AMERICAN OPTION PRICING UNDER STOCHASTIC ...ufdcimages.uflib.ufl.edu/UF/E0/02/26/41/00001/guha_s.pdfOur work involves pricing American options under stochastic volatility. In solving

AMERICAN OPTION PRICING UNDER STOCHASTIC VOLATILITY:EFFICIENT NUMERICAL APPROACHES

By

SUCHANDAN GUHA

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

1

© 2008 Suchandan Guha

2

To my parents, my sister, my newly-born nephew and my wife.

3

ACKNOWLEDGMENTS

My sincere thanks first of all to my advisor Dr. Farid AitSahlia, who guided and directed me

through my research. He has been an inspiration for me. I would also like to thank my committee

members (Dr. Stanislav Uryasev, Dr. Hani Doss and Dr. Liqing Yan) for their cooperation and

valuable insights.

I also extend my gratitude to Manisha Goswami, a fellow student, who I have been working

with for the last few years. It has been a great pleasure working with her.

My deepest thanks go to Mataji Shri Nirmala Devi for her blessings throughout my life.

This effort would not have been possible without the constant support of my family, especially

my grandparents, my parents, my sister and my brother-in-law. I am deeply grateful to my wife

Nandini who stood by me and has been a constant moral and emotional support for the last few

years.

This acknowledgement would remain incomplete without the mention of all my friends in

various parts of the world.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 American Option Pricing Formulations . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 The Free-Boundary Approach . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Integral Representation Approach . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Numerical Methods for Pricing American Options . . . . . . . . . . . . . . . . . 151.3.1 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Lattice Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.3 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.4 Analytical Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 TESTING THE PROPOSED METHOD ON A CONSTANT VOLATILITY MODEL . 20

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Stock Price Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Boundary with Constant Volatility . . . . . . . . . . . . . . . . . . . . . 222.2.3 Using the Boundary in the Decomposition Formula . . . . . . . . . . . . 23

2.3 Numerical Implementation and Results . . . . . . . . . . . . . . . . . . . . . . . 232.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 USING THE PROPOSED METHOD ON A STOCHASTIC VOLATILITY MODEL . 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Heston Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Boundary Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5

4 USING A CONSTANT VOLATILITY BOUNDARY IN A STOCHASTIC VOLATILITYDECOMPOSITION FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Boundary with Constant Volatility . . . . . . . . . . . . . . . . . . . . . 424.2.2 Using the Decomposition Formula to Obtain the Option Price . . . . . . . 45

4.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.1 Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

APPENDIX

A STOCHASTIC VOLATILITY APPROXIMATIONS FOR A SV MODEL . . . . . . . 57

B CONSTANT VOLATILITY APPROXIMATIONS FOR A SV MODEL . . . . . . . . 61

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6

LIST OF TABLES

Table page

2-1 ρ = 0.375, σ = 0.3, Strike = 100: American Put Option . . . . . . . . . . . . . . . . . 24

2-2 ρ = 1.220, σ = 0.4, Strike = 100: American Put Option . . . . . . . . . . . . . . . . . 24

2-3 ρ = 0.305: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2-4 ρ = 1.220: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-1 θ = 0.0225: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3-2 θ = 0.09: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3-3 θ = 0.2: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4-1 σ = θ −√

θσv, θ = 0.0225: Summary Analysis . . . . . . . . . . . . . . . . . . . . . 47

4-2 σ = θ, θ = 0.0225: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4-3 σ = θ +√

θσv, θ = 0.0225: Summary Analysis . . . . . . . . . . . . . . . . . . . . . 48

4-4 θ = 0.0225: Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4-5 θ = 0.0225: Stochastic Volatility Results . . . . . . . . . . . . . . . . . . . . . . . . . 49

4-6 σ = θ −√

θσv, θ = 0.09: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . 49

4-7 σ = θ, θ = 0.09: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-8 σ = θ +√

θσv, θ = 0.09: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . 50

4-9 θ = 0.09: Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4-10 θ = 0.09: Stochastic Volatility Results . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4-11 σ = θ −√

θσv, θ = 0.2: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . 51

4-12 σ = θ, θ = 0.2: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4-13 σ = θ +√

θσv, θ = 0.2: Summary Analysis . . . . . . . . . . . . . . . . . . . . . . . 52

4-14 θ = 0.2: Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4-15 θ = 0.2: Stochastic Volatility Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A-1 θ = 0.0225: American call option prices . . . . . . . . . . . . . . . . . . . . . . . . . 58

A-2 θ = 0.09: American call option prices . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A-3 θ = 0.2: American call option prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7

B-1 σ =√

θ −√

θσv, θ = 0.0225: American Call Option Prices . . . . . . . . . . . . . . 62

B-2 σ =√

θ, θ = 0.0225: American Call Option Prices . . . . . . . . . . . . . . . . . . . 63

B-3 σ =√

θ +√

θσv, θ = 0.0225: American Call Option Prices . . . . . . . . . . . . . . 64

B-4 σ =√

θ −√

θσv, θ = 0.09: American Call Option Prices . . . . . . . . . . . . . . . . 65

B-5 σ =√

θ, θ = 0.09: American Call Option Prices . . . . . . . . . . . . . . . . . . . . . 66

B-6 σ =√

θ +√


B-7 σ =√

θ −√


B-8 σ =√

θ, θ = 0.2: American Call Option Prices . . . . . . . . . . . . . . . . . . . . . 69

B-9 σ =√

θ +√


8

LIST OF FIGURES

Figure page

3-1 1000 and 10,000 Sample Paths: Variation of American Call Option Boundary . . . . . 37

3-2 100,000 and 1,000,000 Sample Paths: Variation of American Call Option Boundary . . 38

3-3 25 and 20 Time Steps: Variation of American Call Option Boundary . . . . . . . . . . 39

3-4 10 and 5 Time Steps: Variation of American Call Option Boundary . . . . . . . . . . . 40

3-5 American Call Option Boundary with T = 0.25, σ = 0.04, θ = 0.1, r = 0.03, q = 0.05 41

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Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

AMERICAN OPTION PRICING UNDER STOCHASTIC VOLATILITY:EFFICIENT NUMERICAL APPROACHES

By

Suchandan Guha

August 2008

Chair: Farid AitSahliaMajor: Industrial and Systems Engineering

We developed two new numerical techniques to price American options when the underlying

follows a bivariate process. The first technique exploits the semi-martingale representation of

an American option price together with a coarse approximation of its early exercise surface that

is based on an efficient implementation of the least-squares Monte Carlo method. The second

technique exploits recent results in the efficient pricing of American options under constant

volatility. Extensive numerical evaluations show these methods yield very accurate prices in a

computationally efficient manner with the latter significantly faster than the former. However, the

flexibility of the first method allows for its extension to a much larger class of optimal stopping

problems than addressed in this paper.

10

CHAPTER 1INTRODUCTION

1.1 Introduction

Derivatives in some form or other have existed since long back in history. However the first

modern, organized futures market, The Chicago Board of Trade was created in North America

in 18481 . This was followed by opening of other organized derivatives markets in the US such

as the Chicago Mercantile Exchange, The New York Mercantile Exchange and the Chicago

Board Options Exchange. Just to give an estimate on the size of the derivatives market, Baba and

Gallardo (2008) from the Bank of International Settlements report that the notional amounts of

all categories of over-the-counter contracts reached $596 trillion at the end of December 2008,

following a 24% increase in the first half of the year whereas the notional amounts of outstanding

credit default swaps (CDSs) was $58 trillion.

Among different types of derivatives, options are contracts which give a holder the right

to buy (for a call option) or sell (for a put option) an asset at some future date by a specified

expiration date (maturity). Options that grant this exercise for the expiration date only are termed

European. Those that allow the user to exercise this right anytime up to the maturity date are

known as American options. Pricing American options is challenging because of this flexibility

of exercise available to the user. Various pricing techniques have been proposed by researchers

over the years. Our work involves pricing American options under stochastic volatility. In solving

this problem we are faced with two major issues. The first one is due to flexibility of exercise,

the second issue is because of stochastic volatility. The current literature is rich with methods

for pricing American options, even with stochastic volatility. However the problem with these

existing approaches is that most of them are very demanding in terms of computational time.

Thus one of our goals of research was to design efficient algorithms which can price American

1 Obtained from http://www.cbot.com/cbot/pub/page/0,3181,942,00.html (Chicago Board ofTrade)

11

options fast with good accuracy. In order to do this it is essential to survey the existing pricing

methods for American options which is done next.

1.2 American Option Pricing Formulations

American option pricing can be dated back to Samuelson (1965) in a model that was a

precursor to the widely adopted Black and Scholes (1973) framework. In this classical context,

the price St of the underlying asset (labeled stock) is assumed to follow a geometric Brownian

motion, that is satisfying

dSt = St(r − δ)dt + σStdWt (1–1)

where r is the prevailing riskless rate in the market, δ the dividend rate of the asset and σ its

volatility. The price Vt = V (St, t) of an option with exercise price (strike) K on this asset is then

the value function of the following optimal stopping problem:

V (S, t) = supτ∈T[t,T ]

E[e−rτg (Sτ ) |St = S

](1–2)

where T[t,T ] is the set of optimal stopping times in [t, T ] and g(S) is the payoff upon exercise,with g(S) = (S − K)+ and g(S) = (K − S)+ for a call and a put, respectively. (For a moredetailed review, refer to Broadie and Detemple (2004).)

