Option Pricing with Stochastic Volatility and Market Price ...

Recent Advances in Finite Differences and Applied & Computational Mathematics

Option Pricing with Stochastic Volatility and Market Price of Risk: An Analytic Approach

NATTAKORN PHEWCHEAN*1, YONG HONG WU2, YONGWIMON LENBURY1 1Department of Mathematics, Faculty of Science, Mahidol University

Centre of Excellence in Mathematics, CHE, 328 Si Ayutthaya Road, Bangkok THAILAND

2Department of Mathematics & Statistics, Curtin University, Perth 6102 AUSTRALIA

*Corresponding author: nattakorn.phe@mahidol.ac.th

Abstract: - The purpose of this paper is to develop a European option pricing model taking into account stochastic volatility under a stochastic market price of risk (MPR) in Black-Scholes framework. Explicit formulae are derived for European call and put prices by following an analytical approach of Abraham Loui.

Key-Words: - European option pricing model, stochastic volatility, stochastic market price of risk, BlackScholes model.

1 Introduction

For decades, an option has been playing a very important role in financial world as a derivative financial instrument for traders who need to gain confidence to make a profit in the stock market. Many researchers have proposed and developed many interesting mathematical models for option pricing. The main goal is to find the best model to estimate the price of an option.

In Black-Scholes structure, option price C(t) is determined by the well-known factors: stock price S(t), strike price of an option K(t), interest rate r, time t, maturity date T and volatility [1]. Many

strong assumptions have been assumed under the Black-Scholes framework such as stock prices are normally distributed with known mean and constant volatility. To make the option pricing model more accurate, some assumptions of the Black-Scholes model are relaxed and some other parameters may be considered following the real-world circumstance.

The volatility assumption has been extended and generalized by allowing the volatility to vary. Scott [2], Hull and White [3], Wiggins [4] and Heston [5] have generalized the option pricing model by considering the stochastic behavior of volatility. The extension of the study of the stochastic volatility has been continued to analyze more properties on parameters by several authors [6-7].

In this paper, we relax the assumption of constant volatility to allow the stochastic property

for volatility and we consider the market price of risk parameter under the stochastic mean-reverting process.

2 The Model and Method

To establish the pricing models, the financial market is assumed to be complete and without arbitrage. Under the probability space (, P, F), is the pricing outcomes space, F is the -algebra denoting

measurable events, and P is the probability measure.

There exists a fixed martingale measure Q which

presumably is equal to the probability measure P

such that the asset price discounted at the risk-free

interest rate is martingale. This assumption

guarantees that the market has no arbitrage

opportunity [6]. All stochastic processes in such any

pricing environment are adapted to the filtration

{Ft }, generated by the Wiener processes. According to Girsanov's theorem with multiple Brownian

motions, there exist Ft -adapted processes k1(t) and k2 (t) . The equivalent martingale measure Q and the measurable probability P are related by the

following Radon-Nikodym derivative equation [8].

dQ

dP

Ft

exp

t

k1 (s)dW1 ( s)

0

t

k2 (s)dW2 (s)

0

1 t

2 0

k1(s)2 k2 (s)2

ds

.

(1)

Defined on a complete probability space (, P,

F), Brownian motions dW1(s) and dW2 (s) are one-

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Recent Advances in Finite Differences and Applied & Computational Mathematics

dimensional uncorrelated Weiner Process. The risky asset price corresponds to a stock process S(t) such that, in an infinitesimal amount of time dt, the infinitesimal amount of stock price dS(t) has mean

S (t, S(t))S(t)dt . The stock price can be modelled by

the following stochastic differential equation

dS (t ) S (t )

S

(t,

S (t ))dt

(t )dW1 (t )

(2)

where S(0) 0 . S (t, S(t)) is stock yield and (t) is

mean-reverting stochastic volatility which follows the process

d (t) (t) dt 1dW1(t) 2dW2(t) (3)

where (0) 0, , , 1 and 2 are constants. Eq. (3) is correlated to (2). Obviously, this

correlation allows the relationship between corresponding parameters S (t) and (t). In a

particular case, the absence of correlation between two processes can be derived by setting the

parameter 1 or 2 equal to zero.

The market price of risk or MPR is considered as

a risk factor ki which follows the mean reverting

process

dki (t) ki (ki ki (t))dt kidWi (t)

(4)

where k(0) 0. ,ki , ki and ki are constants, i= 1,

2.

The models in (2) and (3) are affected by the

MPR; as a result, in the next section, the pricing

models are remodelled to capsulate the risk factors

which yield some advantages for asset pricing.

