QuoteOriginally posted by:Soap

Hi @Alan, I just received your new book. Very well written and thanks for the great book!

Immediately I went to Chapter 3 and 7, as I would like to find the probability transition density for SVJ model or Bates (1996). I wonder is it possible to derive the transition density function for SVJ? Which chapters/sections could help? Thanks.

Hi Soap,

Welcome to the forum and thanks for the kind remarks about my book.

If you bought it at an amazon, I need some reviews so I would much appreciate your posting a few sentences where you bought it.

As far as the probability transition density for the Bates (1996) SVJ model goes, there are two of those.

First there is the marginal one:

[$]p(T,X_T | X_0, V_0)[$], where [$]X_T = \log S_T[$], which is all you need for evaluating terminal payoffs that only depend upon the stock price.

Then, there is the joint one:

[$]p(T,X_T, V_T | X_0, V_0)[$], which you could use for maximum likelihood parameter estimation if you had a good volatility proxy.

The marginal one is relatively easy, since the characteristic function

[$]\Phi(T,V_0;z) = E[e^{i z X_T}|X_0=0,V_0] = \int e^{i z X} p(T,X | X_0=0,V_0) \, dX[$] is known in closed-form.

So you just do a Fourier integral inversion:

[$] p(T,X | X_0=0, V_0) = \int e^{-i z X} \Phi(T,V_0;z) \, \frac{dz}{2 \pi}[$],

which is no harder than doing the Fourier integral to get option values.

Then, because of the conditional translational invariance (the MAP property discussed in the book on pg. 275 and elsewhere),

[$] p(T,X | X_0, V_0) = p(T,X - X_0 | X_0=0, V_0)[$].

The joint one I will have to ponder as these are not so straightforward.

Of course, in the limit of zero jump intensity, the answer must reduce to the joint density for the pure Heston model I give in the Ch. 7 you mentioned.

So that is one clue.

Good question -- I will keep thinking about it.

Edit: erased some 'thinking out loud distractions'; see my subsequent 'Update' for the answer.