       Re: BS PDE

• To: mathgroup at smc.vnet.net
• Subject: [mg54997] Re: [mg54961] BS PDE
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 9 Mar 2005 06:34:23 -0500 (EST)
• References: <200503081003.FAA23271@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 8 Mar 2005, at 11:03, GLP wrote:

> Hi would to like to solve a PDE (exact or num) for the black & scholes
> model. it is
> D1(t) f(s,t)+r*s*D1(s)f(s,t)+0.5*b^2*s^2*D2(s)f(s,t)=r*f(s,t)
> But i dont know how to put this final condition
>
> lim f(s,t)-s=const. when s goes to + Inf.
>
> any suggest?
>
> thnks
>
> giacomo
>
>
>

Life is not as easy as that.

1. Mathematica's DSolve can't solve directly parabolic PDEs. It can't
even solve the Heat Equation. Mathematica can be helpful in solving the
BS equation but you have to guide it. You need to use a method like the
Fourier transform, which current DSolve does not use automatically.
Alternatively you can apply the Feynman-Kac theorem, in which
Mathematica can again be helpful  but it won't do it for you by itself.

2. NDSolve in version 5 and later can solve parabolic PDEs. But in any
case, solving directly the BS equation is, in general, the wrong
approach because of severe numerical instability. What you should do is
to transform the equation and the initial and boundary conditions into
the heat equation using standard transformations, NDSolve the heat
equation and finally transform the solution into a solution of the BS
equation. You will of course still have the problem with boundary
conditions at Infinity (and also -Infinity) for the heat equation. You
simply use suitable finite values for the endpoints of the grid over
which the solution is computed.

Andrzej Kozlowski

```

• References:
• BS PDE
• From: "GLP" <cigen@hotmail.com>
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