A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

http://dx.doi.org/10.12988/ams.2014.46424

A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

Joseph Ackora-Prah Department of Mathematics

Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Perpetual Saah Andam

Department of Mathematics

Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Samuel Asante Gyamerah

Department of Mathematics

Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Daniel Gyamfi

Department of Mathematics

Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Copyright © 2014 Joseph Ackora-Prah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited.

Abstract

Evolutionary computation have been used in different areas of re-

search in finance. The more the perfect price of option we obtain the

more attractive it becomes to the investors. Investors have developed

much interest in option investment but when the option is exercised at

a wrong time, it can lead to massive loss for the investor. This paper is

mainly focused on pricing a European put option when the underlying

security price is geometric mean reverting with the assumption that the

Girsanov change of measure has already been implemented and it has

a constant interest rate. We provide a Genetic Algorithm which gives a

perfect option price needed to be redeemed by the option buyer so as the option seller gets some profit rather than the asset expiring worthless.

Keywords: European put option, Geometric mean reverting model, Ge- netic Algorithm

1 Introduction

An option investment can turn into great massive gains for the investor. This is because an option allows an investor to control the profit potential of an investment many times the size of the actual amount the investor has at risk in the market [4]. When an investor invests in options, the investor protects himself or herself from total loss by taking positions on the option market that minimize risk through hedging. We use the Geometric mean reverting model to simulate the underlying asset price. It determines the proper valuation of an option and sets accurate prices for the options using available information obtained. In this paper we consider European option style.

A European option is an option that gives the right to the holder to trade an underlying asset S for prescribed price K at the expiry date without being obliged to do so. There are two types of European options namely; Euro- pean put option and European call option. European call option provides the holder the right to buy an underlying asset at the expiry date for the strike price without being under obligation to do so. European put option provides the holder the right to sell an underlying asset at the expiry date for the strike price without being under obligation to do so.

Genetic Algorithms evolved from both natural and artificial genetics. John Henry Holland was the key brain behind Genetic Algorithms. It has brought up many insight in using Genetic Algorithms to solve practical problems. He published a book known as “Adaptation in natural and artificial systems” in 1975 [6].

Ackora-Prah et al (2014) [4] presented a Genetic Algorithm to price a fixed

term American put option when the underlying asset price is Geometric Brow-

nian Motion. They used Genetic Algorithm and Black Scholes model to calcu-

late the option price and the optimal stopping time. They compared the per-

formances of the Genetic Algorithm and the Black Scholes model and found a

perfect price for the American put option using Genetic Algorithm which was

lower than that of Black Scholes model under the same condition. They con-

cluded that the Genetic Algorithm approach performed better than the Black

Scholes model.

Shu-Cheng and Lee (1997) [4] [9] presented the use of Genetic Algorithm in option pricing in which they concentrated on the European call option. They found the price of European call options whose exact solution was known from the Black Scholes option pricing theory using Genetic Algorithm. They use GENESIS 5.0 software and they noticed the boundary conditions using the Genetic Algorithm was arbitrarily imposed and it only satisfied the case when the stock price was greater than the exercise price. The solutions that they found using the basic Genetic Algorithm were compared to the exact solution and their results showed that Genetic Algorithm was a powerful tool for option pricing.

Investing in an asset that follows Geometric mean reverting model is a dif- ficult decision to take. This is because the price of the asset is always going down. What if the asset follows the Geometric Mean Reverting model, will Genetic Algorithm still perform better? This leaves decision to the investors as to whether to invest in it or not.

2 Preliminaries

Definition 2.1 Itˆ o Integral.

An Itˆ o integral is defined as, Z T

0

S _t d ˜ B _t =

n−1

X

i=0

S _i ( ˜ B _t

− ˜ B _t

),

where ˜ B _t is a standard Brownian motion adapted to the filtration F _t [8].

Lemma 2.2 Itˆ o Lemma.