By following Abraham Lioui's analytic approach

[3], the following proofs are derived. With the

assumptions of arbitrage-free and frictionless

market, the formulae of European options are

proved analytically.

When MPR ki (t) is considered, the kernel of movement is changed. As a result, the Wiener

processes Wi (t) of the models (2)?(4) are replaced

by W i (t) where

t

W i (t) Wi (t) ki (s)ds .

(5)

0

The dynamic models (2)?(4) are changed. The

equation of the stock price (2) becomes

dS (t ) S (t )

S

(t, S(t))dt

(t)dW 1(t)

.

(6)

Because of the fact that MPR has an influence on the directional movement of the underlying asset, the stochastic volatility follows the process

d (t)

(t) 1

1k1

(t

)

2

k2

(t

)

dt

1dW 1(t) 2dW 2 (t)

(7)

when MPR is defined by Ornstein-Uhlenbeck

process

dki (t) ki ki ki (t) dt kidW i (t) (8)

where ki ki (1 ki ) and ki ki / 1 ki .

European call option is valued first in order to obtain European put option formula by using the concept of put-call parity. The European call option C(t) is defined as [3]

C(t) EQ er(T t)

S(T ) K

|

Ft

(9)

er(Tt)EQ S(T)1S(T)K | Ft Ker(T t)EQ 1S(T )K | Ft (10)

when

1S (T )K

1 0

if if

S(T ) K otherwise

.

We assume the distribution of stock price S(T) is

modeled as follows.

S(T)S(t)expr12Tt(s)ds2Tt

T(s)dsT dW1(s)

t

t

(11)

By using the analytic approach of Abraham Lioui

[3] and applying simple integration techniques, we

can derive integral term

T (s)ds (t) 1e (Tt)

t

(T t) 1

1 e (Tt)

1

k1

(T

t)

1 k1

1 e k1 (T t )

2

k1 k1

1 e (T t )

1 k1

1 e k1 (T t )

2 k 2

(T

t)

1 k2

1 e k 2 (T t )

2

k2 k 2

1 e (T t)

1 k 2

1 e k 2 (T t )

e e (Tt)

k1(T t)

1

k1

1

k1

1 e k1 (T t)

k1

(t

)

2

1 k2

e e (Tt) k2(Tt)

1

k2

1ek2(Tt)

k2(t)

1

T t

1

k1 k1

k1 k1

e k1 (T s)

e (T s)

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Recent Advances in Finite Differences and Applied & Computational Mathematics

k1 k1

e e (T t )

k1 (T t )

dW

1

(s)

2

T t

1

k k

2 2

k2 k2

e k 2 (T s)

e (T s)

k2 k 2

e e (T t)

k 2 (T t )

dW

2

(s)

.

(12)

By substituting (12) into (11), the formula for

stock pricing is derived. However, since the integral

in (12) is in a very complicated form. We will

derive the formula for stock prices and option prices

by simplification.

Now we define functions as follows.

M1

(t

)

(t

)

1 e (T t )

M 2 (t)

(T t) 1

1 e (T t )

M3(t)

1 k1

(T

t)

1 k1

1 e k1 (T t )

2

k1 k1

1 e (T t )

1

k1

1 e k1 (T t )

M4 (t)

2 k 2

(T

t)

1 k2

1 e k 2 (T t )

2

k2 k 2

1 e (T t)

1

k 2

1 e k 2 (T t )

M5(t)

1

1 k1

e e (T t )

k1 (T t )

1

k1

1 e k1 (T t )

k1

(t

)

M6 (t)

2

1 k2

e e (T t)

k 2 (T t )

1

k2

1 e k 2 (T t )

k2

(t

)

P1 ( s)

1

1

k1 k1

k1 k1

e k1 (T s)

e

(T s)

k1 k1

e e (T t )

k1 (T t )

P2 (s)

2

1

k k

2 2

k 2 k2

e k 2 (T s)

e (T s)

k2 k 2

e e (T t )

k 2 (T t )

From (11), we consider the integral term

T

(s)ds2

R(t)

T

P1(s)dW1(s)2

T

P2 (s)dW 2 (s)2

t

t

t

2

6 i1

Mi

(t)

T t

P1(s)dW1(s)

2

6 i1

Mi

(t)

T t

P2 (s)dW 2

(s)

T

T

2 P1(s)dW 1(s) P2 (s)dW 2 (s) . (13)

t

t

By applying It isometry, we have

T

(s)ds

2

R(t)

T

P12 (s)ds

T

P22 (s)ds

t

t

t

2

6 i1

Mi

(t)

T t

P1(s)dW1(s)

2

6 i 1

Mi

(t)

T t

P2 (s)dW 2 (s)

