help with pde

*To*: mathgroup at smc.vnet.net*Subject*: [mg16110] help with pde*From*: "Will Holt" <will at aventura.freeserve.co.uk>*Date*: Thu, 25 Feb 1999 08:25:03 -0500*Organization*: Customer of Planet Online*Sender*: owner-wri-mathgroup at wolfram.com

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.