There are a number of different approaches to solve this optimal stopping problem. Among

the most common and successful are the free-boundary, variational inequalities, and integral

representation approaches, which are reviewed below.

1.2.1 The Free-Boundary Approach

The Free-Boundary approach tries to price American options using the fundamental

valuation equation for European options along with a few more conditions suitable for American

options. Using Ito’s lemma (Karatzas and Shreve (1988)), the equation can be derived as

dVt =

(∂V

∂t+

∂V

∂SSt(r − δ) + 1

2

∂2V

∂S2S2t σ

2

)dt +

∂V

∂SStσdWt (1–3)

12

In order to solve this fundamental valuation equation for American options, the following

conditions need to be added

V (S, T ) = g(S) on

where α = (r − δ − 12σ2)/σ2, β = α2 + 2r/σ2, the pricing problem is first transformed into

a linear complementarity form. After a few steps of integration and relaxing some conditions,

the variational inequality form is obtained, wherein the problem is to find u ∈ V such that thefollowing equation

0 ≤∫ ∞−∞

∂u

∂τ(ν(x, τ)− u(x, τ))dx +

∫ ∞−∞

∂u

∂x(ν́(x, τ)− ú(x, τ))dx (1–5)

holds for all t ∈ [0, T ], for all test functions ν ∈ V , where V is the set of test functions ν(x, τ)that are continuous, continuously differentiable in τ , differentiable almost everywhere in x and

satisfy the following conditions

ν(x, 0) = eαxg(Kex) for x ∈ <

limx→−∞

e−αx−βτu(x, τ) = g(0) for τ ∈ [0, 12σ2T

]

limx→∞

e−αx−βτu(x, τ) = g(∞) for τ ∈ [0, 12σ2T

]

ν(x, t) ≥ ĝ(x, τ)

(1–6)

where ĝ(x, τ) = eαx+βτu(x, τ)g(Kex)

1.2.3 Integral Representation Approach

The previous two sections have presented formulation for American option pricing in the

broad realm of partial differential equations. As is common, these formulations do not lead to

closed-form solutions and numerical methods based on finite-differences have been adapted

for this purpose. Their major drawback is their limitation to small-dimension state spaces and

numerical stability. (see e.g, AitSahlia and Carr (1997) for a review).

An alternative approach, based on fundamental probabilistic considerations is that of integral

representation (also called the price decomposition formula) where the price of an American

option is expressed as the sum of the corresponding European option price and an integral term

representing the value of early exercise. For example, the price C(St, t; K) of an American call

14

option (strike = K, maturity = T and spot price = St) at time t is given as

C(St, t; K) =c(St, t; K) +

∫ Tt

δSte−δ(ν−t)N(d(St; B(ν), ν − t))dν

−∫ T

t

rKe−δ(ν−t)N(d(St; B(ν), ν − t)− σ√

ν − t)dν(1–7)

where

d(X; K, t) =1

σ√

t

[log

(StK

)+

(r − δ + 1

2σ2

)t

]

c(St, t; K) is the price of the corresponding European option and N(x) is the cumulative standard

normal distribution function. This decomposition is attributed to various authors such as Kim

(1990), Jacka (1991) and Carr, Jarrow, and Myeni (1992), who derive it through different

techniques (see also Karatzas and Shreve (1998)).

As the boundary B() is unknown in this equation, we use the knowledge that the option is

exercised when St = Bt and thus C (B(t), t; K) = B(t) −K to get an integral equation for thisboundary:

B(t)−K =c(Bt, t; K) +∫ T

t

δBte−δ(ν−t)N(d(Bt; B(ν), ν − t))dν

−∫ T

t

rKe−δ(ν−t)N(d(Bt; B(ν), ν − t)− σ√

ν − t)dν(1–8)

1.3 Numerical Methods for Pricing American Options

All the methods described in the previous section give us different ways of approaching

the pricing problem. However none of them yields a closed form solution for American options.

Thus we need to revert to numerical techniques to obtain the price. A significant amount of

research has been done with respect to numerical pricing methods for American options, which

can be grouped in four categories

1. Finite Difference Methods

2. Lattice Methods

3. Monte Carlo Methods

4. Analytical Approximations

15

1.3.1 Finite Difference Methods

Finite difference methods correspond to solving of the fundamental valuation PDE (1–

3) along with the boundary conditions given in equation 1–4 numerically. The advantages

associated with the finite difference method is the availability of a wide range of numerical

techniques in the existing literature and the flexibility of the method regarding to solving for the

price for more complex processes. The method, first proposed by Brennan and Schwartz (1977)

is often used to calculate benchmark values for different models. Randall et al. (1997) have

developed a higher level language for automatic code generation for finite difference methods

making it more easier to use this tool.

1.3.2 Lattice Methods

First started by Parkinson (1977) and Cox, Ross, and Rubinstein (1979), the basic idea

behind lattice methods is to discretize the state space into a grid and calculate values at each grid

point through dynamic programming techniques. The binomial option pricing model, proposed

by Cox et al. (1979), uses a discrete time framework to calculate values of the state variable

through a binomial tree for a given number of time steps. Each node in the tree represents a

possible price at that point of time. Thus if the stock price at time t is St there are two possible

movements for the next time step: up or down with the possible prices given as

St+1(up) = St.u and St+1(down) = St.d

where u and d are given as

u = eσ∆t

d = e−σ∆t

In this way, starting with a single stock price of S0 at time 0, the binomial tree is formed through

the passage of time. Once the tree is formed, the calculations proceed backwards, starting from

the expiration date T , where the node values are equal to the payoff values. The value Vt−1,i of

16

node i at time t− 1 is calculated from the nodes i and i + 1 at time t as

Ct−1,i = max{e−rt(pCt,i+1 + (1− p)Ct,i), g(Si)} (1–9)

where the probability of up movement p is given as

p =e(r−δ)∆t − d

u− d

and g(Si) is the payoff value at the price Si associated with node i.

For a trinomial lattice there are 3 movements possible for the next time step instead of 2 in

binomial case. The trinomial lattice generally produces results with greater accuracy, although

at the cost of greater computational time. (c.f. Jarrow and Rudd (1983), Amin (1991) and Boyle

(1986))

Both lattice and finite-difference techniques are known to be computationally extensive with

lattice methods taking O(mn) time for computation, m where is the number of time steps while

n is the number of asset price levels. For various models we have m = O(n) thus making the

total computation time O(n2).

1.3.3 Monte Carlo Methods

Monte Carlo simulation methods are quite useful for the calculation of an expected value.

From expression 1–2, the American option price is actually the expected value of the discounted

payoff occuring at the optimal stopping time. Boyle et al. (1997) and Glasserman (2004) survey

the various Monte Carlo methods used in the pricing of options and other financial derivatives.

Various advantages of using Monte Carlo techniques in this regard are as follows:

• Monte Carlo methods impart a great deal of flexibility to the calculation of option pricesas a great variety of processes and complex payoffs can be simulated using Monte Carlosimulation. Examples include stochastic volatility, jump processes and various other pathdependent payoffs.

• A good amount of saving in time can also be achieved by Monte Carlo methods if usedcarefully. Convergence rate for most Monte Carlo simulation methods are obtained asO(1/

√n) by the central limit theorem. Significant computational improvements can also

be obtained by using variance reduction techniques. We use antithetic sampling in ourwork for the same purpose.

17

The application of Monte Carlo methods to optimal stopping problems such as American

option pricing has increased significantly ever since the publication of Longstaff and Schwartz

(2001). In their approach, the value to continue, which is the most difficult to obtain, is approxi-

mated by least-squares regression. In fact this concept has also appeared in a somewhat different

manner in Carrière (1996). Further variations on this same theme were also derived in Tsitsiklis

and Van Roy (1999, 2001) whereas the somewhat related stochastic mesh concept is developed in

Broadie and Glasserman (2004) to address the issue of exponential order computation time.

1.3.4 Analytical Approximations

Another method that has gained wide acceptance in the last few years is the analytical

approximation of the early exercise premium for the American option. Geske and Johnson

(1984) evaluated American options as compound European options and used the Richardson

extrapolation for the purpose of numerical approximation. AitSahlia and Lai (2001), Ju (1998)

and Ingersoll (1998) review various approximation methods for the early exercise premium. As

knowledge of the boundary is necessary for the calculation of early exercise premium, Huang,

Subrahmanyam, and Yu (1996) proposed to approximate it as a piecewise constant function

and then used a three-point Richardson extrapolation scheme to obtain the price of the option.

