, the plotsexhibit a smile effect, which refers to the resulting U-shaped curve, as opposed to a straight line that one wouldexpect if asset prices had constant volatility. Implied volatil-ities for in- and out-of-the-money options are observed tobe higher than at-the-money options. Assuming constantvolatility therefore leads to considerable mispricing. Hence,models that allow the volatility of the underlying assetprice to be stochastic are needed to better capture marketbehavior.

Several models have been proposed to better model theevolution of volatility. One approach has been to use ARCHmodels and their variants (Bollerslev et al. 1992). Using

another diffusion process to model volatility, however, hasbecome a more popular choice. Hull and White (1987),Scott (1987), Stein and Stein (1991), and Heston (1993)each propose different diffusion processes to represent thedynamics of asset price volatility.

1.1. Relevant Literature

The pricing of American options under constant volatil-ity and European options under stochastic volatility haveboth received considerable attention in literature. Numer-ical methods available to compute the price and/or

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Chockalingam and Muthuraman: Stochastic Volatility794 Operations Research 59(4), pp. 793809, 2011 INFORMS

exercise boundary for American options under constant

volatility include simulation-based techniques (Broadie and

Glasserman 1997, Longstaff and Schwartz 2001), bino-

mial lattices (Cox et al. 1979), partial differential equa-

tion (PDE) solution methods (Brennan and Schwartz

1977, Muthuraman 2008), front-fixing methods (Wu and

Kwok 1997, Nielsen et al. 2002), front-tracking meth-ods (Pantazopoulos et al. 1998), and integral methods

(Kim 1990, Jacka 1991, Carr et al. 1992). Myneni (1992),

Karatzas and Shreve (1998), and Broadie and Detemple

(2004) provide an overview of American option pricing

under the Black-Scholes setting.

The pricing ofEuropean options under stochastic volatil-

ity has been looked at for various choices of diffusion

dynamics to represent the volatility process. Hull and White

(1987) use a lognormal process, while Scott (1987) and

Stein and Stein (1991) use a mean-reverting Ornstein-

Uhlenbeck (OU) process. Closed-form solutions have been

obtained in Heston (1993) when volatility is modeled as

a square-root process. Ball and Roma (1994) also use thesame square-root process.

Relatively little attention has been paid to the problem

of pricing an American option under stochastic volatility.

Literature on American options under stochastic volatil-

ity can be classified into PDE-based and non-PDE-based

approaches. The PDE methods solve the free-boundary

problem arising from the use of classical dynamic pro-

gramming arguments and provide the entire price function

and the optimal-exercise policy explicitly. Non-PDE-based

approaches compute the price for any given time, asset

price and underlying volatility by computing the condi-

tional expectation under a suitable martingale measure.A popular approach to solve the related free-boundary

PDE problem is to reformulate it as a linear complemen-

tarity problem (LCP). The projected successive over relax-

ation (PSOR) method proposed by Cryer (1971) is widely

used to solve these LCPs. Clarke and Parrott (1999) use

a stretching transformation and an adaptive-upwind finite

difference approximation to discretize the LCP, resulting

in the need to solve many discrete complementarity prob-

lems. A multigrid iteration method is developed to solve

these problems. Oosterlee (2003) states that the projected

line Gauss-Seidel smoother used in Clarke and Parrott

(1999) is too involved, and studies alternate smoothers

that can be used in conjunction with the multigrid itera-tion method, finding that an alternating line Gauss-Seidel

smoother is a better choice, bringing about better conver-

gence. Oosterlee (2003) also improves upon Clarke and

Parrott (1999) using a recombination of iterants. Ikonen and

Toivanen (2004) solve each of the discrete complementar-

ity problems obtained after time and space discretization

using operator splitting methods. The method divides each

time step into two fractional steps, integrates the PDE with

an auxiliary variable over the time step, then updates the

solution to satisfy the linear complementarity conditions

due to the early-exercise constraint. Componentwise split-

ting methods are utilized in Ikonen and Toivanen (2007a) to

solve the discrete complementarity problems. The compo-

nentwise splitting method splits each discretized LCP into

three LCPs, and the use of Strang symmetrization further

decomposes the three LCPs into five LCPs. Each LCP con-

sists of tridiagonal matrices and is solved using the Brennanand Schwartz (1977) algorithm. Another technique, known

as the penalty method, replaces the unknown free-boundary

with a nonlinear penalty term and solves the resulting non-

linear fixed-boundary problem. Zvan et al. (1998) use a

standard Galerkin finite element method to discretize the

arising PDE and use penalties to force the discrete prob-

lems to satisfy the early-exercise constraint. The authors

highlight the equivalence of the penalty method and the

linear complementarity formulation. Ikonen and Toivanen

(2007b) compare the five methods for pricing American

options under stochastic volatility and find that, while the

error in prices computed by any of these methods is com-

parable, the componentwise splitting method is consider-ably faster. They do, however, acknowledge that the PSOR

method is the easiest to implement, while the component-

wise splitting method is the most difficult to implement.

