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MathGroup Archive 1999

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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.





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