Ju (1998) did the same analysis using piecewise exponential function approximation for the

boundary yielding better accuracy than Huang, Subrahmanyam, and Yu (1996) and Geske and

Johnson (1984) and better computational efficiency. AitSahlia and Lai (2001) improve upon

Ju (1998) by solving a one-dimensional integral equation to determine the exercise boundary,

instead of Ju’s two dimensional equation, which can be prone to numerical stability issues.

AitSahlia and Lai (1999), as alluded to in Huang et al. (1996) and Ju (1998) also establish that an

accurate evaluation of the boundary is not essential for calculating the price of the option.

1.4 Research Motivation

This thesis deals with the problem of pricing American options when the underlying process

is such that its volatility is stochastic and follows a diffusion process, in contrast to the constant

18

volatility of the classical Black-Scholes model. This is evidently a more complex optimal stop-

ping problem than the one reviewed earlier. The goal of this thesis is to present two approaches

to solve efficiently this problem. The first one is a combination of the corresponding integral

representation formula (obtained by Chiarella and Ziogas (2005)) with a coarse implementation

of the least-squares Monte Carlo described above. The latter technique is used to determine

quickly an approximation of the optimal stopping surface to be used in the price decomposition

formula. For this purpose, a significantly small number of sample paths are generated: a few

thousands, compared to the tens or hundreds of thousands required for accurate Monte Carlo

pricing. Additionally, only a very limited (around five) exercise dates are allowed, which is an

order of magnitude less than the original formulation. In so doing a coarse approximation of the

exercise surface is generated and then used in the integral representation formula. This approach

is in the spirit of AitSahlia and Lai (1999) who demonstrated that one need not know the exercise

boundary accurately to price American options when volatility is constant. This new approach is

first tested in chapter 2 on a special case of stochastic volatility, namely that of constant volatility.

Chapter 3 then contains a full-fledged test on a genuine stochastic volatility model.

The second approach is to approximate the stopping surface via the integral equation for a

corresponding constant volatility for which the efficient method of AitSahlia and Lai (2001) is

applied. Chapter 4 contains extensive numerical results indicating that it is very accurate. It is

also significantly faster than the first. However, the latter can be extended more readily to higher

dimensions thanks to its (partial) reliance on Monte Carlo simulation.

19

CHAPTER 2TESTING THE PROPOSED METHOD ON A CONSTANT VOLATILITY MODEL

2.1 Introduction

In this chapter, we test the idea of approximating the early exercise boundary via coarse

Monte Carlo simulation on a model with constant volatility, as a prelude to its original intent to

be used for stochastic volatility in the next chapter. An advantage of Monte Carlo simulation is

its flexibility in handling large state spaces and complex option payoffs. While it appeared to be

impractical to apply to American options initially, this method has enjoyed significant interest

since the publication of the papers of Carrière (1996) and Longstaff and Schwartz (2001). Their

basic breakthrough is to approximate the continuation value in the associated optimal stopping

problem by an efficient regression calculation. The comparison of this value with the stopping

payoff at every state and time is at the heart of the backward dynamic programming algorithm

used to solve this problem. In this chapter, the approach is followed to the extent that it uses

significantly less sample paths and over a greatly reduced number of exercise dates. This is

because of our goal of coarsely approximating the early exercise boundary only, while foregoing

any interest in computing the option through this method as it will through the decomposition

formula, which requires knowledge of said boundary.

This chapter is organized as follows. Section 2.2.1 simulates the stock price while section

2.2.2 outlines the steps for the evaluation of the early exercise boundary. Finally section 2.2.3

calculates the price of the option using the decomposition formula using the boundary obtained in

section 2.2.2. The results are shown and analyzed in section 2.3.

2.2 Method

2.2.1 Stock Price Simulation

Assumed to follow the Black and Scholes (1973) model, the behavior of the stock price St at

time t is given as

dSt = (r − q)Stdt + σStdWt (2–1)

20

where r is the riskless interest rate, q is the dividend rate and σ is the volatility of the asset.

{Wt} follows a standard Brownian motion, which makes the stock price process {St} follow ageometric Brownian motion.

For this chapter we focus on American puts under the given constant volatility model (call

options can be studied similarly). The price of an American option at time t is given as the

optimum value in the following optimum stopping problem

U(S, t; K) = supτ∈Tt,T

E[e−r(τ−t)f(Sτ )|St = S

](2–2)

where T is the time the option matures. The payoff function f(Sτ ) with a strike price of K is

given as

f(x) =

max(0, K − x) for put

max(0, x−K) for Call(2–3)

For pricing the American put option we use the decomposition formula wherein the

American option price is expressed as the European option price plus an early exercise premium.

Kim (1990), Jacka (1991), Carr et al. (1992) have derived the American put price U(S, t; K) as:

U(S, t; K) = UE(S, t; K) +

∫ Tt

[rKe−r(T−t)N(d2(S,B(s), T − s))

− qSe−q(r−s)N(d1(S, B(s), T − s))]ds

(2–4)

N(.) denotes the standard normal distribution (cumulative) and {Bt} denotes the early exerciseboundary. The corresponding European option price UE in equation 2–4 is given as

UE(S, t; K) = Ke−r(T−t)N(d2(S,K, T ))− qSe−q(r−s)N(d1(S, K, T, )) (2–5)

where

d1(x, y, τ) =ln x

y+ (r − q + 1

2σ2)τ

σ√

τ

d2(x, y, τ) = d1(x, y, τ)− σ√

τ

21

However, in order to evaluate the option price using equation 2–4, we need to know the

value of {Bt}, the early exercise boundary. As stated in chapter 1, we want to calculate thisboundary efficiently using simulation. Thus in order to obtain the price of the option we need to

do the following in the given order:

1. Evaluate the early exercise boundary using a range of number of time steps and samplepaths (done in section 2.2.2).

2. Use the obtained boundary in the decomposition formula to obtain the price of theoption(done in section 2.2.3).

2.2.2 Boundary with Constant Volatility

The stock price process (equation 2–1) can be discretized as

S(t + ∆t) = S(t) exp

((r − 1

2σ2

)∆t + σ∆W1(t)

)(2–6)

The time domain [0, T ] is divided into n equidistant time steps such that

ti =i× T

n∀i = 0, 1, 2, . . . , n

where n is the number of time steps used in the discrete approximation 2–6 of 2–1.

Next, the early exercise boundary is obtained using the LSM algorithm. This is done starting

from the point of maturity and going backwards using the following steps:

1. At time t = T, B(t) is set as (c.f. Kim (1990))

B(T ) =

{K if r ≤ q(r/q)K if r > q

2. At each time step, the payoff for each sample path is calculated. Looking at the payoffs atthe subsequent paths, the payoff from continuation is calculated for each in-the-money path(c.f. Longstaff and Schwartz (2001)). This array of payoffs from continuation is regressedagainst the corresponding stock prices to obtain the function for the expected future payoff.

3. Using the values of the stock prices at each path, the expected future payoff is calculatedfor each path which is compared to the current payoff to determine whether exercisehappens or not (Longstaff and Schwartz, 2001).

22

4. The boundary value at time t is obtained as the maximum (for Put) or minimum (for Call)value among the stock prices for different sample paths.

b(t) =

{max{S(t)|S(t) ∈ Et} for Putmin{S(t)|S(t) ∈ Et} for Call

where Et is the set of stock prices at time t resulting in exercise.After this calculation, we obtain a 2 dimensional boundary represented as {B(t0), B(t1), . . . , B(tn)}.2.2.3 Using the Boundary in the Decomposition Formula

Once the boundary {Bt} is evaluated, it is applied to the decomposition formula (2–4).Although UE is easily calculated, we need to revert to numerical approximation techniques in

order to calculate the integral in equation 2–4. The integral in this case is a bounded one and

is thus calculated using a simple trapezoidal rule. We also tried the Simpson’s rule, however

without much extra benefits and therefore decided to remain with the simpler trapezoidal rule.

2.3 Numerical Implementation and Results

As a benchmark we use the values reported in AitSahlia and Lai (1999) for American put

option prices. The normalized prices were multiplied by the strike value to get the real price. The

parameter values for these prices, reported in their canonical forms are converted to the normal

form to retrieve σ, r and T . Fixing σ at a particular value, we get the values of r and T from the

reported values of ρ and s as follows:

r = σ2 × ρ

T = − sσ2

Using the method described in section 2.2, prices are calculated for American put options

using the obtained values for the parameters. Tables 2-1 and 2-2 show the prices of the American

put options for ρ = 0.305 and 1.220 respectively, calculated using different methods. The

columns BM and SA serve as our benchmark. BM (Bernoulli method) gives the price obtained

by AitSahlia and Lai (1999) using a convergent dynamic programming algorithm, whereas SA

(spline approximations) gives the price calculated using their interpolation splines. The columns

to their right show the prices obtained using the proposed algorithm. A column showing ”MKN”

23

gives prices obtained by first calculating the boundary using M × 1000 sample paths and N timesteps, and then using the obtained boundary in the price decomposition formula as in equation

2–4.