Let S _t be a stochastic process and f (x, t) be a measurable function with con- tinuous partial derivatives up to the second order then,

df (t, S _t ) = ∂f

∂t (t, S _t )dt + ∂f

∂x (t, S _t )dS _t + 1 2

∂ ² f

∂x ² (t, S _t )(dS _t ) ² .

Let ˜ B : [0, T ] × Ω → R be a Wiener process defined up to T > 0 and let S : [0, T ] × Ω → R be a stochastic process that is adapted to the natural filtration F of a Wiener process . Then

E

"

Z T 0

S _t d ˜ B _t

 ² #

= E

Z T 0

S _t ² dt



.

2.1 Geometric Mean Reverting Model

Models predict the way prices of asset behave and the movement of the asset in the market. The geometric mean reverting Brownian motion is also known as the Black-Karasinski model. Suppose the price of the underlying asset S t

which follows the geometric mean reverting model is given as,

dS _t = k(ρ − ln S _t )S _t dt + σS _t d ˜ B _t , (1) from the model we determine the underlying asset price using the Itˆ o Lemma.

The assumption made is that the Girsanov change of measure has already been implemented and that ˜ B(t) is a Brownian motion with respect to the risk neu- tral measure Q and ρ being the constant force of interest. The degree of Mean Reverting, k and the volatility rate, σ are constants. ˜ B _t is a one-dimensional standard Brownian motion and d ˜ B _t ∼ N [0, dt]. S _t is the underlying asset price at time t.

Let ˜ B _t , 0 ≤ t ≤ T be a Brownian motion on a probability space (Ω, F , Q) and F (t), 0 ≤ t ≤ T be a filtration for this Brownian motion where T is a fixed final time.

We want to solve equation (1) explicitly, We let Y _t = ln S _t . Then we have,

∂Y _t

∂t = 0, ∂Y _t

∂S _t = 1

S _t , ∂ ² Y _t

∂S _t ² = − 1 S _t ² . From the Itˆ o Lemma we obtain,

dY _t = ∂Y _t

∂t dt + ∂Y _t

∂S _t dS _t + 1 2

∂ ² Y _t

∂S _t ² (dS _t ) ² ,

= 0 + 1 S t

dS _t + 1 2

 −1 S _t ²



(dS _t ) ² ,

= 1

S _t dS _t − 1

2S _t ² (dS _t ) ² . We use the fact that (d ˜ B _t ) ² = d ˜ B _t · d ˜ B _t = dt [1].

But we have,

(dS _t ) ² = 

k(ρ − ln S _t )S _t dt + σS _t d ˜ B _t  2

, taking the term having (d ˜ B _t ) ² only we obtain,

σ ² S _t ² (d ˜ B t ) ² = σ ² S _t ² (d ˜ B t ) ² = σ ² S _t ² dt.

Then,

dY _t = 1 S _t

h

k(ρ − ln S _t )S _t dt + σS _t d ˜ B _t i

− 1

2S _t ² σ ² S _t ² dt, dY _t = k(ρ − ln S _t )dt + σd ˜ B _t − σ ²

2 dt.

dY _t = k(ρ − Y _t )dt + σd ˜ B _t − σ ² 2 dt, dY _t =



kρ − σ ² 2



dt − kY _t dt + σd ˜ B _t , dY _t + kY _t dt =



kρ − σ ² 2



dt + σd ˜ B _t .

Using the integrating factor e ^{R kdt} = e ^kt we have, (dY _t + kY _t dt)e ^kt = e ^kt



kρ − σ ² 2



dt + σe ^kt d ˜ B _t , d(Y _t e ^kt ) = e ^kt



kρ − σ ² 2



dt + σe ^kt d ˜ B _t .

