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이학석사 학위논문

Pricing European options withdividends under a stochastic

volatility and stochasticinterest rate

(배당을 고려한 확률론적 변동성과 이자율 하에서의유럽형 옵션의 가격 결정)

2015년 2월

서울대학교 대학원

수리과학부

김 동 준



(배당을 고려한 확률론적 변동성과 이자율 하에서의유럽형 옵션의 가격 결정)

지도교수 강 명 주

이 논문을 이학석사 학위논문으로 제출함

2014년 12월

서울대학교 대학원

수리과학부

김 동 준

김 동 준의 이학석사 학위논문을 인준함

2015년 1월

위 원 장 (인)

부 위 원 장 (인)

위 원 (인)



A dissertation

submitted in partial fulfillment

of the requirements for the degree of

Master of Science

to the faculty of the Graduate School ofSeoul National University

by

Dongjun Kim

Dissertation Director : Professor Myungjoo Kang

Department of Mathematical SciencesSeoul National University

Feburary 2015

Abstract

In this paper, we can derive the first-order price approximation of European

options with dividends under a stochastic volatility and stochastic interest

rate by using Feynman-Kac Theorem and Poisson Equations. Under a risk-

neutral measure, we choose the Ornstein-Uhlenbeck process for a stochastic

interest rate. Lastly, we can obtain the result that corrected price will not

depend on the present volatility and compare a difference with Monte-Carlo

simulations.

Key words: Stochastic volatility, Stochastic interest rate, Feynman-Kac for-

mula, European options with dividend, Ornstein-Uhlenbeck Process, Poisson

equations

Student Number: 2012-20243

i

Contents

Abstract i

1 Introduction 1

1.1 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Multidimensional Stochastic Calculus . . . . . . . . . . . . . . 3

2 Model 7

2.1 Asset price model . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 11

3 Asymptotics for Pricing 13

3.1 The Formal Expansion . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Poisson Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Computation of Aε . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Accuracy of the Approxmation . . . . . . . . . . . . . . . . . 20

4 Numerical Results 22

5 Conclusion 29

Abstract (in Korean) 32

Acknowledgement (in Korean) 33

ii

List of Figures

1.1 Stock paths with S0 = 100, µ = 0.1, σ = 0.2 . . . . . . . . . . . 2

2.1 Interest rate for long run mean is 0.02. . . . . . . . . . . . . . 11

4.1 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 23

4.2 Explicit method . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Implicit method . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Crank-Nicolson method . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Explicit, Implicit method with Various maturity time t . . . . 27

4.6 Crank-Nicolson method with Various maturity time t . . . . . 27

iii

Chapter 1

Introduction

The object of this first section is to review the basic ideas, objects, results of

the classical pricing theory.

1.1 Risk-Neutral Pricing

The main result we want to make in this section is that there is a unique

probability measure P∗ equivalent to P such that, under this probability,

the discounted price e−∫ t0 rsdsSt is a martingale. By fundamental theorem

of asset pricing, the expected value under P∗ of the discounted payoff of a

derivative gives its no-arbitrage price. Such a probability measure describing

a risk− neutral world is called an equivalent martingale measure.[1]

The Black-Scholes basic model for the risky stock price model corresponds

to a continuous process St such that with a constant rate of return µ, and

centered random fluctuations independent of the past up to time t. These

fluctuations are modeled by σdWt, where σ is a positive constant volatility.

The corresponding process for the infinitesimal return is

dStSt

= µdt+ σdWt (1.1.1)

The right-hand side of (1.1.1) has the financial interpretation of a return

term(µ) plus a risk term(σ).

1

CHAPTER 1. INTRODUCTION

In integral form, (1.1.1) is

St = S0 + µ

∫ t

0

Stdt+ σ

∫ t

0

StdWt, (1.1.2)

where the last integral is a stochastic integral and S0 is the initial value.

Figure 1.1: Stock paths with S0 = 100, µ = 0.1, σ = 0.2

A function of the Brownian motion Wt defines a new stochastic process

h(Wt). By the simplest version of Ito’s formula,

dh(t, St) =

(∂h

∂t+ µ

∂h

∂s+

1

2σ2∂

2h

∂s2

)ds+ σ

∂h

∂sdWt. (1.1.3)

Let r is constant interest rate, we can compute the differential of the

discounted stock price g(t, St) = e−rtSt :

d(e−rtSt) = −re−rtStdt+ e−rtdSt

= e−rt(−rSt + µ)dt+ e−rtσdSt

= (µ− r)(e−rtSt)dt+ σ(e−rtSt)dWt.

(1.1.4)

2


The discounted stock price e−rtSt satisfies the same equation as St of

(1.2.1) where the expected return µ has been replaced by µ− r.In order to find a probability measure under which the discounted stock

price St = e−rtSt is a martingale,

dSt = σSt

(dWt +

(µ− rσ

)dt

). (1.1.5)

We set θ =(µ−rσ

), called the market price of stock risk [5]

or risk premium of stock price, and define

W ∗t = Wt +

∫ t

0

θds = Wt + θt, (1.1.6)

so that

dSt = σStdW∗t . (1.1.7)

We denote by ξθT = e−θWT− 12θ2T . By the property of log-normal distribu-

tion, ξθT has an expected value with respect to P equal to 1.

