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


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1 Applied Mathematical Sciences, vol 8, 14, no 143, HIKARI Ltd, wwwmhikaricom A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model Joseph AckoraPrah 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 14 Joseph AckoraPrah et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract Evolutionary computation have been used in different areas of research 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
2 716 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi 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, Genetic 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; European 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 AckoraPrah et al (14 4 presented a Genetic Algorithm to price a fixed term American put option when the underlying asset price is Geometric Brownian Motion They used Genetic Algorithm and Black Scholes model to calculate the option price and the optimal stopping time They compared the performances 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 concluded that the Genetic Algorithm approach performed better than the Black Scholes model
3 A genetic algorithm to price an European put option 717 ShuCheng and Lee ( 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 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 difficult 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 Preliminaries Definition 1 Itô Integral An Itô integral is defined as, T S t d B n 1 t = S i ( B ti+1 B ti, i= where B t is a standard Brownian motion adapted to the filtration F t 8 Lemma Itô Lemma Let S t be a stochastic process and f(x, t be a measurable function with continuous partial derivatives up to the second order then, df(t, S t = f t (t, S tdt + f x (t, S tds t + 1 f x (t, S t(ds t Let B :, T Ω R be a Wiener process defined up to T > and let S :, T Ω R be a stochastic process that is adapted to the natural filtration F of a Wiener process Then ( T T E S t d B t = E St dt
4 718 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi 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 BlackKarasinski 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 Itô 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 neutral measure Q and ρ being the constant force of interest The degree of Mean Reverting, k and the volatility rate, σ are constants Bt is a onedimensional standard Brownian motion and d B t N, dt S t is the underlying asset price at time t Let B t, t T be a Brownian motion on a probability space (Ω, F, Q and F(t, 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 =, From the Itô Lemma we obtain, Y t S t = 1 S t, Y t S t = 1 St dy t = Y t t dt + Y t ds t + 1 Y t (ds S t St t, ( 1 (ds t, = + 1 ds t + 1 S t St = 1 ds t 1 (ds S t St t We use the fact that (d B t = d B t d B t = dt 1 But we have, (ds t = ( k(ρ ln S t S t dt + σs t d B t, taking the term having (d B t only we obtain, σ S t (d B t = σ S t (d B t = σ S t dt
5 A genetic algorithm to price an European put option 719 Then, dy t = 1 k(ρ ln S t S t dt + σs t d S B t 1 σ S t St t dt, dy t = k(ρ ln S t dt + σd B t σ dt dy t = k(ρ Y t dt + σd B t σ dt, dy t = (kρ σ dt ky t dt + σd B t, dy t + ky t dt = (kρ σ dt + σd B t Using the integrating factor e kdt = e kt we have, ( (dy t + ky t dte kt = e kt kρ σ d(y t e kt = e kt ( kρ σ dt + σe kt d B t, dt + σe kt d B t Then finding the underlying asset price S t at time t, have, d(y s e ks = Ys e ks t = e ks k ( e ks kρ σ (kρ σ ds + t + σ σe ks d B s e ks d B s < t T we Substituting the limits and solving it further we obtain, Y t = Y e + (ρ σ (1 + σe e ks d B s, but Y t = ln S t then we have, ln S t = e ln S + { S t = exp e ln S + (ρ σ (1 + σe e ks d B s (ρ σ (1 } + σe e ks d B s (
6 713 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi Then, S t = S e exp (ρ σ (1 + σe e ks d B s (3 e e ks d B s is an Itô integral (1 with respect to Q, then, E e e ks d B s = The formula for finding the variance is given by, ( ( E e e ks d B s E e e ks d B s = Var(S t but E e e ks d B s = = then from ( we have, ( E e e ks d B s = E ( E e e ks d B s = e t e s ds Then integrating and substituting the limits gives, ( E e e ks d B s = 1 ( 1 t Then it implies that, E Hence, e e ks d B s = and V ar e e ks d B s = 1 ( 1 t e e ks d B s N Therefore, we can rewrite S t from (3 as, S t = S e exp, 1 ( 1 t {(ρ σ (1 } + σy, where Y N, 1 ( 1 t (4
