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I have the PDE $$ -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$ with initial and boundary conditions:

  1. $V(0,S)=max(E-S,0) $
  2. $V(t,S^*)=E-S^*(t) $
  3. $V(t,\infty)=0 $

$S^*$ finding from this condition - $\frac{\partial V(t,S)}{\partial S}\bigg|_{S=S^*}=-1$, and for $S\le S^*$ must be performed $\frac{\partial V(t,S)}{\partial t}=0$. Where $\sigma=0.2$, $r=0.08$, $E=100$.

I have to find $V(t,S)$ with $t=0.25$ and $S=100$.

So, I think that this PDE I can solve using finite difference method. Now I did approximation for PDE using forward and backward explicity methods. But I don't know how I can solve boundary condition. What I need to learn for solving this PDE? Or you can solve it, and tell me how you did it. :) Thanx!

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    Let me try to understand. $S^*$ is an unknown point that can be moving?2012-11-06
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    I think you're right. It's PDE with free boundary.2012-11-06
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    Can you elaborate on the deduction of the model, or provide some reference?2012-11-06
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    [Option Pricing Using The Explicit Finite Difference Method](http://www.goddardconsulting.ca/option-pricing-finite-diff-explicit.html). And [C++ in Finance](http://finance.bi.no/~bernt/gcc_prog/). Last link I used, because I need C++ algorithm for solve this PDE.2012-11-06
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    From a quick read, shouldn't the last term $-rS$ be $-rV$?2012-11-06
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    Of course, last term shoud be $-rV$2012-11-06
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    A first thing to consider is that, if $\frac{\partial V}{\partial t}\Big|_{S \le S^*} = 0$, then for $S \le S^*$ the solution does not depend on time, i.e. $V(t,S) = V(S)$ only. You can calculate $$\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0 \qquad 0 \le S \le S^*$$ and determine $S^*$ with that solution. Can you take it from there?2012-11-06
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    I can sovle this like a homogeneous second order differential equation?2012-11-06
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    Yes you can. Once you have the solution, I think you'll have enough to determine the discretization.2012-11-06
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    It should be a waste for using finite difference method on such separable linear PDE. Please use separation of variables first.2012-11-06
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    @doraemonpaul Can you show how to do it?2012-11-06
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    $S^*$ is a constant or what?2012-11-06
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    If you look at the second boundary condition, you will see that $S^*$ depends on $t$2012-11-06
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    @spoluer, you should improve the writing of some of the equations as some of the equations cannot show properly.2012-11-09

2 Answers 2