TT

2 P1(s) P2 (s)dW 1(s)dW 2 (s) .

(14)

tt

T

(s)ds

2

R(t)

T

P12 (s)ds

T

P22 (s)ds

t

t

t

2

6

Mi

(t)

T

P1 ( s)dW

1(s)

i1

t

2

6 i 1

Mi

(t

)

T t

P2 (s)dW

2

(s)

T

2

t

P3 (s)dW

3 (s)

.(15)

where P1(s) P2 (s) P3 (s) , and we assume

TT

T

dW 1(s)dW 2 (s) dW 3 (s) . Next we consider

tt

t

T

(s)ds

T

dW

1

(s)

T

P12

(s)ds

t

t

t

6

M

i

(t

)

T

P1

(s)dW

1(s)

T

P3 (s)dW 3 (s) . (16)

i1

t

t

We substitute (15) and (16) into (11), we obtain

S

(T

)

S

(t)

exp

r

(T

t)

(t

T 2

)

R(t

)

(2

t 2

T

)

T t

P12

(s)ds

t

T 2

T t

P22

(s)ds

(1

t

T

)

6

M

i

(t

)

T

P1

(s)dW

1(s)

i1

t

(t

T)

6 i1

Mi

(t)

T t

P2

(s)dW2

(s)

(1t

T

T)

t

P3(s)dW3(s)(17)

We denote

(t)2

(1

t

T

)

6

M

i

(t)

T

P1 ( s)2 ds

i1

t

(t T)

6

Mi

(t

)

T

P2

(s)2ds

T

(1t T) P3(s)2ds

(18)

i1

t

t

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Recent Advances in Finite Differences and Applied & Computational Mathematics

(t)t2TR(t)

2

t T 2

T t

P12 (s)ds

t

T 2

T t

P22 (s)ds (19)

From (10), we solve EQ 1S(T )K | Ft by letting

be

d2

1 (t)

ln

S (t ) K

r(T

t)

(t)

.

(20)

Since follows a standard normal distribution, thus

EQ 1S (T )K | Ft N (d2 )

(21)

where

d2

1 (t)

ln

S(t) K

r(T

t)

(t)

.

Next

we

solve

EQ S (T )1S (T )K | Ft EQ S (T )1d2 | Ft

S (t) expr(T t) (t) EQ e (t)1d2 | Ft

S (t) expr(T t) (t) e(1/ 2) (t)2 N (d1) (22)

where d1 d2 (t).

By applying (17)?(22) in (10), the European

option price with stochastic volatility and stochastic

market price of risk can be formulated.

When MPR ki (t) is considered, the kernel of movement is changed. As a result, the Wiener

processes Wi (t) of the models (2)?(4) are replaced

by Wi (t) where

t

W i (t) Wi (t) ki (s)ds

(5)

0

The dynamic models (2)?(4) are changed. The

equation of the stock price (2) becomes

dS (t ) S (t )

S

(t,

S (t ))dt

(t)dW

1 (t )

(6)

Because of the fact that MPR has an influence on the directional movement of the underlying asset, the stochastic volatility follows the process

d (t)

(t) 1

1k1 (t) 2 k2 (t)dt

1dW 1(t) 2dW 2 (t)

(7)

when MPR is defined by Ornstein-Uhlenbeck

process

dk i (t) ki ki k i (t)dt ki dW i (t)

(8)

where ki ki (1 ki ) and ki ki / 1 ki .

European call option is valued first in order to obtain European put option formula by using the concept of put-call parity. The European call option C(t) is defined as [3]

C(t) EQ er(T t)

S(T ) K

|

Ft

(9)

er(Tt)EQ S(T)1S(T)K | Ft Ker(Tt)EQ 1S(T)K | Ft (10)

when

1S (T )K

1 0

if if

S(T ) K otherwise

.

We assume the distribution of stock price S(T) is

modeled as follows.

S (T

)

S

(t)

exp

r

1 2

T t

(s)ds

2

T

t

T

(s)ds

T

dW

1

(s)

(11)

t

t

By using the analytic approach of Abraham

Lioui [3] and applying simple integration

techniques, we can derive integral term

T (s)ds

(t)

1 e (Tt)

t

(T t) 1

1 e (T t)

1 k1

(T

t) 1 k1

1 e k1(T t)

2

k1 k1

1 e (T t)

1 k1

1 e k1(T t)

2 k2

(T

t) 1 k2

1 e k 2 (T t)

2

k2 k2

1 e (T t)

1 k2

1 e k 2 (T t)

1

1

k1

e

(T

t)

e k1 (T t)

1

k1

1 e k1(T t)

k1 (t)

2

1 k2

e (Tt)

ek2(Tt)

1

k2

1ek2(Tt)

k2 (t)

1

T t

1

k1 k1

k1 k1

e k1 (T s)

e (T s)

k1 k1

e e (T t )

k1 (T t )

dW

1

(s)

2

T t

1

k k

2 2

k2 k2

e k 2 (T s)

e (T s)

k2 k 2

e e (T t)

k 2 (T t )

dW

2

(s)

(12)

By substituting (12) into (11), the formula for stock

pricing is derived. However, since the integral in

(12) is in a very complicated form. We will derive

the formula for stock prices and option prices by

simplification. Now we define functions as follows.