We now introduce the probability measure P∗. It is the equivalent measure

to P , meaning that both have the same null sets(they agree on which events

have probability 0), which has the density ξθT with respect to P .

Denoting by E∗· the expectation with respect to P∗, for any integrable

random variable X we have

E∗[X] = E[ξθTX].

The process (ξθt )0≤t≤T is called Radon−Nikodym process. [2]

1.2 Multidimensional Stochastic Calculus

This section defines multidimensional Ito formula, Girsanov’s theorem , Feynman-

Kac formula and develops their properties. These are are used to model the

value of a portfolio that results from trading assets in continuous time.

3


We consider the generalization of the stochastic differential equations

dSt = µ(t, St)dt+ σ(t, St)dWt to the case of systems of such equations:

dSit = µi(t, St)dt+d∑j=1

σi,jdWjt , i = 1, · · · , d,

where W jt are d−independent standard Brownian motions, and

St = (S1t , · · · , Sdt ) is a d−dimensional process. The d− dimensional Ito formula

can then be written:

df(t, St) =∂f

∂t(t, St)dt+

d∑i=1

∂f

∂si(t, St)dS

it

+1

2

d∑i,j=1

∂2f

∂si∂sj(t, St)d < Si, Sj >t,

(1.2.1)

where

d < Si, Sj >t=d∑

k=1

σi,k(t, St)σj,k(t, St)dt = (σσT )i,j(t, St)dt (1.2.2)

follows from the cross-variation rules d < t,W jt >= d < W j

t , t >= 0,

d < W it ,W

jt >= d < W j

t ,Wit >= 0 for i 6= j, and d < W i

t ,Wit >= dt.

(1.2.1) can then be rewritten:

df(t, St) =

(∂f

∂t+

d∑i=1

µi∂f

∂si+

1

2

d∑i,j=1

(σσT )i,j∂2f

∂si∂sj

)dt

+d∑i=1

∂f

∂si

(d∑i=1

σi,jdWjt

),

(1.2.3)

where the partial derivatives of f and the coefficients µ and σ are evaluated

at (t, St).

4


In (1.1.6) we have used a change of probability measure so that the

1−dimensional process W ∗t = Wt + θt becomes a standard Brownian mo-

tion under the new probability P∗. We assume that the d−dimensional pro-

cess (θt) is of the form (θj(St), j = 1, · · · , d),. We define the real process

(ξθt )0≤t≤T by:

ξθt = exp

(−

d∑j=i

(∫ t

0

θj(Ss)dWjs +

1

2

∫ t

0

θ2j (Ss)ds

)), (1.2.4)

which satisfies

dξθt = ξθt

d∑j=1

θj(St)dWjt .

and, therefore, is a martingale. Assume that E∫ T0‖θj‖2 ξθt

2ds < ∞, then

Girsanov′s− theorem states that the processes (W j∗t )0≤t≤T , j = 1, · · · , d,

defined by

W j∗t = W j

t +

∫ t

0

θj(Ss)ds, j = 1, · · · , d, (1.2.5)

are independent standard Brownian motions under P∗.Consider the stochastic differential equation

dSt = β(t, St)dt+ γ(t, St)dWt. (1.2.6)

Let h(s) be a Borel-measurable function and let r be constant. Fix T > 0,

and let t ∈ [0, T ] be given. Define the function

f(t, s) = Et,s[e−r(T−t)h(ST )],

assume that Et,s |h(ST )| <∞. By Feynman−Kac Theorem, then f(t, s)

satisfies the partial differential equation

ft(t, s) + β(t, s)fs(t, s) +1

2γ2(t, s)fss(t, s) = rf(t, s) (1.2.7)

and the terminal condition

f(T, s) = h(s).

5


Let h(S(T )) be the payoff at time T of a derivative security whose un-

derlying asset is the geometric Brownian motion

dS(t) = αS(t)dt+ σS(t)dW (t).

We may rewrite this as

dS(t) = rS(t)dt+ σS(t)dW (t),

where W (t)is a Brownian motion under the risk-neutral probability measure

P∗. The price of the derivative security at time t is

V (t) = E[e−r(T−t)h(S(T ))|F(t)].

Because the stock price is Markov and the payoff is a function of the stock

alone, there is a function p(t, s) such that V (t) = v(t, S(t)). Using Feynman-

Kac Theorem, function p(t, s) must satisfy the following discounted partial

differential equation. This is the Black− Shocles−Merton equation

vt(t, s) + rsvs(t, s) +1

2σ2s2vss(t, s) = rv(t, s).

One should never offer prices derived from a model that admit free-arbitrage.

So we define a Complete market model as every derivative security can

be hedged.

Followings are the Fundamental Theorem of Asset Pricing(FTAP).

