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Hello,currently try to undertake some hedging exercises and look into the impact of premia (see Thread) when hedging in a Heston world (or Heston+ jumps world).In order to get a better feeling for differences in the densities under different measures , I simulated prices and tried to approximate the terminal price distributions. However, the "densities" I obtain still vary considerably from simulation run to simulation run. Since there are semi-closed form solutions for the probabilities of ending in-the-money for these models, I thought this might give more stable results.I was able to implement the call-option price under the SVJ model by looking into Bakshi (1997)However, I have issues how to derive the distribution of the log-return (from which I should be able to get the terminal price distribution). The call-option price in [$]t[$] is given by:[$]C(S_0, V_0,T-t) = S_0 P_1 - K P_2 (*)[$]where [$]P_i, \, i=1,2[$] are probabilities (with interest assumed as constant and zero here). I was unable to find out which probabilities these actually are (must be something with log-price greater than log-strike at maturity) and why they are different. Maybe it is just because the authors also have accounted for stochastic interest rates and otherwise these two would be identical?I came across some older threads in which the density [$]f[$] of the log-return is shown to be computed by:[$]f(x) = \frac{1}{2\pi}\int e^{-ixz}\phi(T,z) dz (**)[$]where [$]\phi[$] is some characteristic function (I guess from on of the above probabilities?). Any help on the probabilities used in (*) and the characteristic function in (**) greatly appreciated!Bestobs

Given the characteristic function of the logarithmic returns [$]\ln \left( S_t / S_0 \right)[$], you can use the COS method to obtain the implied distribution. The computation can be neatly vectorized. SeeFang, Fang and Cornelis W. Oosterlee (2008) "A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansion", SIAM Journal on Scientific Computing, Vol. 31, pp. 826-848.Note that commonly you find the characteristic function of the logarithmic terminal stock prices [$]\ln \left( S_t \right)[$]. Going from one to the other is trivial but you shouldn't miss it. SeeLe Floc'h, Fabien (2014) "Fourier Integration and Stochastic Volatility Calibration", Working Paper, Calypso Technologyfor a nice overview of common characteristic functions and the corresponding cumulants (which you also need in the COS method).

@localvolatilityThanks a lot! I think this is exactly what I was looking for.bestobs

Just tried to implement some of the content of the proposed papers.Before starting with the COS-method I wanted to try out the inversion theorem with a standard numerical integration-scheme[$]f(x) = \frac{1}{2 \pi} \int_{\mathbb{R}} e^{-izx}\phi (z) dz[$].. just to try evaluate the density at certain points. However, if I use the above relation I am unable to integrate as there are still complex terms left in the integrand?(also included my R-code if someone is interested)bestobs

Still stuck the the inversion. For a backcheck I tried to evaluate the density of the log-price under geometric Brownian motion and compare that to my inverted characteristic function (obtained by the cosine-expansion):[$]f(x) = \frac{1}{2 \pi}\int_{\mathbb{R}}e^{-iux}\phi(u) du \approx \sum_0^{N-1}\delta_j cos((x-a)u_j)V_j[$]with[$]a= \mu + 10 \sigma*\sqrt{\tau}\\b = \mu - 10 \sigma*\sqrt{\tau}\\\delta_j = 1 \,( \text{except for j=0} \, \Rightarrow \delta_0 = \frac{1}{2})\\u_j = \frac{j \pi}{b-a}\\V_j = \frac{2}{b-a} Re(\phi ( u_j ) e^{i u_j a})\\N = 2^6[$][$]\mu[$] and [$]\sigma[$] in a and b are the mean and standard deviation of the log-price and [$]\tau[$] is the maturity. The characteristic function of the log-price I used is as follows:[$]\phi(u) = e^{iu log(S) + i \tau (r-q) u - \frac{1}{2}\tau \sigma^2 (iu + u^2)}[$]compared the density [$]f(x = log(S0))\approx 1.15[$] to the COS-estimate and only get values which are extremly small (~1e-11).For completeness my R-code

It is a newbe mistake to jump into this Fourier inversion stuff with COS method, FFT, etc.You should *first* set it up in the most standard way, as a single Fourier integration and develop a robust procedure for that. Once you have a bullet-proof routine for that, *then* you can start experimenting withCOS or FFT or whatever. See this mistake on this forum all the time ...

Ok, do you maybe have some good reference? I'm a bit lost.Also, some hint on why the complex-terms do not cancel out in the following would be really helpful:[$]f(x) = \frac{1}{2 \pi} \int_{\mathbb{R}}e^{-iux}e^{iu log(S0) + i \tau (r-q) u - \frac{1}{2} \tau \sigma^2(iu + u^2)}du[$]this should result in a normal density but I do not see how the complex terms cancel out?bestobs

Recovery of BS soln by Fourier inversion is treated in Appendix 2.1 of my book.For your particular problem, it is a std Gaussian integral -- just complete the square.

Ok, I actually have a copy and currently look into it. However, I don't see how the term [$]e^{-iux}\phi(u)[$] ends up being real-valued?

Too much hand-holding, but since you own my book ..If you complete the square as I suggested, your integrand will be[$] e^{-\frac{\sigma^2 \tau}{2} (u - i A)^2} e^{-\frac{\sigma^2 \tau}{2} A^2}[$], where [$]A[$] is purely real.The [$]u[$] integral is then trivial, and the result is a purely real Gaussian (in [$]x[$]) from the second term.

sry for the late reply. Well, thanks for the "subscription"-bonus anyway : )