ISBN: 978-1-61804-184-5

137

Recent Advances in Finite Differences and Applied & Computational Mathematics

M

1

(t

)

(t

)

1 e (T t )

M 2 (t)

(T t) 1

1 e (T t )

M3(t)

1 k1

(T

t)

1 k1

1 e k1 (T t )

2

k1 k1

1 e (T t )

1 k1

1 e k1 (T t )

M4 (t)

2 k 2

(T

t)

1 k2

1 e k 2 (T t )

2

k2 k 2

1 e (T t)

1

k 2

1 e k 2 (T t )

M5(t)

1

1 k1

e e (T t )

k1 (T t )

1

k1

1 e k1 (T t )

k1

(t

)

M6 (t)

2

1 k2

e e (T t)

k 2 (T t )

1

k2

1 e k 2 (T t )

k2

(t

)

P1 ( s)

1

1

k1 k1

k1 k1

e k1 (T s)

e (T s)

k1 k1

e e (T t )

k1 (T t )

P2 (s)

2

1

k k

2 2

k 2 k2

e k 2 (T s)

e (T s)

k2 k 2

e e (T t )

k 2 (T t )

From (11), we consider the integral term

T

(s)ds2

R(t)

T

P1(s)dW1(s)2

T

P2

(s)dW2

(s)2

t

t

t

2

6 i1

Mi

(t)

T t

P1(s)dW1(s)

2

6 i1

Mi

(t)

T t

P2(s)dW2(s)

T

T

2 P1(s)dW 1(s) P2 (s)dW 2 (s) (13) .

t

t

By applying It isometry, we have

T

(s)ds

2

R(t)

T

P12 (s)ds

T

P22 (s)ds

t

t

t

2

6 i 1

Mi

(t)

T t

P1 ( s)dW

1(s)

2

6 i 1

Mi

(t)

T t

P2

(s)dW

2

(s)

T

2

t

P3

(s)dW

3 (s)

(14)

.

where P1(s) P2 (s) P3 (s) , and we assume

TT

T

dW 1(s)dW 2 (s) dW 3 (s) . Next we consider

tt

t

T

(s)ds

T

dW

1(s)

T

P12

(s)ds

t

t

t

6 i 1

Mi

(t)

T t

P1 ( s)dW

1(s)

T t

P3

(s)dW

3 (s)

(15)

We substitute (15) and (16) into (11) to obtain

S(T )

S(t) exp r(T

t)

(t

T) 2

R(t)

(2

t 2

T

)

T t

P12

(s)ds

t

T 2

T t

P22

(s)ds

(1

t

T

)

6

M

i

(t)

T

P1

(s)dW

1

(s)

i1

t

(t

T)

6 i1

Mi

(t)Tt

P2(s)dW2(s)

(1t

T

T)

t

P3(s)dW3(s)(16)

We denote

(t)2

(1

t

T

)

6

Mi

(t)

T

P1 ( s)2 ds

i1

t

(t

T

)

6

M

i

(t)

T

P2

( s)2 ds

T

(1t T) P3(s)2ds (18)

i1

t

t

(t)

(t

T 2

)

R(t)

(2

t 2

T

)

T t

P12

(s)ds

(t

T 2

)

T t

P22

(s)ds

From (10), we solve EQ 1S(T)K | Ft by letting be

d2

1 (t

)

ln

S (t ) K

r(T

t)

(t)

.

(19)

Since follows a standard normal distribution,

EQ 1S (T )K | Ft N (d2 )

(20)

where

d2

1 (t)

ln

S(t) K

r(T

t)

(t)

.

Next

we

solve

EQ S (T )1S (T )K | Ft EQ S (T )1d2 | Ft

S (t) expr(T t) (t) EQ e (t)1d2 | Ft

S (t) expr(T t) (t) e(1/ 2) (t)2 N (d1) (21)

where d1 d2 (t) . By applying (17)?(21) in (10), the European

option price with stochastic volatility and stochastic market price of risk can be formulated.

ISBN: 978-1-61804-184-5

138

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