Questions about BS model

[baozi]

Hello everyone:
I need some hints about bs-model.
So basically stock price follow winer process or ito lemma where I could get the PDE.
The PDE+Boundary condition(payoff) give the solution to vanilla option values.

So what if I change the payoff to be max(S^2-E,0), I think all I need to do is just integrate a different payoff which means I can get a similar formula.


Could anyone give an formula for the call option with payoff max(S^2-E,0) when t=T, other thing unchanged so I can double check I did everything right.

 

[Daniel Duffy]

If I have understood properly, you want to get a solution (exact if possible, or approximate) for a power payoff?

The PDE itself will be the same but the boundary conditions (S =0, S = Smax) will not necessarily be the same AFAIR.

Here is note (the last article in the link) on this from a while back. hth

http://www.datasimfinancial.com/articles.php

 

[koupparis]

Use Ito's lemma. X = S^2 follows GBM, so the payoff of max(X-K,0) has the same closed form solution as a plain vanilla options with the appropriate substitutions for drift and vol.

 

[baozi]

Thx koupparis, at least I know It's called power options. I googled it and get the equation from http://demonstrations.wolfram.com/PricingPowerOptionsInTheBlackScholesModel/
I know I can solve it just using algebra[[[just changing the boundary conditions]]] and it's not that hard. But it takes some time to get the right equation as shown on the website.
So how do you solve the problem by substituting the new drift and vol?
drift becomes(2r-sigma^2) and vol becomes(2sigma)

Use Ito's lemma. X = S^2 follows GBM, so the payoff of max(X-K,0) has the same closed form solution as a plain vanilla options with the appropriate substitutions for drift and vol.

 

[koupparis]

Take original BS solution and replace mu and sigma with the new mu and sigma from the SDE for S^2.