As for non-PDE approaches, a nonparametric approach

is utilized in Broadie et al. (2000). Several simulation meth-

ods such as the least-squares Monte Carlo approach in

Longstaff and Schwartz (2001), the primal-dual simulation

algorithm of Andersen and Broadie (2004), and the stochas-

tic mesh method of Broadie and Glasserman (2004) that

have all been developed for American options under con-

stant volatility can also be adapted to compute American

option prices under stochastic volatility.

1.2. Contribution and Outline

In this paper, we show that the problem of pricing

American options under stochastic volatility can, in fact, be

transformed into a sequence of European-type option pric-

ing problems. More specifically, the free-boundary problem

arising in the pricing of American options can be trans-

formed into a sequence of fixed-boundary problems. By

a European-type option, we mean that the exercise pol-

icy is set a priori in the option contract. It is important

to note that solving fixed-boundary problems is not very

difficult in three dimensions if one leverages on existingPDE solver packages or libraries that are available to solve

fixed-boundary problems. Software packages such as Com-

sol even allow the reuse of contours from one solution as

boundaries for another problemmaking the implementa-

tion of our method very easy. We also prove the conver-

gence of the sequence of fixed-boundary problems. Our

results show the methodology proposed here is faster than

other available methods while being as accurate as the

PSOR method. Moreover, all it involves is solving linear

PDEs in domains with fixed boundaries.


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Chockalingam and Muthuraman: Stochastic VolatilityOperations Research 59(4), pp. 793809, 2011 INFORMS 795

The approach we present differs in several ways fromthe other papers on American option pricing under stochas-tic volatility. First, the model we consider is far moregeneral. The numerical methods listed above are tailoredfor the Heston (1993) stochastic volatility model. Themethodology presented here works for the Hull and White

(1987), Scott (1987), and Stein and Stein (1991) modelsas well. An assumption that is common to most existingpapers is that the market price of volatility risk is zero.As will be discussed in 2.2, the price of an Americanoption is, in general, not unique and depends on the mar-ket price of volatility risk. Most of these papers can beextended to handle nonzero volatility risk premiums. How-ever, because they do not consider nonzero market price ofvolatility risk, they could not study the effects of volatil-ity risk premium on the exercise policy and the price.

The approach described in this paper readily accommodatesnonzero volatility risk premiums, and we study its effects.Several of the existing papers also use a wrong boundary

condition, which we correct.In terms of underlying methodology, the solution tech-

nique proposed in this paper is very different from any thatis available for pricing American options under a stochasticvolatility setting. Methods, like the penalty method, covertthe linear PDE with a free boundary to a nonlinear PDEwith a fixed boundary. On the other hand, the transforma-

tion we propose retains the linear PDE but solves it withinfixed boundaries in each iteration. Because the complexityof dealing with a single nonlinear PDE is much harder thandealing with a sequence of linear PDEs, it is not surprisingthat runtimes are much better in our case. Methods like thecomponentwise splitting method introduce new approxima-

tions other than the ones due to discretization, while oursdoes not. The accuracy depends only on the method usedto solve the fixed-boundary problem.

The method proposed in this paper extends the class of

methods that are being called moving-boundary methods.Such methods were developed initially for singular con-trol problems (Muthuraman and Kumar 2006, Kumar andMuthuraman 2004) and were demonstrated to work numer-ically for higher-dimensional cases. For American optionpricing in the classical setting, Muthuraman (2008) extendsthis method and provides theoretical guarantees. This paperextends the method to optimal stopping problems in ahigher setting. Most importantly, due to the challenges

in dealing with higher dimensions theoretically, there hadbeen absolutely no theoretical guarantees for the methodin any problem with more than one space dimensionality.This paper provides the first set of such guarantees.

The layout of the paper is as follows. Section 2 presentsthe model formulation. The transformation procedure ispresented in 3. A computational illustration of the pro-cedure is provided in 4, together with insights intohow stochastic volatility affects option pricing. Speed andaccuracy comparisons are also presented in 4. We con-

clude in 5. All proofs are collected in Appendix A,

and a detailed discussion of the finite difference schemeused to solve the fixed-boundary problem is presented inAppendix B.

2. Model Formulation

We start this section with a discussion on the use of a

second stochastic process to model the evolution of volatil-ity. We then formulate the free-boundary problem thatthe American option price and the optimal-exercise policyjointly solve.

2.1. Stochastic Volatility Models

In stochastic volatility models, asset price (Xt) evolution isrepresented by

dXt = Xt dt + tXt dWt (1)where is the constant mean rate of return and Wt is astandard Brownian motion (a Wiener process). The instan-taneous volatility at time t is represented by t. The volatil-

ity t = f Yt , where Yt is another stochastic process andf is a nonstochastic function. The evolution of Yt isrepresented by

dYt = YYt dt + YYt dZt (2)where Y and Y are nonstochastic functions. In Equa-tion (2), Zt is another standard Brownian motion correlatedwith Wt. We assume a constant correlation 1 1,i.e., dWt dZt = dt. Hence, Zt can be written as a lin-ear combination of Wt and an independent Wiener processZt such that Zt = Wt +

1 2Zt. Here, we have two

sources of randomness, namely Wt and Zt, but only onetradeable asset, leading to market incompleteness. We refer

readers to Bjrk (2004) and Fouque et al. (2000) for fur-ther discussions on stochastic volatility models and marketincompleteness.