If a market model has a risk-neutral probability measure, then it does not

admit arbitrage(FTAP1). Consider a market model that has a risk-neutral

probability measure. The model is complete if and only if the risk-neutral

probability measure is unique(FTAP2). All section provides conditions that

guarantee that such such a model does not admit arbitrage and that every

derivative security in the model can be hedged.[3]

6

Chapter 2

Model

2.1 Asset price model

In this section, we consider an underlying asset price model given by the

stochastic differential equations (SDEs)

dSt = (µt −D)Stdt+ f(Yt)StdWst , (2.1.1)

dYt =1

ε(m− Yt)dt+

ν√

2√εdW y

t , (2.1.2)

where St is the underlying risky asset price and f is a smooth positive

function that is bounded away from zero and f is the function of fast-mean

reverting process Yt. Also, W st and W y

t are standard Brownian motions mu-

tually correlated such that d〈W s,W y〉t = ρxydt. We note that the solution

of the Ornstein–Uhlenbeck SDE (2.1.2) is a Gaussian process with a distri-

bution given by Yt ∼ N(m + (Y0 − m)e−1εt, ν2(1 − e−

2εt)). So, its invariant

distribution is given by N(m, ν2). Later, we will use notation 〈 · 〉 for the

average with respect to this invariant distribution.

Since the model (2.1.1) and (2.1.2) describes an incomplete market, there

is a possibility of more than one equivalent martingale measure. Using nota-

tion γ (the market prices of volatility risk) for the non-uniqueness and rt for

7

CHAPTER 2. MODEL

an interest rate, the processes defined by

W(s)∗t = W s

t +

∫ t

0

(µt − rt)/f(Ys)ds,

W(y)∗t = W y

t +

∫ t

0

γ(Ys)ds

are Brownian motions under a risk-neutral measure, say P ∗γ , by the Gir-

sanov theorem, where we assume that (µt−rtf(Yt)

, γ(Yt)) satisfies the Novikov

condition. Here, γ is assumed to be smooth bounded functions of y only.

We can find a P (t, s, r, y) by trying to construct a hedged portfolio of

assets which can be priced by the no-arbitrage principle. Unlike in the Black-

Scholes case(Geometric Brownian Motion), it is not sufficient to hedge solely

with the underlying asset St, since the dW(s)∗t terms can be balanced, but

the dW(y)∗t term cannot. So we try to hedge with the underlying asset(P (1))

and another option which has a different expiration date(P (2)), and which

we also trade continuously.

Let P (1)(t, s, r, y) be the price of a European derivative with expiration

date T 1, and P (2)(t, s, r, y) be the price of another European derivative with

expiration date T 2 > T 1. We try to find processes M1t ,M

2t , Bt such that the

self − financing portfolio (No further funds or assets are required after

the initial investment so that if, for example, more of the asset is bought,

then money would have to be obtained by selling bonds to pay for it)

Πt = M1t P

(1)(t, St, r, Yt)−BtSt −M2t P

(2)(t, St, r, Yt)

instantaneously riskless (or hedged) at any time t < T .

By 2-dimensional Ito’s formula,

dg =

(∂g

∂t+ L(S,Y )g

)dt+

(f(Yt)St

∂g

∂s+ ρsy

ν

ε√

2

∂g

∂y

)dW s∗

t

+√

1− ρ2syν

ε√

2

∂g

∂ydW y∗

t

where LS,Y is the infinitesimal generator of (S, Y ) given by

8

CHAPTER 2. MODEL

LS,Y =1

2f 2(y)s2

∂2

∂s2+ ρsy

ν

ε√

2f(y)s

∂2

∂s∂y+ν2

4ε2∂2

∂y2

+ µts∂

∂s+

1

ε(m− y)

∂

∂y

for smooth function g.

By the self-financing property,

dΠt = M1t dP

(1)(t, St, r, Yt)−BtdSt −M2t dP

(2)(t, St, r, Yt),

and applying 2-dimensional Ito’s formula to P (1) and P (2) yields

dΠt = M1t dP

(1)(t, St, r, Yt)−BtdSt −M2t dP

(2)(t, St, r, Yt)

=

(M1

t

[∂

∂t+ LS,Y

]P (1) −BtµtSt −M2

t

[∂

∂t+ LS,Y

]P (2)

)dt

+ Stf(Yt)

[M1

t

∂P (1)

∂s−M2

t

∂P (2)

∂s−Bt

]dW s∗

t

+ ρsyν

ε√

2

[M1

t

∂P (1)

∂y−M2

t

∂P (2)

∂y

]dW s∗

t

+√

1− ρ2syν

ε√

2

[M1

t

∂P (1)

∂y−M2

t

∂P (2)

∂y

]dW y∗

t

.

We want to hedge the risk from W y∗t , we can choose

Σt = M1t

(∂P (2)

∂y

)−1(∂P (1)

∂y

).

To hedge the remaining risk from W s∗t , we choose

At = M(1)t

∂P (1)

∂s− Σt

∂P (2)

∂s.

For there to be no arbitrage, the now riskless portfolio Πt must grow at

the risk-free rate: dΠt = rΠtdt, which gives

9

CHAPTER 2. MODEL

(∂P (1)

∂y

)−1LP (1) =

(∂P (2)

∂y

)−1LP (2),

with the notation

L =∂

∂t+ LS,Y − (µt − r)s

∂

∂s− r · .

