July 8th, 2016, 1:48 pm
QuoteOriginally posted by: AlanUpdate:Working a little more on Soap's question, I'm pretty sure the joint transition density for the SVJ model is a simple modification of my (7.40) in my Vol. II. In fact, the modification is exactly the same as the modification discussed in Gatheral's 'Volatility Surface' book, pg 66, for the characteristic function that prices vanilla options.In other words, suppose we are talking about the risk-neutral Bates (1996) model, where [$]dS_t = b \, S_t \, dt + \sqrt{V_t} S_t \, dW_t + S_t (e^{\alpha + \delta \, \epsilon_t} - 1) dN_t, \quad \epsilon_t \sim \mbox{Normal}(0,1), \quad E[dN_t] = \lambda dt[$],partly using Gatheral's notation. (Throughout '[$]\sim[$]' means 'is distributed as'). The volatility follows the Heston process. Then, you just insert a factor [$]e^{\psi(u) t}[$] inside the integral in the first line of (7.40), where[$] \psi(u) = -\lambda i u \left( e^{\alpha + \delta^2/2} - 1 \right) + \lambda \left( e^{i u \alpha - u^2 \delta^2/2} - 1 \right).[$] Indeed, the modification for any jump distribution [$]\Delta X_t \sim p_J(\cdot)[$], not just a normal one, can beexpressed in terms of the Fourier transform of the density of the jump-size: [$]\hat{p}_J(u) = \int e^{i u x} p_J(x) \, dx[$]. (Notations: [$]X_t = \log S_t[$] and so [$]\Delta X_t[$] is the jump in the log-stock-price). The first term in [$]\psi(u)[$] is the martingale adjustment. If you want the joint pdf of the real-world process (assuming same model structure), you: (i) omit the martingale adjustment,(ii) use [$] \psi(u) = \lambda (\hat{p}_J(u) - 1)[$] for the general SVJ model, and(iii) replace [$]r - q \rightarrow b[$] in (7.40), where [$]b[$] is simply a real-world parameter to be estimated. At some point, I'll double check and prove these assertions in a blog post at my address below. It wouldn't be surprising to find this is a known result, but I couldn't find one by googling. If anyone knows a cite to this result for the SVJ joint transition density, please post it.Thanks for your answer Alan.So if I want the joint density in the log stock price space, does the equation (7.40) become[$]p(t,X_{t},V_{t}|X_{0},V_{0})=\frac{1}{\pi}\int _{0}^{\infty}\Re \left \{ (e^{X_{t}-X_{0}}e^{(q-r)t})^{-iu}G(t;-u,V_{0},V_{t}) \right \}du[$] ?
Last edited by
Soap on July 7th, 2016, 10:00 pm, edited 1 time in total.