Then finding the underlying asset price S _t at time t, ∀ 0 < t ≤ T we have,

Z t 0

d(Y _s e ^ks ) = Z t

0

e ^ks



kρ − σ ² 2

 ds +

Z t 0

σe ^ks d ˜ B _s

Y _s e ^ks  t

0 =  e ^ks k



kρ − σ ² 2

 t 0

+ σ Z t

0

e ^ks d ˜ B _s

Substituting the limits and solving it further we obtain, Y _t = Y ₀ e ^−kt +

 ρ − σ ²

2k



1 − e ^−kt
+ σe ^−kt Z t

0

e ^ks d ˜ B _s , but Y _t = ln S _t then we have,

ln S _t = e ^−kt ln S ₀ +

 ρ − σ ²

2k



1 − e ^−kt
+ σe ^−kt Z t

0

e ^ks d ˜ B _s .

S t = exp



e ^−kt ln S 0 +

 ρ − σ ²

2k



1 − e ^−kt
+ σe ^−kt Z t

0

e ^ks d ˜ B s



. (2)

Then,

S _t = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ Z t

0

σe ^−kt e ^ks d ˜ B _s



. (3)

R t

0 e ^−kt e ^ks d ˜ B _s is an Itˆ o integral (2.1) with respect to Q , then, E

Z t 0

e ^−kt e ^ks d ˜ B _s



= 0.

The formula for finding the variance is given by, E

"

Z t 0

e ^−kt e ^ks d ˜ B _s

 ² #

−

 E

Z t 0

e ^−kt e ^ks d ˜ B _s

 ²

= Var(S _t )

but E

Z t 0

e ^−kt e ^ks d ˜ B _s



= 0 =⇒

 E

Z t 0

e ^−kt e ^ks d ˜ B _s

 ²

= 0 then from (2.2) we have,

E

"

Z t 0

e ^−kt e ^ks d ˜ B _s

 ² #

= E

Z t 0

e ^−2kt e ^2ks ds

 .

Then integrating and substituting the limits gives, E

"

Z t 0

e ^−kt e ^ks d ˜ B _s

 ² #

= 1

2k 1 − e ^−2kt
. Then it implies that,

E

Z t 0

e ^−kt e ^ks d ˜ B _s



= 0 and V ar

Z t 0

e ^−kt e ^ks d ˜ B _s



= 1

2k 1 − e ^−2kt
. Hence,

Z t 0

e ^−kt e ^ks d ˜ B _s ∼ N

 0, 1

2k 1 − e ^−2kt

 . Therefore, we can re-write S t from (3) as,

S _t = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ σY



, where Y ∼ N

 0, 1

2k 1 − e ^−2kt



.

(4)

At maturity we have,

S _T = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kT
+ σY



, where Y ∼ N

 0, 1

2k 1 − e ^−2kT

 . (5)

2.2 Finding the Expectation and Variance of Underly- ing asset Price

From (4) we have,

S _t = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ σY

 , let Y = V Z, V = 1

2k 1 − e ^−2kt
, Z follows the normal distribution that is Z ∼ N [0, 1] then it follows that,

S _t = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ σV Z



. (6)

Then finding the expectation of underlying asset price it follows that, E[S _t ] = E



S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ σV Z



, E[S _t ] = S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt



E e ^{σV Z}  ,

but,

E e ^{σV Z}  = e

, then the expectation of the underlying asset price is,

E[S t ] = S 0 e

exp



ρ − σ ² 2k



1 − e ^−kt
+ σ ² V ² 2



. (7)

Finding variance of the underlying asset price it follows that, V ar[S _t ] = E[S _T ² ] − (E[S _t ]) ² ,