Much of the literature on stochastic volatility focuses ona few specific models (Table 1). Financial data show that < 0 (Fouque et al. 2000). Of the four models, the Hestonmodel is the most popular and the only one that allowsfor a nonzero correlation. Also, Dragulescu and Yakovenko(2002) provide evidence that the Heston model is in agree-ment with real-world data, leading to wider adoption of themodel by researchers and practitioners. Thus, for exposi-tional ease, we will restrict much of our attention to theHeston model and provide additional comments relevant to

the other models when necessary. The transformation pro-cedure developed in this paper works for all models listedin Table 1.

Table 1. Stochastic volatility models.

Yy Yy f y

Heston (1993) m y y y = 0Stein and Stein (1991) m y y = 0Scott (1987) m y ey = 0Hull and White (1987) c1y c2y

y = 0


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2.2. American Option Pricing

Consider an American put option written on an underly-ing asset with price Xt at time t given by Equation (1).The instantaneous volatility t is represented by the Hes-ton model. The put option with strike price K and matu-rity time T written on this asset pays max K Xt 0 K Xt+ at any time t 0 T . The price of an option,pxy, is represented as a function of the time to expiry, T t, the underlying asset price x and the value, y,of the process Yt. As noted earlier, the market consideredis an incomplete one as a result of volatility not being atradable asset.

Assuming that the market selects a unique equivalentmartingale measure , derivatives should be priced withrespect to this measure to disallow arbitrage giving

pxy

= sup

0

E

er KXT+

X=xY=y

(3)

where r > 0 is the risk-free rate of interest. As mentionedearlier, when the volatility of the asset price is constant,the optimal-exercise policy can be represented by a contin-uous, nonincreasing exercise boundary (e.g., Karatzas andShreve 1998). Analogous to the constant volatility case, theoptimal-exercise policy under stochastic volatility can berepresented by a surface (Fouque et al. 2000). The exercisesurface is a continuous, nonincreasing surface x = by,b + +. This boundary partitions the state spaceand dictates the optimal-exercise policy. The region whereit is optimal to hold, known as the continuation region, isdefined as

=xy

2

+

x > by, and the

region where it is optimal to exercise, known as the stop-ping region, is defined as = xy 2+ x by.

In the continuation region, the price of the Americanoption satisfies the PDE

1

22t x

2 2p

x2+ v2t x

2p

xy+ 1

2v22t

2p

y2+ r x p

x

+ m y vtp

y rp p

= 0 (4)

for all xy .In Equation (4), denotes the market price of volatil-

ity risk. Because volatility cannot be traded, we have anincomplete market under the stochastic volatility setting.Although we lose the uniqueness of derivative pricing, forany given there is a unique price p. An infinitesimalincrease in the volatility risk t increases the infinitesi-mal rate of return on the option by . The market priceof volatility risk and its effect on pricing is discussedin 4.3. For a detailed discussion on the interpretationof and its relation to equivalent martingale measureswe refer the reader to Bakshi and Kapadia (2003) andHenderson (2005).

In the exercise region, the price is the payoff

pxy = K x+ (5)

for all xy . For notational convenience, we definethe differential operator such that the LHS of Equa-tion (4) is denoted by p.

To solve Equation (4) in the region , the followingboundary conditions are needed:

p0xy = K x+ (6)pbyy = K by+ (7)

limx

p

x= 0 (8)

limy

p

y= 0 and (9)

p rx px

+ m py

rp p

= 0 at y = 0 (10)

Figure 1 illustrates the state space and boundaryconditions.

Equations (6) and (7) prescribe the value at expiry andat exercise. As the underlying asset price increases, theprobability that the asset price falls below K, before orat expiry, decreases. This increasingly guarantees a zeropayoff. Equation (8) reflects this behavior of the option.Equation (9) captures the argument that when volatility isextremely large, a marginal increase in volatility has littleeffect on the price.

The boundary condition at y = 0 is directly derivedby taking y = 0 in Equation (4). Several papers (includ-ing Clarke and Parrott 1999, Oosterlee 2003, Ikonen andToivanen 2004, Ikonen and Toivanen 2007a) that developinnovative numerical methods to price American optionsunder the Heston model argue that when volatility is zero,since there is no randomness, the payoff as well as theprice are deterministic, leading to px 0 = K x+.However, this is not the case because Yt is a mean-revertingprocess in the Heston model, meaning that if Yt = 0, then

Figure 1. The state space and boundary conditions.

y

0

K

x

p = 0

limx

p

x= 0

limy

py= 0

p= Kx+

p= K by+


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almost surely dYt is positive, and hence Yt+ is greaterthan zero, making the asset price process nondeterministic.Therefore, the use of px 0 = K x+, although eas-ier to handle, is incorrect and needs to be replaced by thePDE in Equation (10).