Table 2-1. American Put Option prices for Strike = 100, ρ = 0.375, σ = 0.3T S0 BM SA 1K5 10K5 10K10 50K10 100K5 100K10 100K25

0.148 80 20.030 19.944 19.944 19.940 20.039 20.045 20.027 20.029 20.03590 10.785 10.800 10.832 10.786 10.849 10.849 10.859 10.821 10.821100 4.385 4.393 4.410 4.392 4.395 4.392 4.404 4.387 4.389120 0.278 0.279 0.281 0.280 0.278 0.279 0.280 0.278 0.278

0.297 80 20.152 20.192 20.278 20.203 20.199 20.192 20.310 20.199 20.23790 11.874 11.907 12.092 11.921 11.895 11.907 11.981 11.888 11.925100 6.075 6.088 6.232 6.107 6.087 6.092 6.118 6.081 6.094120 1.055 1.057 1.082 1.065 1.059 1.058 1.063 1.058 1.057

0.444 80 20.473 20.495 20.463 20.594 20.694 20.619 20.694 20.612 20.59090 12.799 12.821 12.860 12.870 12.897 12.884 12.927 12.845 12.858100 7.310 7.327 7.391 7.362 7.360 7.363 7.379 7.337 7.342120 1.873 1.880 1.914 1.898 1.888 1.889 1.894 1.884 1.882

Columns 1 and 2 give the value of maturity and spot price. Column BM gives the benchmark price obtained usingthe Bernoulli method while column SA gives the one obtained using spline approximation method in AitSahlia andLai (2001). The columns to the right of the benchmark give the prices obtained by using the proposed method. MKNdenotes price obtained using M×1000 sample paths and N time steps

Table 2-2. American Put Option prices for Strike = 100, ρ = 1.220, σ = 0.4T S0 BM SA 1K5 10K5 10K10 50K10 100K5 100K10 100K25

0.021 80 20.002 20.000 19.951 19.960 19.980 19.980 19.960 19.980 19.99290 10.000 10.017 9.845 9.972 9.937 9.958 9.952 9.941 9.927100 2.138 2.151 2.122 2.123 2.114 2.112 2.121 2.112 2.106120 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

0.083 80 20.002 20.002 19.850 19.897 19.943 19.961 19.895 19.947 19.96990 10.354 10.377 10.433 10.523 10.408 10.487 10.502 10.428 10.356100 3.942 3.992 4.011 4.008 3.971 3.974 4.007 3.964 3.937120 0.190 0.226 0.228 0.227 0.223 0.223 0.227 0.223 0.221

0.146 80 20.002 20.075 19.904 20.009 20.007 20.052 19.980 20.017 20.00690 10.842 10.924 11.056 11.205 11.007 11.072 11.150 11.012 10.893100 4.979 5.028 5.080 5.132 5.049 5.043 5.116 5.038 4.979120 0.651 0.660 0.670 0.673 0.660 0.659 0.671 0.659 0.650

Columns 1 and 2 give the value of maturity and spot price. Column BM gives the benchmark price obtained usingthe Bernoulli method while column SA gives the one obtained using spline approximation method in AitSahlia andLai (2001). The columns on the right of the benchmark give the prices obtained by using the proposed method.MKN denotes price obtained using M×1000 sample paths and N time steps

24

Once the prices are obtained, they are analyzed in the following manner: for each price

calculated using the proposed method, the difference from the benchmark price is recorded. The

maximum and average differences for each column are then calculated and reported. Tables 2-3

and 2-4 do the analysis for tables 2-1 and 2-2 respectively. As expected, it is observed that these

values improve as we increase the number of time steps and sample paths.

Table 2-3. Summary Analysis for ρ = 0.3051K5 10K5 10K10 50K10 100K5 100K10 100K25

Maximum Deviation from BM 0.218 0.121 0.221 0.146 0.221 0.139 0.117Average Deviation from BM 0.073 0.042 0.046 0.040 0.071 0.027 0.035

Maximum Deviation from SA 0.185 0.099 0.199 0.124 0.199 0.117 0.095Average Deviation from SA 0.055 0.023 0.040 0.032 0.063 0.025 0.028

For each column, MKN denotes price obtained using M×1000 sample paths and N time steps

Table 2-4. Summary Analysis for ρ = 1.2201K5 10K5 10K10 50K10 100K5 100K10 100K25

Maximum Deviation from BM 0.214 0.363 0.165 0.230 0.308 0.170 0.073Average Deviation from BM 0.083 0.084 0.044 0.057 0.079 0.045 0.020

Maximum Deviation from SA 0.172 0.281 0.083 0.148 0.226 0.088 0.090Average Deviation from SA 0.070 0.070 0.035 0.040 0.067 0.036 0.035

For each column, MKN denotes price obtained using M×1000 sample paths and N time steps

2.4 Conclusion

In this chapter we tested on a constant volatility model our proposed numerical approach to

price American options under stochastic volatility. The reason behind doing so is that constant

volatility prices have been calculated by a number of researchers and so by doing this test we

can at least prove that the model is serving its purpose for known cases. In order to calculate

the price, we first calculate the early exercise boundary and then use it in the corresponding

decomposition formula to compute the option price. As exact knowledge of this boundary is

not required to compute accurately the option price in this manner, we opted to use a coarse

implementation of the LSM to evaluate the boundary. Numerical results for this chapter support

the viability of this approach, which is next tested on a model with stochastic volatility.

25

CHAPTER 3USING THE PROPOSED METHOD ON A STOCHASTIC VOLATILITY MODEL

3.1 Introduction

This chapter tests fully the numerical technique introduced earlier to price an American

option written upon an underlying asset that follows a bivariate diffusion process. The technique

presented here exploits the semi-martingale representation of an American option price together

with a coarse approximation of its early exercise surface that is based on an efficient implementa-

tion of the least-squares method of Carrière (1996) and Longstaff and Schwartz (2001). Extensive

numerical results show that our approach yields very accurate prices in a computationally effi-

cient manner. In addition, the flexibility of the method allows for its extension to a much larger

class of optimal stopping problems than addressed in this chapter.

This chapter is organized as follows. In the next section, the stochastic volatility model

of Heston (1993) is reviewed. Section 3 develops our approximation approach to price Amer-

ican options under this model. Section 4 contains a systematic numerical evaluation of this

approximation and Section 5 concludes.

3.2 Heston Pricing Model

In this model, the volatility of the underlying asset is assumed to be stochastic and follows a

mean-reverting process that indicates its tendency to return to a long-term average. Specifically,

denoting the underlying price process by {S(t)}t and its return volatility process by {V (t)}, wehave the bivariate specification:

dS(t) = (r − q)S(t)dt +√

V (t)S(t)dW1(t) (3–1)

dV (t) = κ(θ − V (t))dt +√

V (t)σv

(ρdW1(t) +

√1− ρ2dW2(t)

)(3–2)

where r and q denote the risk-free rate and the dividend yield respectively, with W1 and W2 two

independent standard Brownian motions defined on a common underlying complete filtered

probability space (Ω, (Ft)t , P ), where P is the risk-neutral measure. The volatility V (t)

26

therefore evolves as a mean-reverting square root process with a rate of mean reversion κ, long-

term mean θ and volatility of volatility σv. This square root process specification is particularly

appealing as it guarantees that V (t) remains positive as long as 2κθ ≥ σ2v .For illustrative purposes, we consider an American call option with strike price K. Let

CA(S, v, τ) denote the price of this option when the underlying has price S and spot volatility

v, with τ units of time left to expiry. Using standard arbitrage arguments, this price CA can be

shown to satisfy the following partial differential equation

∂CA∂τ

=vS2

2

∂2CA∂S2

+ ρσvS∂2CA∂S∂v

+σ2v

2

∂2CA∂v2

+

(r − q)S∂CA∂S

+ (κ[θ − v]− vλ)∂CA∂v

− rCA

in the region D = {0 ≤ τ ≤ T, 0 ≤ S ≤ b(v, τ), 0 ≤ v < ∞} along with the boundaryconditions

CA(S, v, 0) = max(S −K, 0),

CA(b(v, τ), v, τ) = b(v, τ)−K,

limS→b(v,τ)

∂CA∂S

= 1,

limS→b(v,τ)

∂CA∂v

= 0

where b(v, t) denotes the optimal early exercise price (boundary) at time t for spot volatility v,

and λv denotes the corresponding market price of volatility risk, with λ determined empirically.

This market price of risk approach is a common way to address the market incompleteness that is

inherent in the stochastic volatility formulation.

Chiarella and Ziogas (2005) use the method in Jamshidian (1992) to convert the above

homogeneous PDE defined in the region D to an inhomogeneous one in an unrestricted domain:∂CA∂τ

=v

2

∂2CA∂x2

+ ρσv∂2CA∂x∂v

+σ2v

2

∂2CA∂v2

+(r − q − v

2

) ∂CA∂x

+ (α− βv)∂CA∂v

−H(x− ln b(v, τ)){erτ (qex − rK)},(3–3)

27

where α ≡ κθ and β ≡ κ+λ, in the unrestricted domain −∞ < x < ∞, 0 < v < ∞, 0 ≤ τ ≤ T ,subject to the boundary conditions:

CA(x, v, 0) = max(ex −K, 0),

limx→ln b(v,τ)

∂CA∂x

= b(v, τ)erτ ,

limx→ln b(v,τ)

∂CA∂v

= 0,

where H(x) is the Heaviside step function defined as

H(x) =

1, x > 0,

12, x = 0,

0, x < 0

.

