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1

You don't need any assumption about the distributional properties of $S_t$. What matters for the FTAP is the drift only. By definition, the risk neutral measure $Q$ is the measure, equivalent to the natural measure $P$ (*), under which the local rate of return (i.e. the instanteneous drift of the SDE of $S_t$ per unit of $S_t$) of "any" traded asset $S_t$ (...


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You can use this article Probability distribution of returns in the Heston model with stochastic volatility Let $$\begin{align} & d{{S}_{t}}=r{{S}_{t}}dt+\sqrt{{{\nu }_{t}}}\left( \rho dW_{1}^{Q}(t)+\sqrt{1-{{\rho }^{2}}}dW_{2}^{Q}(t) \right) \\ & d{{v}_{t}}=\kappa (\theta -{{v}_{t}}){{d}{t}}+{{\sigma }_{v}}\sqrt{{{\nu }_{t}}}dW_{1}^{Q}(t) \\...


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There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order to avoid having to count the normal of windings. Second, using contour shifts can make the integral much better behaved. eg integrate along the line with $0.5$...


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I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT


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In SV model, it is well-known that the integrand for the call price can sometimes show high oscillation, can dampen very slowly along the integration axis, and can show discontinuities. Remedy The ‘‘Little Trap’’ formulation of Albrecher et al. Also , you can use Fourier transforms Bakshi and Madan (2000) Lewis,(2001). Gatheral (2006) Carr and ...


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Sorry if this is late, but this is the bible of Heston (and it has code) https://www.amazon.co.uk/Heston-Model-Extensions-Matlab-Finance-ebook/dp/B00EMADBN2/ref=sr_1_1?ie=UTF8&qid=1468410988&sr=8-1&keywords=Heston+matlab


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Hint By application of Ito's lemma, we have $$d(e^{kt}v_t)=\kappa e^{\kappa t}v_t\,dt+e^{\kappa t}dv_t+d(e^{\kappa t})dv_t$$ therefore $$v_t=v_0e^{-\kappa t}+\theta(1-e^{-\kappa t})+\sigma\int_{0}^{t}\sqrt{v_s}e^{-\kappa(t-s)}dB_{s}^{v}+\int_{0}^{t}e^{-\kappa(t-s)}J^v\,dN_{s}$$ $J_v$ is random jump size occurring at time $t_i$ and $N_t=N_t-N_0$ is the ...



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