It is important to note that Equations (4)(10) can be

solved for any sufficiently smooth surface, i.e., exercisepolicy b. The optimal-exercise policy, however, is the onlypolicy for which p is smooth across the boundary b. Thiscondition, commonly known as the smooth pasting con-dition, implies that p as well as p/x are continuousacross b and gives rise to Fouque et al. (2000):

limxb

p

x= 1 (11)

for optimality. A solution pxyby to Equa-tions (4)(11) also has to satisfy the condition

pxy K

x+

xy

(12)

in order to be the true price function and optimal-exercisepolicy.

2.3. Other Stochastic Volatility Models

The structure of the pricing problem described by Equa-tions (4)(11) remains the same for the other modelsdiscussed in 2.1. However, the specific PDE and theboundary condition given in Equation (10) are model spe-cific. This is due to the differences in the evolution of Yt,its domain, and the function f that determines thevolatility.

Table 2 summarizes the PDE that replaces Equation (4)and the boundary condition that replaces Equation (10) foreach model.

3. Transforming the Free-BoundaryProblem

In this section, we first present the transformation pro-cedure that solves the free-boundary problem described

Table 2. Boundary conditions.

Model PDE boundary condition, boundary

Stein and Stein (1991) 12

2t x2 2

px2

+ t x 2

pxy

+ 12

2 2

py2

+ rx px

+ m y py

rp p

= 0p

y= 0 y

Scott (1987)1

22t x

2 2p

x2+ t x

2p

xy+ 1

22

2p

y2+ rx p

x+ m y p

y rp p

= 0

1

22

2p

y2+ rx p

x+ m p

y rp p

= 0 y

Hull and White (1987)1

22t x

2 2p

x2+ c23t x

2p

xy+ 1

2c22

4t

2p

y 2+ rx p

x+ c12t c22t

p

y rp p

= 0

p = K x+ y = 0

in 2.2. We then demonstrate the mechanics of the trans-formation procedure with a numerical illustration.

The American option pricing problem is defined byEquations (4)(11) and is satisfied by the price functionpxy and the optimal-exercise policy by. Nowconsider an arbitrary policy defined by a surface b0y.

When such an arbitrary exercise policy is used by an optionholder, the value to the holder will be referred to as theassociated value p0xy. Clearly the price is the asso-ciated value of using the optimal-exercise boundary, thatis, p0xy = pxy when b0 = b and p0xy pxy for any b0.

For an arbitrary exercise policy b0, one can find theassociated price p0 by solving Equations (4)(10), whichcan be done using standard PDE techniques such as finitedifference and finite element schemes. A finite differencescheme for computing the associated values is detailedin Appendix B. For a given b0, the uniqueness of theassociated value function is established by the following

proposition.Proposition 3.1. If pn satisfies Equation (4) with theboundary conditions given by Equations (6)(10) for agiven bny, then pn is unique.

All proofs are collected in Appendix A.Our aim is to construct a transformation procedure that

will converge and provide the price of the option onconvergence. Starting from a guess b0, if we can con-struct a sequence of policies b0 b1 that are monotonicincreasing, i.e., bny < bn+1y for all ( y) and isbounded above, then convergence is inevitable. Exercise-policy improvement would also imply that pnxy K.

An increase in correlation can be thought of as adecrease in overall uncertainty in the system. When

out-of-the-money (x > K), it is only natural that the holder

Figure 10. Effects of changing correlation .

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00

0.5

1.0

1.5

2.0

Stock price, x

Optionprice,p

= 1

= 0

= 1

(a) Price functions (= 1.5 months and t= 0.4)

0 1 2 36.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

Time to expiry,

Stockprice,x

= 1.5 months

(b) Exercise policies (t= 0.4)

= 1

= 0

= 1

prefers more randomness to increase chances of a posi-

tive payoff, hence explaining option price decreases with

increases in correlation when out-of-the-money. When in-

the-money (x < K), the holders preference for less ran-

domness is indicated by the increase in price with increases

in correlation.

Exercise boundaries decrease as correlation increases, as

shown by Figure 10(b). This aligns with the notion that

under optimality, higher prices imply later exercise, becauseat the exercise boundary the payoff is always K x+.

4.3. Impact of Market Price ofVolatility Risk on Option Pricing

Next, lets consider the effect of , the market price of

volatility risk. An interpretation for can be obtained by

considering delta-hedged option portfolios that are con-

structed by buying an option and hedging it with a fraction

of the underlying asset, such that in the Black-Scholes set-

ting, the rate of return matches the risk-free interest rate.