To obtain CA through equation (3–3), one still needs the knowledge of the optimal stopping

(exercise) boundary b(v, t). In the classical context of constant volatility for the underlying asset

return, AitSahlia and Lai (1999) have shown that this boundary is well-approximated by linear

splines with very few knots, typically 3 or 4. When the volatility of the underlying asset itself

follows a stochastic process as in (3–2) above, Broadie et al. (2000) have produced empirical

evidence to suggest that the corresponding optimal stopping surface can be well-approximated in

a log-linear fashion near the long-term variance level; i.e.:

ln b(v, τ) ≈ b0(τ) + vb1(τ), near θ,

thus reducing the determination of b(v, τ) to that of b0(τ) and b1(τ). Under this assumption,

Chiarella and Ziogas (2005) then express the solution for the PDE (3–3) as the following

28

decomposition formula:

CA(S, v, τ) =Se−qτP1(S, v, τ,K; 0)−Ke−rτP2(S, v, τ,K; 0)

+

∫ τ0

qSe−q(τ−ξ)P1(S, v, τ − ξ, eb0(ξ);−b1(ξ))dξ

−∫ τ

0

rKe−r(τ−ξ)P2(S, v, τ − ξ, eb0(ξ);−b1(ξ))dξ,

(3–4)

where

Pj(S, v, τ − ξ, b; w) = 12

+1

Π

∫ ∞0

Re

(fj(S, v, T − ξ; φ,w)e

−iφ ln b

iφ

)dφ (3–5)

for j = 1, 2 and

f1(S, v, τ − ξ; φ,w) = e− ln Se−(r−q)(τ−ξ)f2(S, v, T − ξ; φ,w)

f2(x, v, τ − ξ; φ, ψ) = exp g0(φ, ψ, τ − ξ) + g1(φ, ψ, τ − ξ)x + g2(φ, ψ, τ − ξ)v,

with

g0(φ, ψ, τ − ξ) = (r − q)iφ(τ − ξ)+α

σ2

{(β − ρσiφ + D2)(τ − ξ)− 2 ln

[1−G2(ψ)eD2(τ−ξ)

1−G2(ψ)]}

,

g1(φ, ψ, τ − ξ) = iφ,

g2(φ, ψ, τ − ξ) = iψ + β − ρσiφ− σ2iψ + D2

σ2

[1− eD2(τ−ξ)

1−G2(ψ)eD2(τ−ξ)]

,

D22 ≡ (ρσiφ− β)2 + σ2φ(φ + i)

G2(ψ) ≡ β − ρσiφ− σ2iψ + D2

β − ρσiφ− σ2iψ −D2 .

In order to determine approximately the terms b0(τ) and b1(τ) for every τ , we shall rely on

a coarse implementation of the least-squares Monte Carlo (LSM) algorithm due to Carrière

(1996) and Longstaff and Schwartz (2001). This flexible method uses a combination of Monte-

Carlo simulation with least-squares regression to evaluate American option prices. In our

adaptation of this approach, we shall estimate b0(τ) and b1(τ) over a finite subset of discrete

29

dates τ1, τ2, . . . , τN , with N very small, typically between 5 and 10, and a small number of

sample paths, just a few thousands, compared to simulation runs several orders of magnitude

larger that are required for accurate results through Monte Carlo. In fact, our approach is

motivated in part by the constant volatility results that indicate that the exact knowledge of the

boundary for the integral representation expression is not critical for the accuracy of the option

price calculation (cf. AitSahlia and Lai (1999).) In addition, Glasserman (2004) also shows that

simulation-based valuations of American option prices do not critically depend on an accurate

evaluation of the optimal exercise strategy.

3.3 Boundary Evaluation

In this section we detail the steps of the approach described above. In particular, we show

how the early exercise surface is approximated numerically. Then in the next section we use

this approximation through numerical integration in the American option pricing decomposition

formula. For our purposes, the early exercise surface will be approximately determined by the

LSM method. As a Monte-Carlo based technique, it will generate discrete sample values Ŝi and

V̂i of the stock and its variance, respectively, by discretizing the associated stochastic differential

equation (3–1) - (3–2). A natural choice is the Euler scheme:

Ŝi+1 = Ŝi + µŜi∆t +

√V̂iSt∆W1,

V̂i+1 = V̂i + κ(θ − V̂i)h +√

V̂iσv

(ρ∆W1 +

√1− ρ2∆W2

),

where ∆W1 and ∆W2 are independent standard normal random random variables with variance

h, which is defined as the time mesh-size.

However, we follow Glasserman (2004) who suggests that the second-order scheme of

Milstein (1978) and Talay (1982) given below has a better convergence (less bias) for option

30

pricing applications:

Ŝi+1 =Ŝi

(1 + rh +

√V̂i∆W1

)+

1

2r2Ŝih

2

+

([r +

ρ− κ4

]Ŝi

√V̂i +

[κθ

4− 1

16

]Ŝi√V̂i

)∆W1h

+1

2Ŝi

(V̂i +

ρ

2

)(∆W 21 − h) +

1

4

√1− ρ2Ŝi (∆W2∆W1 + ξ) ,

Vi+1 =

κθh + (1− κh)V̂i +√

V̂i

(ρ∆W1 +

√1− ρ2∆W2

)− 1

2κ2(θ − V̂i)h2

+

([κθ

4− 1

16

]1√V̂i− 3κ

2

√V̂i

)(ρ∆W1 +

√1− ρ2∆W2

)h

+1

4ρ2(∆W 21 − h) +

1

4(1− ρ2)(∆W 22 − h) +

1

2ρ√

1− ρ2∆W1∆W2,where ξ is a random variable independent of ∆W1 and ∆W2 such that P{ξ = h} = P{ξ =−h} = 1/2.Boundary Evaluation through LSM Once stock and volatility paths are generated, the

LSM method consists of approximating the expected value of continuation by least squares.

Specifically, LSM assumes that the option can be exercised at one of N dates, t1, t2, . . . , tN = T ,

along each of the M sample paths generated by Monte Carlo. In order to calculate the boundary,

we perform the following steps for every time step (starting from tN = T and going backwards

tN−1, tN−2, . . . t1)

1. At a particular time step tk, we first calculate the continuation cash flow{C(ω, tj; tk, T ) : k + 1 ≤ j ≤ N} for each sample path as in Longstaff and Schwartz(2001), who also show that the value of continuation F (ω; tk) can be expressed as

F (ω; tk) = EQ

[N∑

j=k+1

e(tj−tk)rC(ω, tj; tk, T )

∣∣∣∣Ftk]

where r is the riskless discount rate and the expectation (with respect to the risk neutralmeasure Q) is taken conditional on the information set Ftk at time tk. However, in orderto obtain F (ω; tk) for our problem we need to approximate it as a linear combination of

31

following basis functions of the stock price and the volatility.

F (ω; tk) ≈ C0 + C1L0(

s(ω)

K

)+ C2L1

(s(ω)

K

)+ C3L0

(s(ω)

K

v(ω)

θ

)(3–6)

where K is the strike price for the option and θ is the long run mean of the volatility asgiven in Heston’s model and

L0(X) = exp

(−X

2

)

L1(X) = exp

(−X

2

)× (1−X)

Thus regressing the continuation cash flows against the basis functions, we are able tocalculate the coefficients C0, C1, C2 and C3 in (3–6). Then for each sample path in thattime step, the expected cash flow from continuation is calculated and compared with thecurrent payoff. The option is exercised for a particular sample path if the current payoffis greater than the expected payoff from continuation. This is done for all the M stockprices (one for each sample path) at this time step. Let ne denote the number of samplepaths out of the total NS where exercise happens. Let these ne points be denoted as(si, vi) i = 1, . . . , ne.

2. The obtained exercise points are then divided into classes C1, C2, . . . , Cnv so that

(si, vi) ∈ Cj if vi ∈ (Vj, Vj+1)

where Vj (j = 1, . . . , nv) are equidistant values of volatility, such that V0 =vMin andVnv =vMax,

Vj = V0 + jVnv − V0

nv∀ j = 1, . . . nv

vMin and vMax are given rough values according to the distribution of volatility inequation (3–1) for Heston’s process. The values nv, vMin, and vMax clearly depend onthe stochastic volatility model and can be set in advance in a number of different ways. Inour particular case, we ran numerical simulations of the volatility process alone ahead ofthe pricing calculations and determined that vMin could be set to its natural value of 0, asit is a variance, and vMax was set to .70, which is conservative value as observed variancevalues are overwhelmingly less .06. Numerical experiments were performed as “dry runs”for the fine-tuning of nv, set to 10, as it was the value a little higher than observed, wherepricing quality started to deteriorate.

