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Hurd Zhou 2009 Basket Option Formula

December 3rd, 2018, 6:34 pm

In Hurd Zhou 2009 part 5 + appendix they develop the Fourier transform of a,  suitably dampened, N-dimensional spread option.

They do not bother to present the actual pricing formula, though. I know that the idea is to take the expectation of the payoff and exchange integrals no problem there.
What I struggle with is proving that the exchanging the integrals are ok. Furthermore, is it obvious that in the general case the Fourier inversion theorem is valid? 

In the two dimensional case I was able to exchange integrals by Fubini using Tonelli to show integrability of absolute value. I also showed that the dampened payoff is both L1 and L2, so it is my understanding that is sufficient, but I got nowhere trying to do it generally.
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Re: Hurd Zhou 2009 Basket Option Formula

December 3rd, 2018, 7:52 pm

Basically, it seems you need a multi-dimensional justification of Parseval's formula for this application. In my paper on the related 1D case, you will see (my Th 3.1) that I rely on a Theorem in Titschmarsh. He comments at the end of his theorem proof that having [$](f(x),g(x)) \in L^1 \equiv L^1(R)[$] and one of them bounded is sufficient, but his general conditions are weaker than that. Here [$]R = (-\infty,\infty)[$]. In my paper, I show how to get/use weaker conditions by introducing generalized Fourier transforms.

Presumably something similar holds in n-dimensions. 

In your application, [$]f(x)[$] is a payoff function and [$]g(x)[$] is the probability density with [$]x \in R^n[$], using log-prices. For n-dimensional GBM, for example, [$]g(x)[$] will be an n-dimensional normal density, so bounded and in [$]L^1(R^n)[$].  Other densities may not be bounded, but (being probability densities) we know they are [$]L^1[$]. 

Arranging the payoff to be bounded should not be too hard: usually it is some put-call parity type trick, removing the + (max) condition from the payoff, valuing that, and then focusing on the payoff that's left. In my paper, I show that Titschmarsh's theorem can be applied as long as [$]f(x)[$] is (i) Fourier integrable in a strip and (ii) bounded. This avoids needing an [$]L^1[$] payoff.* For example, a put option payoff [$](K - e^x)^+[$] is bounded, not in [$]L^1[$], but has a generalized Fourier transform in a [$]z[$]-plane strip. 

So, to summarize, my suggestion is to try to adapt that generalized Fourier transform approach.

* But the strip has to overlap the analyticity strip for the characteristic function of [$]g(x)[$] at [$]-z[$]. This requirement is never a problem for a normal density, but for some others, it may require care.


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