The CEV model: who's good with Bessel functions?

Alan · November 1st, 2009, 4:18 pm

I will use wikipedia notation for the noncentral chi-squared distribution. Let f(x; k, lambda) be the pdf and P(x; k, lambda) be the associated cumulative distribution function.In "Computing the constant elasticity of variance option pricing fomula", Schroder proved the identity,asuming k > 0. Notice the integral is over the non-centrality parameter, not the random variable parameter.Schroder's proof is quite tricky and long. Here is my real question: Can anyone see an alternative, hopefully simpler, proof? I have tried a few times without any luck.You don't necessarily need to look at his paper, although it is hereThe identity is just below his eqn (2).Even without looking, if you can prove this in 10 lines or less, you probably have a simpler proof. If so, please post it. Thanks!

GoGoFa · November 2nd, 2009, 7:27 am

I just looked up the proof. Writing a short proof always needs clever pre-work. So if you have already proven a lot of identities it might be possible to write the proof in ten lines.Note that you also have 2k+2 degrees of freedom on the left-handed side and 2k degrees on the right, a parameter which enters thedensity at a lot of places. In order to get these identities you have to do a lot of basic algebra.So I don't think that there is a really simple proof.

Alan · November 2nd, 2009, 12:42 pm

Thanks for looking. Perhaps some ground rules are needed. Certainly one may assert, without proof, any of thestandard Bessel function identities in, say, Abramowitz and Stegun. Many of these connect Bessel functions withdifferent indicies.

lesniewski · November 2nd, 2009, 6:12 pm

Alan, attached are my (somewhat rough and incomplete) calculations with the CEV model. I believe that what you are asking about is related to formula (31) of my notes. Its proof is written on the upper half of page 5.

Alan · November 2nd, 2009, 8:25 pm

Excellent, Andrew -- thank you! That was exactly the kind of thing I was looking for.

lesniewski · November 2nd, 2009, 9:27 pm

Alan, you are welcome. In my previous posting, I have updated the file with a version that is slightly more readable.

Alan · November 4th, 2009, 2:07 pm

Got it, Andrew, thanks. BTW, when going through the argument, I find a need for some factors of 1/2in the 2 eqns below the sentence "On the other hand, we verify readily that..." on pg. 5. For example,p(x;r,lambda) is proportional to exp(-(x+lambda)/2) and taking the x or lambda derivative willpull down a factor of 1/2. Specifically, I find Partial p(r)/Partial x = -1/2 p(r) + 1/2 p(r-2), suppressing the other args, and similarly with the eqn below that one. This doesn't cause any problems with establishing that d phi/d lambda = 0, so all goes through anyway. It's a nice derivation!

lesniewski · November 4th, 2009, 2:26 pm

Thanks, Alan, I'll fix it. Also, the Neumann problem isn't written out in full detail - the argument should be similar to the Dirichlet boundary condition(Hankel transform).

Enzolee · October 2nd, 2014, 5:13 pm

Thank you for your note. But I am confused by equ (19) on the note. How did you get the A(v)?