11/17


The gain on this portfolio is the difference between the risk-free rate and the earnings from the portfolio. Obviously, ina Black-Scholes setting, when the delta-hedged portfolio iscontinuously rebalanced, the delta-hedged gain is always

zero. Bakshi and Kapadia (2003) study and relate the delta-hedged gains to the market price of volatility risk. They

show that when the market price of volatility risk is pos-itive (negative), the expected delta-hedged gain is positive(negative). It also implies that when volatility is stochastic

and volatility risk not priced, i.e., when = 0, the expecteddelta-hedge gain is zero. Furthermore, using market data,they demonstrate empirically that is negative. The find-ing that is negative is consistent with the notion thatmarket volatility rises when market return drops.

Figure 11(a) shows that as increases, p decreases.

Henderson (2005) shows that for European put options, asthe market price of volatility risk increases, the price ofthe option decreases. The question is whether this trans-lates to American option pricing as well. Figure 11 provides

Figure 11. Effects of changing volatility riskpremium .

8 9 10 11 12 13 140

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Stock price, x

O

ptionprice,p

= 2

= 0

= 2

0 1 2 37.0

7.5

8.0

8.5

9.0

9.5

10.0

Time to expiry,

Stockprice,x

= 1.5 months

= 2

= 0

= 2

(a) Price functions (= 1.5 months and t= 0.4)

(b) Exercise policies (t= 0.4)

numerical evidence that it does hold true for American putoptions as well.

An insight into this behavior is revealed by consideringthe delta-hedged portfolio. When the market does not pricevolatility risk, the delta-hedged gain should be zero, andthe price of the option matches the price of constructing the

portfolio. When the market does price volatility risk, how-ever, the delta-hedged gain is proportional to the volatilityrisk premium. Hence, as increases, the expected gainon this portfolio also increases. In this case, the price of

the option would be the price of constructing the portfoliominus the expected gain.

4.4. On Implied Volatilities

One of the primary arguments favoring stochastic volatil-ity models is that the implied volatilities computed from

observed option prices are not constant and often exhibit asmile curve when plotted against strike prices. We exam-ine the nature of implied volatility curves and their depen-

dence on both correlation and market price of volatility riskin Figure 12. Implied volatilities are computed for variousstrike prices from 8 to 12, at an underlying stock pricex = 11. We use a bisection search in conjunction with thebinomial-tree method to calculate the implied volatilities.

The smile curves are well captured in Figure 12(a). Theypivot at K = 11, which is the underlying stock price atwhich the implied volatilities are computed. When K < x,

the implied volatility decreases as correlation increases.On the other hand, when K > x, a decrease in correla-tion causes a decrease in the implied volatility as well.

From 4.2, we know that for K < x, as increases, p

decreases. This translates directly to decreasing impliedvolatility. A drop in p therefore implies a drop in volatility.By a similar reasoning, because p increases as increases

for K > x, the monotonic relationship between option priceand volatility leads to the implied volatility increasing.

The smile curves in Figure 12(b) demonstrate that as increases, the implied volatility decreases. Recall from

4.3 that as increases, the option price p decreases.This decrease in option price leads to a lower impliedvolatility, because option prices increase monotonically

with volatility.

4.5. Runtime and Accuracy Comparisons

Ikonen and Toivanen (2007b) exhaustively compare the fivemethods available in literature for pricing American optionsunder stochastic volatility. These are the PSOR, multigrid,

operator splitting, penalty, and the componentwise split-ting (CS) method. Of these, they find that CS performsfastest, with comparable accuracy to the other methods.From the standpoint of implementation, the authors note

that the PSOR method is the easiest and CS is the hardest.The remaining three methods fall in between these two interms of speed/accuracy and ease of implementation, high-

lighting a trade-off between ease of implementation and the


12/17


Figure 12. Effects of changing and on impliedvolatilities.

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00.20

0.25

0.30

0.35

0.40

0.45

Strike price, K

Impliedvolatility

= 0.8

= 0.5

= 0

= 0.5

= 0.8

(a) Changing

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

Strike price, K

Impliedvolatility

= 2

= 1

= 0

= 1

= 2

(b) Changing

speed/accuracy of the method. In this section, our objec-

tive is to place the method presented in this paper. To thisextent, we compare the speed and accuracy to that of only

the CS method and the PSOR method, because the other

methods are known to fall between these two, both in termsof speed/accuracy and implementation ease. The compar-

isons were carried out using a C++ implementation withthe GMM

++library on a 2.8 GHz Intel Xeon Mac Pro

with 2 GB RAM and 1.6 GHz bus speed.It is also important to keep in mind that implementa-

tion of the proposed method is easy and straightforward

because it solves only a fixed-boundary PDE problem in

each iteration, and that the PDE is the classical convection-diffusion PDE with a second-order cross-derivative term.

Several off-the-shelf packages can do this readily. We also

provide in Appendix B a simple finite difference schemethat solves the fixed-boundary problem that is used for the

results in this section. In the finite difference scheme, time

stepping is done using an implicit Euler method. Note that

the method proposed here, in its current form, works forall the popular stochastic volatility models and allows forboth positive and negative volatility risk premiums.

