sinc functions with application to finance Ali Parsa 1*, J. Rashidinia 2

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1 sinc functions with application to finance Ali Parsa 1*, J. Rashidinia 1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran , Iran *Corresponding author: aliparsa@iust.ac.ir School of Mathematics, Iran University of Science and Technology, Narmak, Tehran , Iran rashidinia@iust.ac.ir Abstract: In this paper, we consider Black-Scholes equation, that arises in the American option model when the stock price follows a diffusionprocess with jump components. We use the sinc method to solve this equation with its boundary conditions andfind thenumerical solution of the generalized Black Scholes partial differential equation.the advantage of our method is that the sinc functions satisfies the boundary conditions and vanishes in infinity. This method has simple programing and gives the goodresult compared to the previous methods. The quadrature, based on sinc functions, is very accurate and can be used to approximate the neededintegrals. Keywords: American option, Jump diffusion process, sinc function, sinc method. 1. Introduction In financial world, an option gives its holder the right (but not the obligation) to buy or sell a prescribed risky asset from the writer for a prescribed fixed price on or before a prescribed time in the future. The fixed prescribed price is called the exercise price or strike price, and the prescribed time in the future is called the maturity date or expiry date. There are different types of options for various purposes, for example, vanilla options (European call or put option, American call or put option etc.). In contrast to the original Black Scholes framework where for most of the option valuation problems closed-form formulas exist or standard numerical methods could be applied, in nonstandard financial models more complicated and precise techniques are required. We have chosen to extend the Black Scholes model, by exploring the jump diffusion (Poisson) model, see [6] and sinc method for approximation of the option prices.. Mathematical model: The modified Black Scholes PDE The Black-Scholes equation is a partial differential equation, which describes the price of the option over time. The key idea behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset just the rights way and consequently eliminate risk. This hedge implies that there is only one right price gor the option, as returned by the Black-Scholes formula given in following: t rs S 1 σ S V S rv0 (1) As a result the Black Scholes equation (1) is modified as a partial integral differential equation by adding an integral part that reflects the jump diffusion structure of the process []: 1

2 t rλks S 1 σ S V rλvλ S () endowed with initial and boundary conditions: V ( S, T ) max( K S,0) V ( S, T ) 0 ass andv (0, t ) K and with the additional condition for S lim, where η shows the jump amplitude and the function ρ satisfies 1 ln( ) ( ) 0, ( ) d 1, ( ) exp( ) 0 J J (3) where μ is the mean of the lognormal jump process and b(t) is the optimal exercise boundary which should be approximated. It is known that only the region S > b(t) has financial meaning. The financial parameters r, σ, λ, K, σ J, η are constant numbers in this mathematical model. The big letter K is the strike price of the option contract. After rescaling the variables the problem of pricing an American put option becomes []: t 1 y 1 σ V y λ rλk rλv y y, ln exp (4) And V(0,y)=max(1-b(0)e y,0)=0, for 0, as b(0)=1 V(0,τ)=1-b(τ) 0, lim, τ 0 The Eq. (4) is a partial integral differential equation; we first approximate the integral part of Eq. (4) by sinc quadrature and then implement the sinc basis function for the PDE part. 3. Sinc collocation method [5] The approximate solution to (4) is defined by

3 ,,,, (5) Where 1 and 1. The basis functions {,, are given as product of basis functions for the approximate one-dimensional problem, so that,, Where the conformal maps in the spatial domain are given by ln and ln (7) where and 0, that and,,. We define 1, / 0, 0, / 1, /, 3 1, Where. The approximation to the p-th derivative of F is given by (6),,,, (8) The weight function g is chosen relative to the order of the derivative that is to be approximated. For instance, if one wishes to approximate an p-th derivative, then the choice often cuffices. Then we get 1,,, (9) 3

4 ,,,,,, 3 6 (10) (11) The truncated (mapped) quadrature rule is, Then in (4) the integral part approximate by, ln ln exp,exp (13) By substituting (9) to (13) in (4) and collocating in the nods of and,,,,, we have 1 (1) ln, exp (14) 4

5 Then we have equations and unknown coefficient of and unknown values of and unknown values of then with initial condition and by collocating in,,, we obtain: 1),,, (15) ) V(0,τ)=1-b(τ),, By solving the nonlinear system (14), (15) and (16) we can obtain the unknown values. References: (16) [1] Black F, Scholes M.ThePricing of Options and CorporateLiabilities. TheJournal of PoliticalEconomy, 1973; 81(3): [] Golbabai A, Ahmadian D, Milev M.Radial basisfunctionswithapplicationtofinance: American putoption underjumpdiffusion. Mathematical and ComputerModelling, 01; 55: [3] Kadalbajoo M.K, Tripathi L.P, KumarA: A cubic B-spline collocation method for a numerical solution of thegeneralized Black Scholes equation, Mathematical and Computer Modelling, 01;55, [4] Cheng J, Zhang J.E. Princing American options analytically: A homotopy analysis method,journal of Economic Dynamics &control, Article in press. [5] Lund J, Bowers K.L: Sinc Methods for Quadrature and Differential Equations, SIAM,199. [6] Merton R: Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics , [7] Hull J.C.Options Futures and Other Derivatives. 8th ed., Prentice Hall PTR,