7 A genetic algorithm to price an European put option 7131 At maturity we have, S T = S e kt exp {(ρ σ (1 } kt + σy, where Y N, 1 ( 1 T (5 Finding the Expectation and Variance of Underlying asset Price From (4 we have, e S t = S exp {(ρ σ (1 } + σy, let Y = V Z, V = 1 ( 1 t, Z follows the normal distribution that is Z N, 1 then it follows that, S t = S e exp {(ρ σ (1 + σv Z} (6 Then finding the expectation of underlying asset price it follows that, e ES t = E S exp (ρ σ (1 + σv Z, e ES t = S exp (ρ σ (1 E e σv Z, but, E e σv Z = e σ V, then the expectation of the underlying asset price is, e ES t = S exp (ρ σ (1 + σ V (7 Finding variance of the underlying asset price it follows that, ES t = E V ars t = ES T (ES t, { e S exp (ρ σ (1 + σv Z},
8 713 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi but, ESt e = S exp (ρ σ (1 E e σv Z, E e σv Z = e σ V, ESt e = S exp (ρ σ (1 + σ V Also, from (7 we obtain, then we get, (ES t e = S exp (ρ σ (1 + V σ, (ES t = Then the variance of S t is, e S exp (ρ σ (1 + V σ e V ars t = S exp (ρ σ (1 ( e V σ e σ V 1 Using Numerical Values to Simulate the Underlying Asset price Fixed numerical values were used to simulate the underlying asset price (S t over the period of, T using equation (6 The following values were used; S = $1, σ = 35, k = 1, ρ = 1% and T = 1 S 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
9 A genetic algorithm to price an European put option 7133 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, 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 using 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 ( 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 (
10 7134 J AckoraPrah, P S Andam, S A Gyamera and D Gyamfi 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 underlying 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, } We use Roulette wheel selection for the candidates to be drawn independently We use onepoint 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 European put when the underlying asset price is geometric mean reverting We let the initial underlying asset price, S = $1, the strike price, K = $1, the volatility rate σ = 35, the speed of reverting, k = 1, the interest rate, ρ = 1% 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 at maturity time An option price of $14 means that the holder of the option should pay $14 The seller of the option buys assets and bonds at the initial time with the $14 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 asset 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 Geometric Mean Reverting asset The result obtained from exercising a European put option on a mean reverting asset was supportive and this will benefit the
11 A genetic algorithm to price an European put option 7135 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 successfully References 1 B Oksendal, Stochastic Differential Equations: An introduction with applications SpringerVerlag, New York, 6 (3 DA Coley, An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific Publishing, ( E V Mvanda, Pricing of a Perpetual American put option when stock prices are mean reverting University of Dar es Salaam, (1 4 J AckoraPrah, S K Amponsah, P S Andam and S A Gyamerah, A Genetic Algorithm for Option Pricing: The American Put Option, Applied Mathematical Sciences, Hikari ltd, 8 (14, J C Hull, Options, Futures, and other Derivatives, PrinticeHall international limited, ( J H Holland, Adaptation in natural and artificial systems, University of Michigan Press, ( K A Sidarto, On the Calculation of Implied Volatility using a Genetic Algorithm, Seminar Nasional Aplikasi Teknologi Informasi, (6 8 S E Shreve, Stochastic Calculus for finance, pringer Science and Business Media New York, 1 (4 9 SH Cheng and WC Lee, Option Pricing with Genetic Algorithm: The case of European  Style Options Proceedings of the seventh International Conference on Genetic Algorithms, ( Y Chen, S Chang, C Wu, A Dynamic Hybrid Option Pricing Model by Genetic Algorithm and BlackScholes Model, World Academy of Science,7 (1, Received: June 1, 14
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