As earlier, for our method, the first boundary guess isb0y = 1 for all and y, and we take the parameter setthat has been used in the illustration. The relaxation param-eter and stopping criterion used for the PSOR method arethe same as those found in Ikonen and Toivanen (2007b).Tables 3 and 4 list the prices obtained using the respectivemethods for five initial asset prices and two volatilities forvarious grid sizes. For the true values, we use the valueslisted in Ikonen and Toivanen (2007b), which the authorsobtain from using the CS method in conjunction with avery fine grid.

Figure 13 plots the root-mean-square errors (RMSE) andruntimes for the three different methods for the variousgrid sizes listed in Tables 3 and 4. The performance of theCS method in relation to the PSOR method is comparableto the relation found in Ikonen and Toivanen (2007b). Asthe figure shows, for the same accuracy the transformationprocedure is on average 10 times faster than the PSORmethod and more than twice as fast as the componentwisesplitting method.

As the figure shows, the transformation procedure hasbetter accuracy than the CS method, particularly on coarsergrids. This is understandable because the CS method intro-duces additional errors when the splitting is performed. Atthe same time, the speed of our scheme is greater thanthat of the PSOR method and the CS method. As men-tioned before, the CS method is harder to implement thanthe PSOR method, but this leads to the better performanceof the CS method. It is interesting to note, however, thatimplementing our transformation procedure is no more dif-

ficult than implementing the PSOR method because ourtransformation procedure is essentially a standard finite dif-ference implementation, modified to update the boundaryduring each iteration.

The speed and accuracy of option prices obtained usingthe transformation procedure are highly dependent on thechoice of the fixed-boundary problem solver. As Figure 13shows, however, even with simple finite differences on auniform grid, the scheme performs well, obtaining accu-rate option prices quickly. In our C++ implementation,we have used the generalized minimal residual method(GMRES) provided by the GMM++ library to solve theresulting system of linear equations. The use of special-

ized solvers such as finite element methods would increasethe speed and accuracy of the scheme by a significant fac-tor; however, it would do so by complicating implementa-tion if off-the-shelf packages are not taken advantage of.This is illustrated in Muthuraman (2008) when comparingAmerican option pricing methodologies in a Black-Scholessetting.

5. Concluding Remarks

Much of the literature on option pricing under stochasticvolatility focuses on European options. We considered in


13/17


Table 3. Option prices at y = 00625.X0

Method Grid size 8 9 10 11 12

PSOR 40 16 8 20000 10952 04966 02042 0083860 32 66 20000 11037 05142 02105 00815

120 64 130 20000 11064 05182 02126 00819240 128 258 20000 11071 05193 02133 00820

Componentwise 40 16 8 20004 11003 04991 02035 00828splitting 60 32 66 20000 11043 05147 02104 00813

120 64 130 20000 11066 05183 02126 00819240 128 258 20000 11073 05194 02133 00820

Transformation 40 16 8 20000 10952 04966 02042 00838procedure 60 32 66 20000 11035 05142 02105 00815

120 64 130 20000 11063 05181 02126 00819240 128 258 20000 11071 05193 02133 00820

True value 20000 11076 05200 02137 00820

Table 4. Option prices at y = 025.

X0

Method Grid size 8 9 10 11 12

PSOR 40 16 8 20691 13139 07720 04293 0232460 32 66 20760 13292 07908 04442 02405

120 64 130 20775 13320 07940 04467 02419240 128 258 20779 13329 07951 04476 02424

Componentwise 40 16 8 20676 13094 07646 04232 02297splitting 60 32 66 20758 13287 07900 04435 02401

120 64 130 20774 13317 07936 04463 02417240 128 258 20780 13328 07949 04474 02423

Transformation 40 16 8 20691 13140 07721 04294 02325procedure 60 32 66 20760 13291 07908 04442 02405

120 64 130 20775 13319 07940 04467 02419

240 128 258 20780 13329 07951 04476 02424True value 20784 13336 07960 04483 02428

this paper the harder problem of pricing American options

in a stochastic volatility setting. It was shown that com-

puting the price of an American option under stochasticvolatility is only as difficult as computing the price of a

Figure 13. A comparison of RMSE and computingtime.