E[S _t ² ] = E



S ₀ ^2e

exp

 2



ρ − σ ² 2k



1 − e ^−kt
+ σV Z



,

E[S _t ² ] = S ₀ ^2e

exp

 2

 ρ − σ ²

2k



1 − e ^−kt



E e ^{2σV Z}  , but,

E e ^{2σV Z}  = e ^2σ

^V

,

E[S _t ² ] = S ₀ ^2e

exp

 2

 ρ − σ ²

2k



1 − e ^−kt
+ 2σ ² V ²

 . Also, from (7) we obtain,

(E[S _t ]) ² =



S ₀ ^e

exp



ρ − σ ² 2k



1 − e ^−kt
+ V ² σ ² 2

 2

, then we get,

(E[S t ]) ² =

 S 0 2e

exp

 2

 ρ − σ ²

2k



1 − e ^−kt
+ V ² σ ²



. Then the variance of S _t is,

V ar[S _t ] = S ₀ ^2e

exp

 2

 ρ − σ ²

2k



1 − e ^−kt



e ^V

^σ



e ^σ

^V

− 1  .

Using Numerical Values to Simulate the Underlying As- set price

Fixed numerical values were used to simulate the underlying asset price (S t ) over the period of [0, T ] using equation (6). The following values were used;

S ₀ = $100, σ = 0.35, k = 1, ρ = 10% and T = 1. S ₀ is the initial underlying

asset price, σ is the volatility rate, ρ is the interest rate, k is the degree of

Mean Reverting and T is the maturity time.

Simulated underlying asset price

Figure 1: Simulated Geometric Mean Reverting Underlying Asset The graph in figure 1 displays a simulated underlying asset price over the period of [0, T ]. It can be observed from the graph that the underlying asset price is decreasing when there is an increase in time.

3 The Geometric Mean Reverting Model us- ing Genetic Algorithm

In European call option the holder expects the price of the underlying asset to rise at the expiry date. Let S _T be the underlying asset at the expiry date and K be the strike price then the holder of this European call option expects the payoff of S T − K for S T > K when the right is exercised and if the right is not exercised then it is zero (0).

For European put option, the holder expects the price of the underlying asset

to fall at the expiry date. Then we expect the holder of this European put

option’s payoff to be K − S _T for S _T < K when the right is exercised and if the

right is not exercised then it is zero (0).

We denote the payoff of a European put option as P _T = (K − S _T ) ⁺ . Let OP _T = OP (S _T ) be the price of the option at time T , then at maturity the payoff is OP _T = P _T .

We use this brief procedure to price a European put option when the un- derlying asset is geometric mean reverting.

We first generate random asset price using equation (5) and because we are writing a European put, we use a fitness function of max{K − S _T , 0}. We use Roulette wheel selection for the candidates to be drawn independently. We use one-point crossover and flip bit mutation because we decoded it into binary form.

4 Results and Discussion

We assign numerical values to the parameters to find the option price of a Eu- ropean put when the underlying asset price is geometric mean reverting. We let the initial underlying asset price, S ₀ = $100, the strike price, K = $120, the volatility rate σ = 0.35, the speed of reverting, k = 1, the interest rate, ρ = 10% and the maturity time T = 1 year. We used python software to price the European put option when the underlying asset price follows the Black- Karasinski model as given in equation (5).

We obtain the option price as $ 14.00 at maturity time. An option price of $14.00 means that the holder of the option should pay $14.00. The seller of the option buys assets and bonds at the initial time with the $14.00 received to make the same profit as the option buyer. This is because the movement of the prices is always going down and an investor will be at risk if the investors exercises a European call on this asset. A European put is supposed to be exercised on this asset and the investor is suppose to sell out the asset to make profit or else the asset will expire worthless.

5 Conclusion

We have used Geometric mean reverting model to simulate the underlying as-

set with Python software for the programming. Our Genetic Algorithm was

used to calculate for the option price under a European put. We found a

perfect option price for exercising a European put option that follows a Geo-

metric Mean Reverting asset. The result obtained from exercising a European

put option on a mean reverting asset was supportive and this will benefit the

option seller because if this asset is not sold off it will expire worthless.

Acknowledgements. We express our gratitude to the Almighty God and the Department of Mathematics, Kwame Nkrumah University of Science and Technology for providing us resources to help complete this research success- fully.

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Received: June 1, 2014