The left-hand side contains terms depending on T 1 but not T 2 and vice

versa for the right-hand side. Thus both sides must be equal to a function

that does not depend on expiration date T 1, T 2. We denote this function

Λ(y) =ρsy(µt − rt)

f(y)+ γ(y)

√1− ρ2sy

and γ(t, s, y) is an arbitrary function.

The Vasicek model is chosen as an interest model for rt. Then under the

risk-neutral measure P ∗γ the evolution dynamics of (2.1.1) and (2.1.2) are

transformed into

dSt = (rt −D)Stdt+ f(Yt)StdW(s)∗t , (2.1.3)

drt = (b− art) dt+ σdW(r)t = a

(b

a− rt

)dt+ σdW

(r)t , (2.1.4)

dYt =

(1

ε(m− Yt)−

ν√

2√ε

Λ(Yt)

)dt+

ν√

2√εdW

(y)∗t , (2.1.5)

where a (mean reversion rate of interest), b (average direction of interest

rate movement) and σ (volatility of interest rate) are constants and we assume

that W (r) is correlated with W (s)∗ such that d〈W (s)∗,W (r)〉t = ρsrdt but it

is independent of W (y)∗. Figure (2.1) shows the paths of interest rate from

(2.1.4). As one can notice from figure, if we assume more large volatility,

then interest rate tends to the long run mean more sensitive. Note that the

market price of interest rate risk is absorbed by the term b(t) based on an

argument in Pelsser (2000).

As opposed to stock prices for instance, interest rates cannot rise in-

definitely. This is because at very high levels they would hamper economic

10

CHAPTER 2. MODEL

activity, prompting a decrease in interest rates. Similarly, interest rates can

not decrease below 0. As a result, interest rates move in a limited range,

showing a tendency to revert to a long run value. So we choose OU-process

to model long-run interest rates.

Figure 2.1: Interest rate for long run mean is 0.02.

2.2 Problem Formulation

Considering the Markovian property of the joint process (St, rt, Yt) and by

using notation E∗{·} for the expectation with respect to the risk-neutral

11

CHAPTER 2. MODEL

measure, the no-arbitrage price of a European option with a payoff function

h(s) is given by

P ε(t, s, r, y) = E∗{e−∫ Tt rsdsh(ST )|St = s, rt = r, Yt = y}, (2.2.1)

where the dependence on the small parameter ε is expressed explicitly. As

usual, by the application of the Feynman–Kac formula (cf. Oksendal, 2002

for instance) to (2.1.3 - 2.1.5), we obtain a characterization of P ε(t, x, r, y)

as a solution of the terminal value problem

LεP ε(t, s, r, y) = 0, t < T,

P ε(T, s, r, y) = h(s),(2.2.2)

where the partial differential operator Lε is given by

Lε =1

εL0 +

1√εL1 + L2, (2.2.3)

by using the notation

L0 = (m− y)∂

∂y+ ν2

∂2

∂y2, (2.2.4)

L1 = ν√

2

(ρsyf(y)s

∂2

∂s∂y− Λ(y)

∂

∂y

), (2.2.5)

L2 =∂

∂t+

1

2f 2(y)s2

∂2

∂s2+ (r −D)(s

∂

∂s)− r·

+ ρsrσf(y)s∂2

∂s∂r+ (b− ar) ∂

∂r+

1

2σ2 ∂

2

∂r2

(2.2.6)

with the terminal condition P (T, s, r, y) = h(s).

12

Chapter 3

Asymptotics for Pricing

In this part, a multi scale asymptotic method(Fouque, Papanicolaou, Sir-

cac, and Solna(2011)) is used to derive an approximated European option

price for the model given by (2.1.3)-(2.1.5). Also, the error of the corrected

price is obtained.

3.1 The Formal Expansion

The method is to expand the solution

P ε = P0 +√εP1 + εP2 + ε

√εP3 + · · · , (3.1.1)

where P0, P1, · · · are functions of (t, s, r, y) to be determined such that

P0(T, s, r, y) = h(s). We are primarily interested in the first two terms

P0 +√εP1. The terminal condition for the second term is P1(T, s, r, y) = 0.

Also, we called P0 +√εP1 as corrected price.

Plugging (3.1.1) into (2.2.2), the leading order term P ε0 is given by the

unique solution to the problem(1

εL0 +

1√εL1 + L2

)P ε0(t, s, r, y) = 0 (3.1.2)

13

CHAPTER 3. ASYMPTOTICS FOR PRICING

with the terminal condition P ε0(T, s, r, y) = h(s).

To derive an explicit expression for P0 and P1, respectively, we substitute the

expansion (3.1.1) into (3.1.2) and obtain

1

εL0P0

+1√ε(L0P1 + L1P0)

+ (L0P2 + L1P1 + L2P0)

+√ε(L0P3 + L1P2 + L2P1)

+ · · · = 0.

(3.1.3)

Equating terms of order 1ε, we must have

L0P0 = 0. (3.1.4)

The operator L0 acts only on the y variable; hence P0 must be a constant

with respect to that variable, which implies that

P0 = P0(t, s, r), (3.1.5)

a function of (t, s, r) only.

