On the Solution of the Black-Sholes Equation with Jump Process

The research is financed by Center of Excellence in Mathematics Postgraduate Education and Research Development Office Faculty of Science at Mahidol University for financial support.(Sponsoring information) Abstract In this paper, we study the well known equation that is the Black-Scholes equation by considering its solution for the case of jump processes, particularly the jumps of prices of the stock models can be interpreted by the concepts of distribution theory. AMS Subject Classification: 91B28


Introduction
In the year 1973, F. Black and M. sholes has first introduced the well known equation that can be solved for the call option of the stocks.Such equation is named the Black-Scholes equation which is given by with the terminal condition u(s T , T ) = (s T − k) +  (2) denotes (s T − k) + = max(s T − k, 0), for 0 ≤ t ≤ T where u(s, t) is the option price at time t, r is the interest rate, s = s(t) is the price of stock at time t, s T is the price of stock on the expiration date at time T , k is the strike price and σ is the volatility of stock.They obtain such solution u(s, t) =sΦ which is call the Black-Sholes Formula where Φ denote by Michael, 2001).In this paper, we study such solution for the case of the jumps of s.Let s = s(t) be the prices of the stock having the jumps of magnitude a 1 , a 2 , a 3 , . . ., a m at time t = t 1 , t 2 , t 3 , . . ., t m respectively for 0 < t 1 < t 2 < t 3 , . . .< t m < T. By using the jumps in the distributional sense (I.M, 1964, p22) and the method of option prices with the stock model.We let where is a Heaviside function, and x(t) is a continuous functions for all t.Thus we obtain the Black-Sholes in (1) for the jump processes in the form with the terminal condition we obtain as a solution of (3) for the case of jump processes.For the case of no jumps that is a i = 0, equation (6) reduces to (1) and (5) reduces to (2).

Preliminaries
Let us consider the stock model where s = s(t) is the price of a stock at time t, μ is a drift and σ is a volatility of the stock s and B is the Wiener process or Brownian motion.Suppose s(t) has jumps of magnitude a 1 , a 2 , a 3 , . . ., a m at time t = t 1 , t 2 , t 3 , . . ., t m respectively.Let where is a Heaviside function.Now x(t) is a continuous functions for all t and the derivative dx(t) dt = ds(t) dt except t = t 1 , t 2 , t 3 , . . ., t m .Differentiable both sides of ( 8) and obtain where ds(t) dt is the derivative of s(t) in the distributional sense.Now from ( 7), ( 8) and ( 9) we obtain Now from (1) we have u(s, t) is the option price at time t.Let By using Itô chain rule, we have By ( 11), Since, we have (dB) 2 ≈ dt, thus dtdB ≈ (dt) 3/2 .Now, (dt) 2 and (dt) 3/2 are not first order, so we discard the terms (dt) 2 and dtdB.
From (10), we have since (dB) 2 ≈ dt and the terms (dt) 2 and dtdB are cancelled, Thus Now consider the term dtJ(t) and dBJ(t), the same as before, dtJ(t) = 0 and Published by Canadian Center of Science and Education Let u(s, t) = φs + ψp where φ is a number of shares of stock and ψ is a number of bonds and p is the value of a bond.Now, we have du(s, t) = φds + ψdp and where dp = rpdt, r is the interest rate.We set u(s, t) = v(x(t), t) thus we equate the dv from ( 12) and ( 13) and and choose φ = ∂v ∂x , we then obtain From ( 14) and ( 15) Equation ( 16) is the Black-Sholes P.D.E with jump processes.Now Put which is the Black-Scholes equation with R = x(t) + m i=1 a i H(t − t i ) and 0 ≤ t ≤ T .From (2), we have the terminal condition u(s, T ) = (s T − k) + , k is the strike price.Since ) is the price of stock at time t = T .In this work, we study the equation ( 17) with the terminal condition (18).So, we can say that the equation (1) with the terminal condition (2) is the Black-Sholes equation with no jump.If we have jump processes, we obtain ( 16).If we put R = x(t) + m i=1 a i H(t − t i ), we obtain (17) with the terminal condition (18).We next study the Black-Sholes formula with jump process.Recall that, for the call option price today is which is called the Black-Sholes formula where 2 T , k is the strike price and N is the cumulative standard normal distribution function For the jump processes, we have s(t) = x(t) + m i=1 a i H(t − t i ).Now, for t = T , s(T ) = s T = x(T ) + m i=1 a i H(T − t i ) and t = 0, s(0) = x(0), since m i=1 a i H(−t i ) = 0 for t i > 0 or s 0 = x 0 = x(0).Thus for the call option to day, we obtain Thus, we see that the call options to day for the jump and no jump are the same and can be computed from the same Black-Sholes formula.
We next study the solutions of the Black-sholes equation given by ( 17) with the given terminal condition (18).From the Black-Sholes formula we try the solution where 0 ≤ t ≤ T .We show that v(R, t) satisfies ( 17).At first we can verify that ke see [1, p.193], where R = x(t) + m i=1 a i H(t − t i ), N (d ± ) is the derivative of N(d ± ).From ( 19), ( 20)and ( 21), we can compute ∂v(R, t) and also ∂v(R, t) thus lefthand side are cancelled to be zero.That implies (17) holds.It follows that v(R, t) given by ( 19) is the solution of (17).

Main results
The work in the preliminaries section, starting form ( 7)-( 21) leading to the main theorem.
− t i ) and R T = x(T ) + m i=1 a i H(T − t i ) is the price of stock at time t = T for the jump processes, r is the interest rate and σ is the volatility of the stock.Then we obtain the unique solution of (22) satisfies (23) which is given by v(R, t) = RN(d + (T − t, R)) − ke −r(T −t) N(d − (T − t, R)), as the solution of the Black-Sholes equation with jump processes.Proof.Now, we haveH(t − t i ) = 1, for t i < t; 0, for t < t i .i = 1, 2, . . ., m