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prodiptag
Topic Author
Posts: 124
Joined: September 12th, 2008, 4:41 pm

### simple option price with uncertain vol

Hi all, just wondering this .... suppose I price a vanilla options using a simple black type normal model, i.e. dF = drift + V0.dW, where V0 is known and constant. I get a price, say BS(V0). Now suppose another case where I am sure the sigma is constant, but do not know what it is. However I know that it is normally distributed with a mean = V0 and SD of say sigma0. The new price, say is BS(V0,sigma0). I think BS(V0) <= BS(V0, sigma0), given BS is a convex function of vol. One way to estimate BS is to price a series of BS with different vols and take an expectation. But is there any analytical formula for this price? even any rough approximation can we work out? The second part, I don't know if the eqivalent vol (i.e. the constant vol that equate the expected BS for a given strike K in a vanilla BS formula) will depend on explicitly K or not and in what form, but if it does, can we then use the current market smile to back out the priced-in distribution? (just imagination, say the market smile is perfectly captured by 2 parametrs - vol levels + skew - then could we get back a distribution in 2 params (like say a normal dist, mean and sd?? V here can be thought of as breakeven vol!!)thanks vm for any inputs...

prodiptag
Topic Author
Posts: 124
Joined: September 12th, 2008, 4:41 pm

### simple option price with uncertain vol

After a long wait, I think this comes close to an answer. Posted for informationhttp://www.fabiomercurio.it/uncertainvol2.pdf

Alan
Posts: 10263
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### simple option price with uncertain vol

Just (numerically) integrate your vanilla formula over your density for V0, and you're done. In some computer algebra system, like Mathematica, this is a one-liner and a 5 min project. Then, use that little piece of code for whatever ...Suppose I tell you that integral has a closed form in terms of some tricky special function. How is that easier?The answer is: it isn't. You need to have a sense of when to declare victory. Sorry for the rant ...
Last edited by Alan on November 22nd, 2012, 11:00 pm, edited 1 time in total.

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