103 102

101

100

101

102

103

RMSE

CPUtimeinseconds

Transformation procedure

Componentwise splitting

PSOR

40168

603266

12064130

240128258

series of European-type options when the exercise policiesare predetermined. A computational procedure for calculat-

ing the price as well as the optimal-exercise boundary wasdeveloped. Using this procedure, we have sought insightsinto the dependence of American option prices, exercise

policies, and implied volatilities on factors such as the mar-ket price of volatility risk and the correlation between stockprice and the volatility process. The method was demon-

strated to be as accurate as the PSOR method, while havingbetter speeds than other existing methods in our numeri-

cal experiments. An avenue for future research would be toextend such a pricing methodology for other American-typederivatives. Specifically, pricing options on multiple assetswould be interesting because there are also inherently mul-

tidimensional problems. The theoretical guarantees estab-lished for the proposed method critically depend on themaximum principle (Theorem A.1). This begs the ques-

tion of whether the methodology can be extended to higherdimensions if one can establish the maximum principle. Fora problem in higher dimensions, if one can determine the

boundary update conditions and also prove the maximum


14/17


principle for the difference in value between two iterations,then certainly the critical part of establishing convergenceis in place. The other details would be problem dependent.Empirical studies that explore questions of investor exercisebehaviors against those computed by the optimal-exercisepolicy would be interesting as well.

Appendix

A. Proofs

The proofs of theorems and propositions are collected here.We begin with the statement and proof of Theorem A.1,which establishes the maximum principle that is needed forsubsequent proofs.

Theorem A.1. For a given T 0 and a continuousgy> 0 for all y 0 T +, let= xy 0 T 2+ x > g y. Also, let h be the solution to

h = 0 in (A1)( is defined in Equation (4)), with boundary conditionsgiven by

h0xy = 0 (A2)hgyy = Fy (A3)

limx

h

x= 0 (A4)

limy

h

y= 0 and (A5)

rxh

x + mh

y rh h

= 0 at y = 0 (A6)

If r > 0 and Fy > 0 for all y 0 T +,then the maxima of h are attained only on the boundarygyy, and the minima of h are attained on theboundary 0xy.

Proof. For notational convenience, let A, B, and C repre-sent 2h/x2, 2h/xy, and 2h/y2, respectively.

We show that the maxima of h is attained only onthe boundary gyy by ruling out other possibili-ties. First, say an internal maxima exists and is attained atsome x y. Now, this maxima will be no less thanhxy for all , x, and y, meaning that h x y hgyy. This implies that h x y > 0 becauseF y > 0. Also, by the necessary conditions for an inter-nal maxima, we have that h/x = h/y = h/ = 0.Substitution into Equation (A1) yields

rh = 12

Ax2y + Bxy + 12

C2y (A7)

The Hessian needs to be a negative semidefinite matrix.This implies that the determinant of the leading nn prin-cipal minor of the Hessian is nonnegative (nonpositive)

when n is even (odd); hence at x y, AC B2 0,and A 0. Thus at this point if A = 0, the second-orderdifferential terms in Equation (A7) can be rearranged as

1

2Ax2y + Byx + 1

2C2y

= 12

Ay

x2 + 2BA

x + C2

A

= 12

Ay

x + B

A

2+

AC B22A2

2

(A8)

Now, because 2 1, we have AC B22 > AC B2 > 0.Hence, from Equation (A7) we have rh < 0, implying

h x y < 0, a contradiction. If A = 0, it follows thatB = 0, leading to a similar contradiction. Therefore, themaxima cannot be attained in the interior.

Now assume that the maximum is attained at someT x y, i.e., on the boundary = T. We must have thathT x ymax y F y > 0, with h/x = h/y =0and h/ 0. Substitution into Equation (A1) now yields

rh = 12

Ax2y + Bxy + 12

C2y h

(A9)

Considering the function h along the cut taken at T, wemust again have A 0 and AC B2 0. Rearranging (ifA = 0), we have

h = 1r

1

2Ay

x + B

A

2+

AC B22A2

2

1

r

h

(A10)

Using arguments similar to those used earlier, even when

A = 0 we have hT x y < 0, again a contradiction. Themaxima cannot be attained on the boundary = T, either.

Next, say that the maximum is attained at some

x 0, i.e., on the boundary y = 0. Again, at x 0,by conditions of maxima, we have h/x = h/= 0 andh/y 0. Substitution into Equation (A6) gives

h = 1r

mh

y (A11)

leading to h x 0 0, another contradiction.

Finally, we need to rule out the possibility of a max-ima being reached as y . Consider a boundary at aY > max/22 m. Now assume that the maxima isachieved at some x Y on the boundary y = Y. At thispoint, the coefficient of hy is negative and because this isa maxima, the Hessian is negative semidefinite, implying anegative LHS, therefore resulting in a contradiction.

Now because the maxima clearly cannot be attained onthe boundary = 0, this leaves us with the result. Argu-ments and reasoning as in the above will establish that the

minima of h is achieved on the boundary 0xy.


15/17


Proposition A.1. If F y = 0 for all y 0 T +, then hxy = 0 for all xy .Proof. The proof follows directly from the proof ofTheorem A.1. If F y = 0 for all y 0 T +, the maximum of h is also 0, because the maximaof h is attained at the boundary gyy, where

hgyy = F y . Because the minima of h is also0, this must mean that h = 0 for all xy . Proof of Proposition 3.1. Assuming there exist two solu-tions h1 and h2, considering the equations solved by h =h1 h2 and using Theorem A.1 directly gives the result.Proof of Proposition 3.2. Let 0 = xy 0 T

2+ x > b 0y and b = xy 0 T 2+ x >

by. In the region 0 b , it is optimal to exercise,but the exercise policy dictated by b0 chooses to subop-timally hold. Hence in 0 b , p0 < p = K x+. Onthe boundary b0, we have that p0 = K x+. Also, byTheorem A.1, the maxima of p0 is attained on b0. Becausep0 = K x on b0 and p0 < K x in 0 b , we musthave that p0/xb0y+ y < 1.