Moreover, in order to eliminate the term in 1√ε

in (3.1.3), we must have

L0P1 + L1P0 = 0. (3.1.6)

The operator L1 given by (2.2.5) takes derivatives with respect to y,

and we therefore deduce from (3.1.5) that L1P0 = 0 and consequently that

L0P1 = 0. P1 also does not depend on y so that

P1 = P1(t, s, r), (3.1.7)

a function of (t, s, r) only. This implies in particular that the corrected

price P0 +√εP1 will not depend on the present volatility y.

14


3.2 Poisson Equations

From the O(1) terms in (3.1.3) and the y- independence of P1, we already

know L1P1 = 0 and have

L0P2 + L2P0 = 0 (3.2.1)

The variable x being fixed, L2P0 is a function of y since L2 involves f(y).

Focusing on the y dependence only, equation (3.2.1) is of the form

L0χ+ g = 0, (3.2.2)

which is known as a Poisson equation for χ(y) with respect to the

operator L0 in the variable y. Let us now consider Yt an Ornstein-Uhlenbeck

process as the solution of the stochastic differential equation (2.1.2)

dYt =1

ε(m− Yt)dt+

ν√

2√εdW y

t ,

where dW yt is a standard Brownian motion. We wrote the explicit solution

for Yt, which showed that Yt is a Gaussian process. Define the infinitesimal

generator of the Markov process Yt is the differential operator

L =1

ε(m− y)

∂

∂y+ν√

2√ε

∂2

∂2y.

We look for an invariant distribution, which will be a distribution for

Y ∗ such that ELg(Y ∗) = 0 for any smooth and bounded g. We denote the

density function of Y ∗ as Φ(y), which satisfies∫∞−∞Φ(y)Lg(y)dy = 0 for any

g. Using integration by parts yields∫∞−∞ g(y)L∗Φ(y)dy = 0, where L∗, the

adjoint of L, is L∗ = −1ε∂∂y

((m − y)·) + ν√2√ε∂2

∂2yand there are no boundary

terms because Φ,Φ′ → 0 as y → ±∞. If equation holds any smooth test

function g, then Φ should satisfy

L∗Φ =2ν2

εΦ′′ − 1

ε((m− y)Φ)′ = 0,

where differentiations are with respect to the variable y. Solving this

second-order ordinary differential equation with the constraint∫∞−∞Φ = 1

15


gives

Φ(y) =1√

2πν2exp

(−(y −m)2

2ν2

).

The centering condition of Poisson equation

< g >=

∫g(y)Φ(y)dy = 0 (3.2.3)

is necessary for (3.2.2) to admit a solution. We average (3.2.2) with respect

to the invariant distribution of Yt, integrate by parts, and use the adjoint

operator L∗ and its property L∗Φ = 0 :

< g > = − < L0χ >

= −∫

(L0χ(y))Φ(y)dy

=

∫χ(y)(L∗0Φ(y))dy

= 0,

(3.2.4)

where there are no boundary terms because Φ,Φ′ → 0 as y → ±∞.Using the bracket notation < · > denoting integration with respect to the

Gaussian N (m, ν2) density, the centering condition of (3.2.1) is expressed as

< L2P0 >= 0.

Since P0 does not depend on y, this is < L2 > P0 = 0. Therefore, we

obtain a terminal value problem for the first term of (3.1.1) as follows.

< L2 > P0(t, s, r) = 0, t < T,

P0(T, s, r) = h(s),(3.2.5)

where

< L2 > =∂

∂t+

1

2< f 2(y) > s2

∂2

∂s2+ (r −D)(s

∂

∂s)− r·

+ ρsrσ < f(y) > s∂2

∂s∂r+ (b− ar) ∂

∂r+

1

2σ2 ∂

2

∂r2:= Lsvsi

(3.2.6)

16


So, < L2 > exactly corresponds to a pricing operator for an asset pricing

model with a constant volatility replaced by the effective volatility√< f 2 > = σ. Then, from the already known formula in Brigo and Mercu-

rio(2007), P0 is given by the explicit solution

P0(t, s, r) = se−D(T−t)N(d1)−KP (t, T )N(d2),

d1 :=

12∆1 +

(1 + b

aβ

)∆2 − (r −D)

(e−a(T−t)−1

a

)+ ln( s

K)

12∆1 + 2∆2 + ∆3

,

d2 := d1 −√

∆1 + 2∆2 + ∆3

P (t, T ) := exp

(1

2∆3 −

b

β∆2 + (r −D)

(e−a(T−t) − 1

a

)),

∆1 := σ2 =< f 2 >=

∫ ∞−∞

f 2(y)Φ(y)dy,

∆2 :=β

a

(T − t+

e−a(T−t) − 1

a

),

∆3 :=σ2

a2

((T − t)− 3 + e−a(T−t)(e−a(T−t) − 4)

2a

)β :=

√< f 2 >σρsr

(3.2.7)

where N is the usual standard normal distribution function. The de-

tailed derivation of (3.2.7) uses Mellin−Transform. Definition of Mellin-

transform is given by Appendix. Also we note that P (t, T ) represents the cur-

rent price of pure discount bond from time t to maturity T under stochastic

interest rate.

