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Re: help with pde

• To: mathgroup at smc.vnet.net
• Subject: [mg16151] Re: help with pde
• From: "Lawrence A. Walker Jr." <lwalker701 at earthlink.net>
• Date: Sat, 27 Feb 1999 03:23:13 -0500
• Organization: Morgan State University
• References: <7b357t$21s@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hello,

I don't know of an automatic way of doing it.  I never got a chance to check
this out but I think that in order to solve it you must:

1. Take the Laplace transform of the equation to change t into another variable.

2. Now we have a 2nd order DE in LaplaceTransform[V[s,t],t,s], so we can solve
this automatically or take the fouier transform, then apply the inverse fouier
after algebraically manipulating the equation.
3. Take the inverse LaplaceTransform.

Take Care,
Lawrence

Will Holt wrote:

> Hi everyone,
>
> I am trying to solve the Black-Scholes pde in mathematica subject to the
> usual boundary conditions.
>
> 0.5*sigma[0]^2*S^2*D[V[S,t],S,S]+r[0]*S*D[V[S,t],S]-r[0]*V[S,t]+*D[V[S,t],t]
> =0
>
> V[0,T]=0,V[S,T]=Max[S-K,0], for some particular S[0], K and T.
>
> The Mathematica command "NDSolve" requires that the first argument must have
> both an equation and an initial condition.
> The problem is that the law motion for S is
>
> dS=alpha[0]*S*dt+sigma[0]dZ
>
> where dZ is a Wiener process that can be substituted by epsilon*SQRt[dt],
> with epsilon a random drawing from a standardised normal distribution.
>
> I thought that I did not have to include this law motion for S in NDSolve,
> but in case I do have to include it, how can I do this?
>
> Also, on a different issue, how can I generate three series of normal random
> numbers that are correlated amongst each other: rho12, rho23, rho13? e.g.
> dX1=rho12dX2+SQRT[1-rho12^2]*de12, with de12 a standard Wiener process
> independent of dX2, and so on for dX2 and dX3.
>
> Any help will be greatly appreciated.
>
> Will.