3. The boundary values for this time step tk corresponding to volatilities {v̄1, v̄2, . . . , ¯vnv}(v̄j =

Vj+Vj+12

) are obtained as b1(tk), b2(tk), . . . , bnv(tk), where

bj(tk) =

{max{si|(si, vi) ∈ Cj} for put optionmin{si|(si, vi) ∈ Cj} for call option

32

Thus at each time step we are able to obtain the boundary values for different ranges ofvolatility.

4. Empirical evidence in Broadie et al. (2000) point to the existence of a linear relationshipbetween ln b(v, t) and v, a relationship also used by Tzavalis and Wang (2003):

ln b(v, t) ≈ b0(t) + vb1(t) (3–7)

Thus for our time discretized version, we need to evaluate the values of b0(tk) and b1(tk) inthe equation

ln bj(tk) ≈ b0(tk) + v̄jb1(tk) (3–8)In order to calculate b0(tk) and b1(tk), the obtained boundary values{b1(tk), b2(tk), . . . , bnv(tk)} for different classes C1, C2, . . . , Cnv are regressed against themidpoints of the corresponding classes {v̄1, v̄2, . . . , ¯vnv}.

3.4 Numerical Implementation

The price of the American call option is computed by applying the values of b0(τ) and b1(τ)

obtained from the previous section in the pricing expression (3–4). The values of the parameters

of Heston’s model are assumed to be known. As is seen in (3–4), in order to evaluate the price

of the option, we need to compute the values of some integrals. The outer integrals in (3–4)

for the early exercise premium are time integrals and are computed using Simpson’s rule. The

number of points for the integration is same as the number of time steps over which the boundary

is calculated. However, within these integrals and for the calculations of the European option,

additional numerical integration is required, over unbounded intervals. We proceed with the use

of Gauss-Laguerre quadrature, which approximates an integral of the form∫∞0

f(x)dx using

optimally chosen n points xi with weights wi, resulting in the approximation

∫ ∞0

f(x)dx ≈i=n∑i=1

f(xi)wi.

These points xi are the roots of orthogonal Laguerre polynomials Ln(x) = ex

n!dn

dxn(e−xxn) (cf.

William H. Press (1992).) For our code we used readily available routines from the QuantLib

library (http://quantlib.org).

To compute the benchmark values against which to compare those generated through the

proposed numerical approach, we follow the steps below:

33

1. Apply LSM algorithm to calculate the early exercise surface.

2. Stop simulated finer paths at this boundary to obtain option price.

In step 1, we first simulate the stock price paths following the second-order scheme described

in section 3. LSM is then applied to these simulated paths to calculate the boundary using the

procedure given in 3.1. The calculated values of the boundary are stored for the next step, which

generates a fresh set of sample paths and stops them at the boundary to get the price for each

path. In order to check if a particular path can be stopped at a particular time, the stock price

is checked with the boundary price for the corresponding volatility class. The average over

all sample paths gives the price of the option. Since this step does not require any complex

calculations (regressions in LSM), it is not much computationally intensive. As a result, we can

increase the number of sample paths considerably without affecting the computational time a lot

and thereby obtaining a price with a much narrower confidence interval than the original LSM.

The number of potential dates however has to be the same as in step 1 above. The benchmark

boundary is generated using 1,000,000 sample paths and time increments of 0.01 year each.

Prices are subsequently calculated by generating a fresh set of 10,000,000 sample paths and

stopping them at the boundary obtained in the previous step.

In our numerical experimentation, the parameter values are set as T = 0.25, ρ = 0.0, κ =

1.0, θ = 0.09, σ = 0.1, as well as various combinations of sample sizes and numbers of time

steps, are considered. In addition, different maturities τ ∈ {0.2, 0.4, 0.6, 0.8, 1.0} and spot pricesare used. For each value of τ , the spot prices s0 are taken from the set {90, 95, 100, 105, 110}.

In order to study the effect of changing the number of sample paths, we evaluate the

boundary surface using 10K, 50K, 100K and 1000K number of sample paths, keeping the

number of time steps constant at 25. As shown in figures 3-1 and 3-2, starting with 1,000 sample

paths, the approximated boundary surface becomes progressively better. In fact, there is little

noticeable difference between the one generated with 100,000 sample paths and that which is

generated with 1 million.

34

To study the effect of changing the number of time steps, the boundary surface is again

evaluated using 5, 10, 20, 50 time steps, keeping the number of sample paths constant at 100K

and the set of parameters as in the previous setting. The obtained boundaries are shown in

Figures 3-3 and 3-4, which shows that piecewise linear approximations with only a few time

steps and sample paths are warranted.

Figure 3-5 shows the boundary obtained by Chiarella and Ziogas (2005) by solving

the integral equation system for the boundary using numerical integration techniques for

Volterra integral equations. As it is apparent from the figures, the boundary in Figure 3-5 is well

represented by the boundary in Figure 3-4B. Next, we compute option prices according to our

method. Table A-1 shows corresponding results compared against benchmark values. Since

the latter are generated via simulation, the corresponding column labelled ”BM” contains both

averages and 95% confidence intervals. The column heading ”MkN” indicates that prices are

calculated on the basis of an approximate boundary obtained with N time steps and M × 1, 000sample paths.

To assess the accuracy of our approach, for each column we determine whether a price

falls in the 95% benchmark confidence interval (CI). If it misses the CI, we record the amount

by which it misses and an average is taken for the corresponding column. Tables A-1, A-2, A-3

give the prices obtained for values of θ of 0.0225, 0.09, and 0.2, respectively, with all the other

parameters remaining the same as above. Tables 3-1, 3-2, 3-3 are for the resulting analysis. It

can be observed that the proposed method works generally well, and even better for low standard

volatility values (i.e. when the long run mean of the volatility θ is low). Thus the results for

θ = 0.0225 are better than that with θ = 0.09, which are in turn better that the one with θ = 0.2.

However θ = 0.09 corresponds to a volatility of 30% which by itself is fairly common market

volatility value. For each value of θ, we observe that the price estimate improves as the number

of sample paths and the number of time steps are increased.

Also reported in tables 3-1, 3-2, 3-3 are the computation time for each column. It is clear

that a reduction in computation time comes at the cost of increasing value of the error. However

35

it is evident that good computation speeds are achieved with very little loss in accuracy, even for

the small values of sample paths and time steps.

3.5 Conclusion

In this chapter we presented a novel and efficient numerical technique to price American

option prices when the underlying asset follows a diffusion process indexed by a volatility

that itself follows a stochastic process. Extensive numerical tests indicate that this approach is

very efficient and accurate. Future work is planned to extend it to stochastic volatility models

with random jumps, where the underlying price process also is subject to random shocks (as

a result, for example, of unforseen economic developments.) In fact, since the method relies

on the combination of a very general result regarding for optimal stopping (the Doob-Meyer

decomposition of Snell envelopes) together with a very flexible Monte-Carlo approach, further

expansions into larger classes of problems can be envisioned.

36

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

A Boundary calculated using 1000 sample paths

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

B Boundary calculated using 10,000 sample paths

Figure 3-1. Approximate Boundary for American call option with T = 0.25, σ = 0.04, θ = 0.1,r = 0.03, q = 0.05, evaluated using 25 time steps

37

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

A Boundary calculated using 100,000 sample paths

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

B Boundary calculated using 1000,000 sample paths

Figure 3-2. Approximate Boundary for American call option with T = 0.25, σ = 0.04, θ = 0.1,r = 0.03, q = 0.05, evaluated using 25 time steps

38

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

A Boundary calculated using 25 time steps

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

B Boundary calculated using 20 time steps

Figure 3-3. Approximate Boundary for American call option with T = 0.25, σ = 0.04, θ = 0.1,r = 0.03, q = 0.05, evaluated using 100,000 sample paths

39

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

A Boundary calculated using 10 time steps

0

0.1

0.2

0.3

00.05

0.10.15

0.2

100

110

120

130

140

150

160

170

Volatility

Time to maturity

Bou

ndar

y V

alue

B Boundary calculated using 5 time steps

Figure 3-4. Approximate Boundary for American call option with T = 0.25, σ = 0.04, θ = 0.1,r = 0.03, q = 0.05, evaluated using 100,000 sample paths

40

Figure 3-5. Boundary obtained by Chiarella and Ziogas (2005) with T = 0.25, σ = 0.04, θ = 0.1,r = 0.03, q = 0.05 by solving the integral equation system for the boundary using thenumerical approximation techniques for Volterra equations

Table 3-1. Summary Analysis for θ = 0.02251k5 10k5 10k10 50k10 100k5 100k10 100k25

Max distance from CI 0.439 0.169 0.029 0.028 0.076 0.026 0.030Average distance from CI 0.084 0.058 0.008 0.010 0.040 0.011 0.002

% in CI 0% 4% 16% 12% 0% 12% 68%Computation Time (sec) 0.08 0.32 0.56 2.98 3.04 6.6 16.8

For each column, MkN denotes price obtained using M×1000 sample paths and N time steps









41

CHAPTER 4USING A CONSTANT VOLATILITY BOUNDARY IN A STOCHASTIC VOLATILITY

DECOMPOSITION FORMULA

4.1 Introduction

In the previous chapter we observed that we do not need to have a very accurate estimate of

the early exercise boundary in order to calculate the price of an option. The results in table A-1,

A-2 and A-3 show that prices of options obtained by first calculating the boundary using different

(lesser) number of sample paths and different (lesser) number of time steps and then using the

boundary in the decomposition formula were fairly close to the chosen benchmark for the prices.

