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```Option Pricing Formula for Gao’s Stock Model
Xin Gao
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
[email protected]
Abstract
The option pricing problem is one of central contents in modern finance. In this paper, European option pricing formula for Gao’s stock model is formulated in fuzzy financial
market.
Keywords: fuzzy process, option pricing, Liu process
1
Introduction
Brownian motion was introduced to finance by Bachelier [1]. Samuelson [18] [19] proposed the
argument that geometric Brownian motion is a good model for stock prices. In the early 1970s,
Black and Scholes [3] and, independently, Metron [14] used the geometric Brownian motion to
determine the prices of stock options. Stochastic financial mathematics was founded based on
the assumption that stock price follows geometric Brownian motion. The Black-Scholes formula
has become an indispensable tool in today’s daily financial market practice.
Different from randomness, fuzziness is another type of uncertainty in real world. In order
to deal with the evolution of fuzzy phenomena with time, Liu [12] proposed a fuzzy process,
a differential formula and a fuzzy integral. Later, the community renamed them Liu process,
Liu formula and Liu integral due to their importance and usefulness, just like Brownian motion,
Ito formula and Ito integral. Some researches surrounding the subject have been made. You
[20] studied differential and integral of multi-dimensional Liu process. Qin [16] considered some
properties of analytic functions of complex Liu process. Dai [4] gave a reflection principle related
to Liu process.
As a different doctrine, Liu [12] presented an alternative assumption that stock price follows
geometric Liu process. Moreover, a basic stock model for fuzzy financial market was also proposed
by Liu [12]. We call it Liu’s stock model in order to differentiate it from Black-Scholes stock model.
Qin and Li [17] presented the European options pricing formula for Liu’s stock model. Gao [6]
presented a new stock model incorporating the mean reversion, which is a general economic
phenomenon. Considering the option pricing problem is a fundamental problem in financial
market, we investigate the European option pricing formula for Gao’s stock model.
The rest of this paper is organized as follows. Some basic concepts about Liu process are
recalled and Liu’s stock model is introduced in Section 2. Gao’s stock model is recalled and we
solve it in Section 3. European call and put option price formulas are derived in Sections 4 and
5. Finally, some conclusions are listed.
2
Liu’s Stock Model
Definition 1 (Liu [12]) A fuzzy process Ct is said to be a Liu process if
(i) C0 = 0,
(ii) Ct has stationary and independent increments,
(iii) every increment Ct+s − Cs is a normally distributed fuzzy variable with expected value et and
1
variance σ 2 t2 whose membership function is
&micro;
&micro;
π|x − et|
√
6σt
&micro;(x) = 2 1 + exp
&para;&para;−1
, −∞ &lt; x &lt; ∞.
Liu process is said to be standard if e = 0 and σ = 1. If Ct is a Liu process, then the fuzzy
process Xt = exp(Ct ) is called a geometric Liu process.
It was assumed that stock price follows geometric Brownian motion, and Black-Scholes stock
model was then founded based on this assumption. Liu [12] presented an alternative assumption
that stock price follows geometric Liu process. Liu [12] presented a basic stock model for fuzzy
financial market in which the bond price Xt and the stock price Yt follow
(
dXt = rXt dt
(1)
dYt = eYt dt + σYt dCt
where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion, and Ct is a
standard Liu process. It is just a fuzzy counterpart of Black-Scholes stock model [3].
3
Gao’s Stock Model
Gao[6] presented a model as a counterpart of Black-Karasinski model [1]. Let Xt be the bond
price, and Yt the stock price.
(
dXt = rXt dt
(2)
dYt = a(b − Yt )dt + σdCt
This model incorporates a general economic behavior: mean reversion. Mean reversion means
that the stock prices appear to be pulled back to some long-run average level over time. That is,
when the stock price Yt is high, mean reversion tends to cause it have a negative drift; when the
stock price Yt is low, mean reversion tends to cause it have a positive drift. In Gao’s model, the
stock price is pulled to a level b at rate a. Superimposed upon this pull is a normally distributed
fuzzy term σdCt .
We rewrite dYt = a(b − Yt )dt + σdCt as
exp(at)dYt + a exp(at)Yt dt = ab exp(at)dt + σ exp(at)dCt .
(3)
Here it is tempting to relate the left hand side to d(exp(at)Yt ). Using Liu’s formula, we obtain
d(exp(at)Yt ) = a exp(at)Yt dt + exp(at)dYt
(4)
Substituted in equation (3) this gives
d(exp(at)Yt ) = ab exp(at)dt + σ exp(at)dCt .
Then we have
Z
Z
t
t
exp(as)dCs
exp(as)ds + σ
exp(at)Yt − Y0 = ab
(5)
(6)
0
0
or
Z
Yt = exp(−at)(Y0 + σ exp(at)Ct + a
exp(as)(b − σCs )ds)
0
by integration by parts.
2
t
(7)
4
European Call Option Pricing Formula
A European call option gives the holder the right, but not the obligation, to buy a stock at a
specified time for a specified price. In this section, we consider European call option pricing
problem for our stock model.
Considering the new stock model, we assume that a European call option has strike price
K and expiration time T . Then the payoff from buying a European call option is (YT − K)+ .
Considering the time value of money, the present value of this payoff is exp(−rT )(YT − K)+ .
Definition 2 European call option price f for Gao’s stock model is defined as
f (Y0 , K, a, b, σ, r) = exp(−rT )
&Atilde;
Z
E  exp(−aT )(Y0 + σ exp(aT )CT + a
T
!+ 
.
exp(at)(b − σCt )dt) − K
0
where K is the strike price at time T .
5
European Put Option Pricing Formula
A European put option gives the holder the right, but not the obligation, to sell a stock at a
specified time for a specified price. In this section, we consider European put option pricing
problem for our stock model.
Definition 3 European put option price f for Gao’s stock model is defined as
f (Y0 , K, a, b, σ, r) = exp(−rT )
&Atilde;
Z
E  K − exp(−aT )(Y0 + σ exp(aT )CT + a
T
!+ 
exp(at)(b − σCt )dt)  .
0
where K is the strike price at time T .
6
Conclusions
In this paper, we investigated the option pricing problems based on Gao’s stock models in fuzzy
financial market. European call and put option price formulas for Gao’s stock model were defined.
Acknowledgments
This work was supported by National Natural Science Foundation of China Grant No.60425309.
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