In our situation, the centering condition of (3.2.1) gives

< L2P0 >= 0

Since P0 does not depend on y, this is < L2 > P0 = 0.

By (3.2.6), LsvsiP0 = 0 and we can write

L2P0 = L2P0− < L2P0 >=1

2

(f(y)2 − σ2

)s2∂2P0

∂s2+

1

2ρsrσ(f(y)− < f(y) >)s

∂2P0

∂s∂r.

17


The second-order correction P2, solution of the L0P2 = −L2P0, is then given

by

P2(t, s, r, y)

= −1

2L−10

((f(y)2 − σ2)s2

∂2P0

∂s2+ ρsrσ(f(y)− < f(y) >)s

∂2P0

∂s∂r

)=

1

2

((φ(y) + c(t, s, r))s2

∂2P0

∂s2+ ρsrσ(ψ(y) + d(t, s, r))s

∂2P0

∂s∂r

) (3.2.8)

where φ(y), ψ(y) is a solution of the Poisson equation

L0φ = f(y)2 − σ2,L0ψ = f(y)− < f(y) >

and c(t, s, r), d(t, s, r) is a constant in y that may depend on (t, s, r).

Next, we derive an expression for P1. The order√ε terms in the expansion

(3.1.3) give the following Poisson equation for P3:

L0P3 + L1P2 + L2P1 = 0 (3.2.9)

The solvability condition for this equation is

< L1P2 + L2P1 >= 0. (3.2.10)

P1 does not depend on y, we can rewrite this centering condition as

< L1P2 > + < L2 > P1 = 0. (3.2.11)

Using the fact that < L2 > P0 = 0 and (3.2.1), we can write

P2 = −L−10 (L2− < L2 >)P0 + c(t, s, r). (3.2.12)

But c(t, s, r) does not depend on y. So L1c(t, s, r) = 0, and then

< L1P2 >= − < L1L−10 (L2− < L2 >)P0.

To write the problem defining P ε1 =

√εP1, it is convenient to intro-

duce the operator Aε =√ε < L1L−10 (L2− < L2 >). With this notation,

LsvsiP ε1(t, s, r) = AεP0(t, s, r) with a terminal condition P ε

1(T, s, r) = 0

18


3.3 Computation of Aε

From (3.2.8) to (3.2.12), we deduce that

< L2 > P1 = LsvsiP1

= − < L1P2 >

=1

2< L1

(φ(y)s2

∂2P0

∂s2+ ρsrσψ(y)s

∂2P0

∂s∂r

)> .

(3.3.1)

First, operator Aε has the following explicit representation

AεP0(t, s, r) = LsvsiP ε1

=√εLsvsiP1

= −√ε < L1P2 >

= V ε3 s

3∂3P0

∂s3+ V ε

2 s2∂

2P0

∂s2+ U ε

3s3 ∂

3P0

∂s2∂r+ U ε

2s∂2P0

∂s∂r,

V ε3 :=

√εν√2ρsy < f

∂φ

∂y>

V ε2 := −

√εν√2< Λ

∂φ

∂y>

U ε3 :=

√εν√2ρsrσρsy < f

∂ψ

∂y>

U ε2 := −

√εν√2ρsrσ < Λ

∂ψ

∂y>

(3.3.2)

Second, the operator Lsvsi satisfies the following identities.

Lsvsi(−(T − t)f) = f − (T − t)Lsvsif

Lsvsi(sn∂nf

∂sn

)= sn

∂n

∂snLsvsif

Lsvsis∂

∂s

(sn∂nf

∂sn

)= s

∂

∂s

(sn∂n

∂sn

)Lsvsif

(3.3.3)

19


Using above two results, P ε1 is given explicitly in terms of P0.

P ε1 = −(T − t)AεP0

+ w1(T − t)(U ε3s∂

∂s

(s2∂2

∂s2

)+ U ε

2s2 ∂

2

∂s2

)P0

+ w2(T − t)(U ε3s∂

∂s

(s∂2

∂s∂r

)+ U ε

2s∂2

∂s∂r

)P0

(3.3.4)

where P0 is given by (3.2.7). Also, w1, w2 are functions of (T − t) and

given by

w1(T − t) =1

a

(T − t+

1− ea(T−t)

a

),

w2(T − t) = T − t+1− ea(T−t)

a.

By (3.2.7), (3.3.4) the corrected price is given explicitly by

P0 + P ε1 = P0 − (T − t)AεP0

+ w1(T − t)(U ε3s∂

∂s

(s2∂2

∂s2

)+ U ε

2s2 ∂

2

∂s2

)P0

+ w2(T − t)(U ε3s∂

∂s

(s∂2

∂s∂r

)+ U ε

2s∂2

∂s∂r

)P0

(3.3.5)

3.4 Accuracy of the Approxmation

Our object in this section is to show that the approximation corrected price

to the price P is of order ε, in the sense that

|P − (P0 + P ε1)| ≤ constant · ε.

In order to do this, one can use the expansion (3.1.1) up to order 3 in√ε

and write

20


P ε = P0 +√εP1 + εP2 + ε

√εP3 − Zε.