The different approximate forms of the boundary that we calculated as a result are shown in

figures 3-1, 3-2, 3-3 and 3-4.

Looking at these figures and using our result that we need to have only a rough estimate

of the boundary, our intuition took us one step further by considering an approximation for the

boundary based on the constant volatility model. Once this boundary is approximated, it is then

used in the decomposition formula for stochastic volatility to obtain the price of the option.

4.2 Method

The first problem we face while trying to use a boundary obtained using constant volatility

is that the obtained boundary is just a 2-dimensional curve (no volatility axis) instead of the 3

dimensional surface that we need to have in order to use in the decomposition formula. This

problem is tackled by simply replicating the curve along the volatility axis so that the boundary

values does not change with volatility. Thus, similar to chapter 2 and 3 here also, the price is

calculated in two broad steps as follows:

1. Evaluate the early exercise boundary approximately using a constant volatility model.

2. Calculate the price of the option for the stochastic volatility model using the aboveboundary in the decomposition formula

4.2.1 Boundary with Constant Volatility

In chapter 3, the boundary was calculated using the LSM method from Longstaff and

Schwartz (2001) and in chapter 2, the same idea was being tested with constant volatility.

42

However in this chapter we again need a boundary for a constant volatility model and for this

purpose we can use existing ”tried and tested” methods of finding the boundary. A significant

research effort has been spent determining the constant volatility boundary by solving the integral

equation derived from the decomposition formula given in Kim (1990), Jacka (1991) and Carr

et al. (1992). This boundary was approximated by Huang, Subrahmanyam, and Yu (1996) as

piecewise constant functions and was calculated using n = 1, 2 & 3 pieces. Ju (1998) did the

same analysis using piecewise exponential function approximation for the boundary instead

of the piecewise constant approximation. A number of different calculations for boundaries

carried out by AitSahlia and Lai (1999) showed that a piecewise exponential boundary can very

well approximate the real boundary. AitSahlia and Lai (2001) solve the integral equation for

the boundary by first undergoing a change in variables and then solve it using a numerically

stable root finding algorithm as their approach is one-dimensional in contrast to the two-

dimensional approach of Ju (1998). In addition, their boundary approximation is the only one

that is continuous, which is in conformity with its theoretical characterization. We use the same

method as in AitSahlia and Lai (2001) to approximate the boundary with a few time steps.

Let S, K represent the stock price and the strike price, respectively. We denote by r, σ, T the

parameters for risk-free rate, volatility and maturity respectively. Applying the following change

of variables (as in AitSahlia and Lai (1999))

s = σ2(t− T ), z = log(S/K)− (ρ− αρ− 12)s

(where ρ = r/σ2 and α = µ/r) to equation 1–7 we obtain the following integral formula for the

boundary (z̄(s)) in the canonical form

ez(s)+(ρ−αρ−12)s =eρs

[ez(s)−

12sN

(z(s)√−s +

√−s)−N

(z(s)√−s

)]

− ρeρs∫ 0

s

[αe−αρu−

12s+zN

(z − z(u)√

u− s +√

u− s)− e−ρuN

(z − z(u)√

u− s

)]du

(4–1)

43

Dividing the interval [s, 0] into m subintervals as s = sm < . . . < s0 = 0 and proceeding in

the same way as in AitSahlia and Lai (2001), the boundary z̄() is solved recursively starting from

z(0) =

0 if α ≥ 1

− ln α if 0 < α < 1(4–2)

As the approximating boundary is piece-wise linear, each intercept is determined by

the previous piece and thus only the corresponding slope needs to be determined as root of a

non-linear equation as explained next.

With z̄j = z̄(sj) and τj = sj − sm, once z̄0, . . . , z̄m−1 are determined, z̄m can be determinedby solving the following equation for z

ez+(ρ−αρ−12)sm = eρsm

[ez−

12smN

(z√−sm +

√−sm)−N

(z√−sm

)]

+ e−ρτm−1N(b(z)τ

1/2m−1

)− 1

2− b(z)

a(z)N

(a(z)τ

1/2m−1 −

1

2

)−

m−1∑i=1

Ai(z)

+ ez+(ρ−αρ−12)sm

[b̃(z)

ã(z)N

(ã(z)τ

1/2m−1 −

1

2

)− 1

2− e−ρτm−1N

(b̃(z)τ

1/2m−1

)+

m−1∑i=1

Ãi(z)

](4–3)

where Ai(z) and Ãi(z) are given by the RHS of equation numbers 9 and 10 of AitSahlia and

Lai (2001) and

b(z) =z − zm−1sm−1 − sm

a(z) = [b2(z) + 2ρ]1/2

b̃(z) = b(z) + 1

ã(z) = [b̃2(z) + 2αρ]1/2

To solve equation 4–3 we use the bisection method, for which we use the lower and upper

bounds for the call option boundary (AitSahlia and Lai (1999)) as starting points

zu(s) = ln

(β̄

β̄ − 1

)−

[ρ(1− α)− 1

2

]s (4–4)

zl(s) = z(0) (4–5)

44

where β̄ = − [ρ(1− α)− 12

]+

{[ρ(1− α)− 1

2

]2+ 2ρ

}1/2

The obtained boundary values z̄(0), z̄(1), . . . , z̄(m) are then converted from the canonical

form to the standard form as

b(t) = Kez(s)+(ρ−αρ−12)s (4–6)

where s = σ2(t− T ).After this calculation, we obtain a 2 dimensional boundary represented as

{b(t0), b(t1), . . . , b(tk)}. In order to stretch it to a 3-dimensional one, the values are just re-peated over the volatility axis. Thus if the {v1, v2, . . . , vnv} are points along the volatility axis,the resulting 3-dimensional boundary is formed as

b(ti, vj) = b(ti) ∀i, j

From Chapter 3, we know that empirical evidence in Broadie et al. (2000) points to the

existence of a linear relationship between ln b(v, t) and v, a relationship also used by Tzavalis and

Wang (2003):

ln b(v, t) ≈ b0(t) + vb1(t) (4–7)

Thus, unlike last chapter, we do not have to perform a regression to evaluate b0(t) and b1(t).

Instead, these values are easily calculated from the obtained boundary b(t) as:

b0(ti) = log b(ti), b1(ti) = 0 ∀i = 1, 2, . . . , nt

4.2.2 Using the Decomposition Formula to Obtain the Option Price

It has to be noted that although we are using a constant volatility model to obtain the

boundary, we use the decomposition formula for stochastic volatility to finally calculate the price

of the option. However once we have the values of b0(t) and b1(t) from the subsection 4.2.1, we

45

can readily use them in the decomposition formula

CA(S, v, τ) =Se−qτP1(S, v, τ, K; 0)−Ke−rτP2(S, v, τ,K; 0)

+

∫ τ0

qSe−q(τ−ξ)P1(S, v, τ − ξ, eb0(ξ);−b1(ξ)φ)dξ

−∫ τ

0

rKe−r(τ−ξ)P2(S, v, τ − ξ, eb0(ξ);−b1(ξ)φ)dξ,

(4–8)

where

Pj(S, v, τ − ξ, b; w) = 12

+1

Π

∫ ∞0

Re

(fj(S, v, T − ξ; φ,w)e

−iφ ln b

iφ

)dφ (4–9)

for j = 1, 2 and

f1(S, v, τ − ξ; φ,w) = e− ln Se−(r−q)(τ−ξ)f2(S, v, T − ξ; φ,w)

f2(x, v, τ − ξ; φ, ψ) = exp g0(φ, ψ, τ − ξ) + g1(φ, ψ, τ − ξ)x + g2(φ, ψ, τ − ξ)v

where

g0(φ, ψ, τ − ξ) = (r − q)iφ(τ − ξ)+α

σ2

{(β − ρσiφ + D2)(τ − ξ)− 2 ln

[1−G2(ψ)eD2(τ−ξ)

1−G2(ψ)]}

g1(φ, ψ, τ − ξ) = iφ

g2(φ, ψ, τ − ξ) = iψ + β − ρσiφ− σ2iψ + D2

σ2

[1− eD2(τ−ξ)

1−G2(ψ)eD2(τ−ξ)]

D22 ≡ (ρσiφ− β)2 + σ2φ(φ + i)

G2(ψ) ≡ β − ρσiφ− σ2iψ + D2

β − ρσiφ− σ2iψ −D2

The method of using numerical integration to calculate the value of the integrals is the same

as in chapter 3. We use Simpson’s rule to calculate the time integral and Gaussian quadrature to

calculate the other integrals.

