Re: BS PDE

• To: mathgroup at smc.vnet.net
• Subject: [mg55005] Re: BS PDE
• From: "Alan" <info at optioncity.REMOVETHIS.net>
• Date: Wed, 9 Mar 2005 06:34:30 -0500 (EST)
• References: <d0ju6b\$n1p\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"GLP" <cigen at hotmail.com> wrote in message news:d0ju6b\$n1p\$1 at smc.vnet.net...
> 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?

I assume you are trying to solve for the value of a call option.

Writing the problematic condition, in your notation, as
D1(s)f(smax,t) == 1 should work in NDSolve.
Even better might be to use x = Log[s] as the spatial
variable. Pick say smax = 3 times the strike price (or the corresponding
xmax) and then increase it until the boundary effects disappear.

In reality that boundary is a singular one and 'no boundary conditions
are necessary' -- this means, in practice, that you can do various things
at some truncated boundary and it won't matter in the limit.

Of course, the PDE can be solved exactly, too, as Black & Scholes did
in terms of the error/cumulative normal function. But that is best
done 'by hand'.

regards,
alan

```

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