At time T, P (T, s, r, y) = P0(T, s, r, y) = h(s, r) and P1(T, s, r) = 0,

Zε(T, s, r, y) = ε(P2(T, s, r, y) +√εP3(T, s, r, y))

Let Lε = 1εL0 + 1√

εL1 +L2, then P ε solves the original equation LεP ε = 0.

We can compute LεZε:

LεZε = Lε(P0 +√εP1 + εP2 + ε

√εP3 − P ε)

=

(1

εL0 +

1√εL1 + L2

)(P0 +

√εP1 + εP2 + ε

√εP3)− LεP ε

=1

εL0P0 +

1√ε(L0P1 + L1P0)

+ (L0P2 + L1P1 + L2P0) +√ε(L0P3 + L1P2 + L2P1)

+ ε(L1P3 + L2P2 +√εL2P3)

= ε(L1P3 + L2P2) + ε32L2P3 = O(ε).

(3.4.1)

We also known that Zε = O(ε), define

F ε(t, s, r, y) = L1P3(t, s, r, y) + L2P2(t, s, r, y) +√εL2P3(t, s, r, y)

Gε(s, r, y) = P2(T, s, r, y) +√εP3(T, s, r, y),

(3.4.2)

then LεZε = εF ε and Zε(T, s, r, y) = εGε. Under the smoothness and bound-

edness assumption on the payoff function h and the boundedness of f(y)

and Λ(y), F ε and Gε are bounded uniformly in s, r and are at most linearly

growing in |y|; Consequently,

P (t, s, r, y) = P0(t, s, r) + P ε1(t, s, r) +O(ε),

which shows that the error in our approximation is of order ε.

21

Chapter 4

Numerical Results

In this section, we’ll compare Call option price using the various meth-

ods such as Explicit Finite Difference method, Implicit Finite Difference

method, Crank-Nicolson Finite Difference method, Monte-Carlo simulation

with closed solution P0(3.2.7) and Black-Scholes Formula. Also, we assume

that S0 = 970, K = 1000, r = 0.03, D = 0.01, t = 0, T = 1, < f 2(y) >= 0.01.

For a simple and convenient calculation, we assume that interest rate r is

constant. In this situation, (3.2.6) is convert to

< L2 >=∂

∂t+

1

2< f 2(y) > s2

∂2

∂s2+ (r −D)

(s∂

∂s

)− r· = 0

Therefore, we obtain a terminal value problem for the first term of (3.1.1) as

follows.

< L2 > P0(t, s) = 0, t < T,

P0(T, s) = h(s),

h(s) =

{ST −K if ST > K

0 if ST ≤ K

(4.0.1)

22

CHAPTER 4. NUMERICAL RESULTS

1. Monte−Carlo Simulation

Monte−Carlo simulation is based on random number generation; ac-

tually, we must speak of pseudo-random numbers, since nothing is random

on a computer. If we feed random numbers into a simulation procedure, the

output will be a sequence of random numbers. Given this output, we use sta-

tistical techniques to build an estimate of a quantity of interest. Intuitively,

the more replications we run, the more reliable our estimates will be. [6]

Figure 4.1: Monte-Carlo Simulation

23


2. Explicit Finite Difference Method

Starting from the initial conditions(j = 0), we may solve the equation for

increasing values of j = 1, 2, · · · ,M . Since the unknown values are given by

explicit expression, this approach is called explicit. [4]

∂P

∂t=Pi,j − Pi,j−1

∆t+O(∆t)

∂P

∂s=Pi+1,j − Pi−1,j−1

2∆s+O(∆s2)

∂2P

∂s2=Pi+1,j − 2Pi,j + Pi−1,j

∆s2+O(∆s2)

Figure 4.2: Explicit method

24


3. Implicit Finite Difference Method

If we use a forward approximation for the derivative with respect to time,

we get an explicit method. We get a completely different scheme if we use

a backward approximation. Thus, the unknown values are given implicitly,

which is where the implicit method. [4]

∂P

∂t=Pi,j+1 − Pi,j

∆t+O(∆t)

∂P

∂s=Pi+1,j − Pi−1,j

2∆s+O(∆s2)

∂2P

∂s2=Pi+1,j − 2Pi,j + Pi−1,j

∆s2+O(∆s2)

Figure 4.3: Implicit method

25


4. Crank−Nicolson Finite Difference Method

We have seen methods involving three points on one time layer and one on

a neighboring layer. It is natural to wonder if a better scheme may be obtained

by considering three points on both layers. The Crank−Nicolson scheme

may be analyzed in a more general framework. [4] We may think of using a

convex combination of two approximations of the second-order derivative in

the finite difference scheme(Assume that λ = 0.5):

∂P

∂t=Pi,j+1 − Pi,j

∆t+O(∆t)

∂P

∂s= λ

Pi+1,j+1 − Pi−1,j+1

2∆s+ [1− λ]

Pi+1,j − Pi−1,j2∆s

+O(∆s2)

∂2P

∂s2= λ

Pi+1,j+1 − 2Pi,j+1 + Pi−1,j+1

∆s2

+ [1− λ]Pi+1,j − 2Pi,j + Pi−1,j

∆s2+O(∆s2)

Figure 4.4: Crank-Nicolson method

26


(a) Explicit method (b) Implicit method

Figure 4.5: Explicit, Implicit method with Various maturity time t

(c) Crank-Nicolson method

Figure 4.6: Crank-Nicolson method with Various maturity time t

27


S0 = 970, K = 1000, r = 0.03, D = 0.01, < f 2(y) >= 0.01, σ = 0.1

Black-Scholes P0 Explicit F.D.M. Implicit F.D.M. Crank-Nicolson F.D.M.