4.3 Numerical Implementation

As in chapter 3, here also we divided our problem into three different scenarios depending

on the value of the long run mean of volatility θ.

46

1. Scenario 1: Low value of θ (= 0.0225 or√

θ = 15%), and v0 = 0.0225

2. Scenario 2: Mid value of θ (= 0.09 or√

θ = 30%), and v0 = 0.04

3. Scenario 3: High value of θ (= 0.2 or√

θ ≈ 45%), and v0 = 0.2In order to determine the boundary using a constant volatility model, we need to assume

a value for this parameter, which can be chosen from a wide range of different possible values.

Since θ represents the long run mean of volatility and since the standard deviation for volatility

is approximately√

vσv (c.f. 3–1), it is reasonable to calculate the early exercise boundary by

assuming σ to be equal to one of the three different values given by

σ2 = θ −√

θσv, θ, θ +√

θσv

Thus for each of the three scenarios, we obtain three different tables of prices evaluated using

boundary obtained by using different values of the constant volatility. We then analyze these

tables in the same way as in chapter 3. For each column we calculate the distance from the

benchmark 95% confidence interval (CI) for each entity and finally the maximum and average

distance from the CI for the entire column. Each of the three analysis tables (one for each of the

three values of θ) has 4 sub-tables. The first three (a, b and c) correspond to the three different

values of σ (in the constant volatility model) used for calculation of the boundary. The last

sub-table (d) is where we apply the method from the previous chapter on a full-fledged stochastic

volatility model. The results are obtained as follows.

4.3.1 Scenario 1

Table 4-1. Summary Analysis for σ = θ −√

θσv, θ = 0.0225

Number of Time Steps3 5 10 25 50

Max distance from CI 1.172 1.204 1.196 1.188 1.185Average distance from CI 0.353 0.352 0.350 0.349 0.348

% in CI 0% 0% 0% 0% 0%

Tables B-1, B-2 and B-3 give the prices of American put options with a long run mean of

volatility θ = 0.0225. The values of σ used in the constant volatility model to calculate the

47

Table 4-2. Summary Analysis for σ = θ, θ = 0.0225Number of Time Steps

3 5 10 25 50Max distance from CI 0.449 0.439 0.435 0.435 0.434

Average distance from CI 0.161 0.152 0.147 0.146 0.146% in CI 0% 0% 4% 4% 4%

Table 4-3. Summary Analysis for σ = θ +√

θσv, θ = 0.0225



% in CI 0% 0% 0% 4% 8%

boundary in these tables are given respectively as

σ2 =

θ −√

θσv = 0.0225− 0.15× 0.1 = 0.0075 for table B-1

θ = 0.0225 for table B-2

θ −√

θσv = 0.0225 + 0.15× 0.1 = 0.0375 for table B-3

Tables 4-1, 4-2 and 4-3 summarize these three tables by calculating the maximum and

average distances from the 95% CI of the Benchmark. Table 4-1 shows that prices in table B-1

are as much as 117¢ away from the 95% CI. In table 4-2 the maximum deviation from the CI

is 45¢. Most of the entries in this table show better results than the previous table. From table

4-3, it can be noted that the prices are closest to the benchmark for table B-3, both in terms of the

maximum deviation as well as the average deviation. In this case, even the fastest approximation

with 3 time steps produces a maximum deviation of 24¢ from the benchmark 95% CI.

However, we note interesting results when tables 4-1, 4-2 and 4-3 are compared to 4-5.

Although table 4-5 is better than 4-1 and 4-2 for all values, such is not the case while comparing

it with table 4-3. Table 4-3 has better values than 4-5 when 3 or 5 time steps are used to calculate

the boundary. Thus, it is observed that with a suitably chosen value for σ, we get better results by

calculating the boundary using a constant volatility model, when we are using a lesser number of

time steps. For higher number of time steps, stochastic volatility gives a better result. However

48

Table 4-4. Computational time (in seconds) for θ = 0.02253 5 10 25 50

Computation Time 0.015 0.01624 0.02684 0.10372 0.34872

Table 4-5. Stochastic Volatility Results for θ = 0.02251k5 10k5 10k10 50k10 100k5 100k10 100k25


% in CI 0% 4% 16% 12% 0% 12% 68%Computation Time 0.08 0.32 0.56 2.98 3.04 6.6 16.8

it has to be noted that the results obtained using constant volatility are also accurate enough

(average difference of ∼ 7-8¢ ) and take approximately half the amount of computational time.4.3.2 Scenario 2


θσv, θ = 0.09



% in CI 0% 0% 0% 0% 0%

Tables B-4, B-5 and B-6 give the prices of American put options with a long run mean

of volatility θ = 0.09. The values of σ used in the constant volatility model to calculate the

boundary in these tables are given as

σ2 =

θ −√

θσv = 0.09− 0.3× 0.1 = 0.06 for table B-4


θ −√

θσv = 0.09 + 0.15× 0.1 = 0.12 for table B-6

In this case, one difference with the previous analysis for scenario 1 is that here we have

a situation where the spot volatility (= 0.04) is different from the long run mean of volatility

θ (= 0.09). Tables 4-6, 4-7 and 4-8 analyze these three tables by calculating the maximum and


are as much as 18¢ away from the 95% CI. In table 4-7 the maximum deviation from the CI is

49





θσv, θ = 0.09



% in CI 4% 16% 16% 20% 20%

8¢. All the entries in this table show better results than the previous table. From table 4-8, it can

be noted that the prices are closest to the benchmark for table B-6, both in terms of the maximum

deviation as well as the average deviation. In this case even the fastest approximation with 3 time

steps produces a maximum deviation of 7¢ from the benchmark 95% CI.

Next we compare tables 4-6, 4-7 and 4-8 to 4-10. Here we find that tables 4-6 and 4-7 are

better than table 4-10 for boundaries calculated with 3 or 5 time steps while the reverse is true

for other time steps. However, in terms of average deviation from the CI, both tables 4-6 and 4-7

seem to do very well. Similarly, table 4-8 also has better values than 4-10 for the 3, 5 or 10 time

steps. Thus similar to section 4.3.1, it is observed that with a suitably chosen value for σ, we get

better results by calculating the boundary using a constant volatility model, when we are using

a fewer time steps (3, 5 or 10). For a higher number of time steps, stochastic volatility gives a

better result. However, this time the results obtained using constant volatility are closer than in

section 4.3.1 (difference of ∼ 2-4¢ ) and again take less than half the computational time.

50



Table 4-10. Stochastic Volatility for θ = 0.091k5 10k5 10k10 50k10 100k5 100k10 100k25



4.3.3 Scenario 3


θσv, θ = 0.2



% in CI 4% 8% 16% 16% 16%

Tables B-7, B-8 and B-9 give the prices of American put options with a long run mean of

volatility θ = 0.2. The values of σ used in the constant volatility model to calculate the boundary

in these tables are given as

σ2 =

θ −√

θσv = 0.2− 0.4474× 0.1 = 0.1553 for table B-7


θ −√

θσv = 0.0225 + 0.4474× 0.1 = 0.2447 for table B-9

Tables 4-11, 4-12 and 4-13 analyze these three tables by calculating the maximum and


are as much as 36¢ away from the 95% CI. In table 4-12 the maximum deviation from the CI is

25¢. Again all entries in this table show better results than the previous table. From table 4-13,

it can be noted that the prices are closest to the benchmark for table B-9, both in terms of the

maximum deviation as well as the average deviation. In this case even the fastest approximation

51





θσv, θ = 0.2



% in CI 28% 44% 56% 60% 64%

with 3 time steps produces a maximum deviation of 22¢ from the benchmark 95% CI while the

average deviation is just 5¢.

However when compared with table 4-15 it is observed that all the tables 4-11, 4-12 and

4-13 do better than stochastic volatility thereby showing that we get a better value when the

constant volatility boundary is used for calculating prices of options with a higher long run mean

of volatility.

4.4 Conclusion

In this chapter we continued our work from chapter 3, where we had found that calculating

the price of American options using rough approximations of the early exercise boundary bore

good results. A substantial amount of time was saved at the cost of a little loss in accuracy. This

approximation of the early exercise boundary was taken a step further in this chapter by using a

constant volatility model to calculate the boundary. The first problem that came up was that the

boundary obtained was only a two-dimensional one rather than the three-dimensional one needed

for the decomposition formula (Chiarella and Ziogas (2005)) given in equation 4–8 This problem

was solved by replicating the boundary along the missing volatility axis, so that it does not vary

along this axis. Once the boundary is thus obtained, it is plugged in to the stochastic volatility

decomposition formula to compute the price of the American option.

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Table 4-15. Stochastic Volatility Results for θ = 0.21k5 10k5 10k10 50k10 100k5 100k10 100k25



The performance of this technique was assesse

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AMERICAN OPTION PRICING UNDER STOCHASTIC ...ufdcimages.uflib.ufl.edu/UF/E0/02/26/41/00001/guha_s.pdfOur work involves pricing American options under stochastic volatility. In solving