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>