For the rest of this section, subscripts denote derivatives.

Proof of Theorem 3.1. Because pnx bny+y bny. The definition of bn+1 also implies bn+1 > bn.

From the definition of bn+1, for all x bnybn+1y, , and y, we have pnx xy < 1. Thisimplies

pnbn+1yy

pnbnyy

< bn+1y bnypnbn+1yy

< pnbnyy + bny bn+1y= K bn+1y= pn+1bn+1yy

Thus, pn+1 > pn on bn+1. Now, the difference p = pn+1 pnsolves

p = 0 in n+1

p0xy = 0pbn+1yy> 0

limx

pxxy = 0lim

ypyxy = 0 and

p = 0By Theorem A.1, p attains its maxima on bn+1 and its min-ima of 0 on the boundary 0xy for xy bn+1 +. This implies that p > 0 in

n+1, i.e., pn+1 > pn.

Finally, we show that pn+1x bn+1y+ y < 1.

Because pn+1x = pnx + px and pnx bn+1yy = 1by the definition of bn+1, it suffices to show that pxbn+1y+y < 0. Assume that pxbn+1y+ y 0 instead. Now because limx pxxy = 0,pxb

n+1y+ y 0 implies that the maxima of pis attained in

n

+1

. But this contradicts Theorem A.1,which states that the maxima of p is attained on bn+1.Therefore, we must have that pxb

n+1y+ y < 0 pn+1x b

n+1y+ y < 1.

B. Finite Difference Implementation

The fixed-boundary problem defined by Equations (4)(10)

can be solved using the finite difference method. In this sec-

tion, we discuss relevant implementation issues using thefinite difference scheme for the Heston stochastic volatility

model.

For the sake of numerical implementation, the time axis,

asset-price axis, and variance axis are truncated to 0 T ,0 X, and 0 Y , respectively, for some large enough

T X Y +. The boundary conditions for x andy are applied at X and Y, respectively.

The time axis, asset-price axis, and variance axis are

discretized into l, m, and n pieces yielding grid steps

= T /l, x = X/m, and y = Y /n, respectively. Fork = 0 l, i = 0 m, and j = 0 n, the price ofthe option at node kij is denoted by pk xi yj =pk xi yj = pkij. Using this notation, a finite dif-ference discretization based on central differences can beobtained as follows:

p/x = pki+1 j pki1 j/2xp/y = pki j+1 pki j1/2y2p/x2 = pki+1 j 2pki j + pki1 j/2x2p/y2 = pki j+1 2pki j + pki j1/2y2p/xy=pki+1 j+1 pki1j+1 pki+1j1 +pki1j1/4xy

We obtain from Equation (4)

D1i jpki1 j1 +D2jpki j1 +D3i jpki+1j1 +D4i jpki1j +D5i jpki j

+D6i jp

ki

+1jD

7i jp

ki

1j

+1

+D8jp

ki j

+1

+D9i jp

ki

+1j

+1

=pk1i j

(B1)

where

D1i j = ji

4

D2j =m

2y j

2

2j

2y

j

2

y

D3i j =ji

4


16/17


D4i j =r i

2 yji

2

2

D5i j = 1 + yji 2 + r +

2j

y

D6i j=

yji2

2 ri

2

D7i j =ji

4

D8j =j

2+

j

2

y

2j

2y m

2yand

D9i j = ji

4

Similarly, p = 0 is discretized to yield

pk1i j =ri

2pki1 j +

1 + r +

my

pki j

ri2

pki+1 j my pki j+1

Given an exercise policy b, the remaining boundary condi-tions are represented as follows:

p0i j = K xi+ for i = 0 m (B2)pki j = K xi+ if xi bk yj (B3)pki j pki1 j = 0 if i = m xi = X and (B4)pki j pki j1 = 0 if j= n yj = Y (B5)

Given the price for a time to expiry k1, i.e., pk1i j i j,

the price at time k can be obtained by solving a systemof linear equations Dpk = pk1, where pk is a mn-vectorthat represents the option prices for all asset prices andvolatilities at time step k, and the mn mn matrix D isassembled using Equations (B1)(B5) at each time step.This set of equations is solved for each k = 1 l. Theresulting matrix p is then the value function associated withthe exercise policy b.

Acknowledgments

We thank the associate editor, Mark Broadie, Haolin Feng,Jose Figueroa-Lopez, Stanley Pliska, Bruce Schmeiser,Stathis Tompaidis, and two anonymous referees for theircomments, suggestions, and feedback.

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