33.9703 34.1212 34.0740 33.8393 34.1517

99 % confidence interval [33.0603 , 35.0927]

Yes Yes Yes Yes Yes


Yes Yes Yes Yes Yes

Table 4.1: Call option price with S0 = 970 using various method

S0 = 1000, K = 1000, r = 0.03, D = 0.01, < f 2(y) >= 0.01, σ = 0.1

Black-Scholes P0 Explicit F.D.M. Implicit F.D.M. Crank-Nicolson F.D.M.

44.8482 44.7856 44.9703 44.6806 44.4856


Yes Yes Yes Yes Yes


Yes Yes Yes Yes Yes

Table 4.2: Call option price with S0 = 1000 using various method

28

Chapter 5

Conclusion

In Conclusion, we can obtain formal derivation of the first-order price

approximation of European options with dividends under a stochastic volatil-

ity and stochastic interest rate. We choose the Ornstein-Uhlenbeck process

and Hull-White process for a stochastic volatility model and interest rate

model, respectively. Based on this chosen model, we can conclude two main

results. First, we can obtain a closed formula for the first-order European

option price with dividend under stochastic volatility and stochastic interest

rate by applying the centering condition of Poisson Equation and method

of Mellin transform. This formula is simply given by the cumulative density

function of standard normal distribution. We can find not only the closed

formula to European put options by using Put-Call Parity, but also the price

that shaped similar to European options by using the Mellin transform. Sec-

ond, we can observe that the result brings out a minor difference by using

Black-Scholes formula and other numerical methods: finite difference method

with stochastic volatility and Monte-Carlo Simulation. However, in Black-

Scholes formula, drift and diffusion terms of stochastic differential equations

are constant. So, we can conclude that the stochastic model brings out better

results than pricing that using a constant volatility and interest rate. And

for Monte-Carlo Simulation, the more replications we run, the more reliable

our estimates we can derive.

29

Appendix

To derive the (3.2.7), we first need to define Mellin transform. For

a locally Lebesgue integrable function f(y), y ∈ R+, the Mellin transform

M(f(y), w), w ∈ C, is defined by

M(f(y), w) := f(w) =

∫ ∞0

f(y)yw−1dy,

and if a < Re(w) < b and c such that a < c < b exist, the inverse of the

Mellin transform is expressed by

f(y) =M−1(f(w)) =1

2πi

∫ c+i∞

c−i∞f(w)y−wdw.

If we define the sequence of the payoff function hn(s) such that limn→∞ hn(s) =

h(s) and the call option price with the payoff function hn(s) = pn(T, s, r) as

pn(t, s, r) = E∗[exp

(−∫ T

t

r∗t dt∗)hn(ST )|St = s, rt = r

],

pn(t, s, r) satisfies the PDE given by (3.2.6). If we find the solution pn(t, s, r)

by using the Mellin transform, we can obtain the formula of the option price

P (t, s, r) from P (t, s, r) = limn→∞ pn(t, s, r). If we define pn(t, w, r) as the

Mellin transform of Pn(t, s, r), then

pn(t, s, r) =1

2πi

∫ c+i∞

c−i∞pn(t, w, r)s−wds.

30

Bibliography

[1] Shreve, Steven E. stochastic calculus for finance II Continuous-time

models, (2004).

[2] Jean-Pierre Fouque, George Papanicolaou, K. Ronnie Sircar, Multiscale

Stochastic Volatility for Equity Interest Rate and Credit Derivatives.

(2011), pp. 157-175.

[3] Jean-Pierre Fouque, George Papanicolaou, K. Ronnie Sircar, Derivatives

in financial markets with stochastic volatility (2000), pp. 501-556.

[4] Paolo Brandimarte, Numerical methods in finance and economics

(2006), pp. 1779-1799.

[5] Robert L. McDonald, Derivatives markets (2006), pp. 161-171.

[6] Don L. McLeish, Monte Carlo simulation and finance (2005), pp. 57-

108.

31

국문초록

이 논문에서 우리는 파인만-칵 정리와 포아송 방정식을 이용하여 확률론

적 변동성과 확률론적 이자율 모형 하에서의 배당을 고려한 유럽형 옵션의

가격을 유도한다. 확률론적 변동성은 오스틴-유린벡 과정을 통해 모형화하고

이를위험중립측도하에서유도한다.마지막으로변동성과독립인옵션가격의

근사치를 구하고 그 차이를 몬테-카를로 시뮬레이션과 비한다.

주요어휘: 확률론적 변동성모형, 확률론적 이자율모형, 파인만-칵 정리, 배당

이 존재하는 유럽형 옵션, 오스틴-유린벡 과정, 포아송 방정식

학번: 2012-20243

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