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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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If you want to calibrate on time series, then you have a 'non linear filtering' problem, since volatility is latent. There have been papers from late 90s/ early 00s that do that: Google for Heston together with Ghysels, Gallant, Renault, Chernov, Tauchen, Pan, Bates, Shephard, MCMC, unscented Kalman filter/ particle filter. Given the significant complexity ...

3

I do not know what are the exact steps you followed, but here's my 2 cents: \begin{align*} C(S_t,t) &= P(t,T) E^Q [ (S_T - K)^+ ] \\ & = P(t,T) E^Q [ S_T 1_{S_T \geq K} ] - K P(t,T) E^Q [ 1_{S_T \geq K} ] \end{align*} Which can be re-written (using a change of numéraire) \begin{align*} C(S_t,t) &= S_t E^{Q^S} [ 1_{S_T \geq K} ] - K P(t,T) E^Q [...

2

Two volatility processes yield a higher flexibility of the model. This is of greater importance if one tries to price derivatives with different maturities in one single model. A additional volatility component helps to capture the term structure of volatility, which can depend greatly on time to maturity. See for example the VIX term structure from CBOE: ...

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I think,the additional volatility factor,$v_2(t)$, provides more flexibility in modeling the volatility surface.We know $\rho$ controls the slope of the implied volatility.In the single-factor Heston model, $\rho$ is constant over maturities,In deed $$Corr[{dS}/{S\,,\,dv]}\;=\rho \,$$ which means that model has trouble providing an adequate fit to market ...

2

Loosely speaking, it can be seen as inserting an additional degree of freedom in the underlying's dynamics. This can be useful from a static perspective: with an additional lever to play on, one can hope to better capture the short term implied volatility smile, which "naive" stochastic volatility models (single volatility factor, no jumps) are known to be ...

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You should not use the Feller condition as a constraint. In many cases its violation will be required for a good fit to the market data.

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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 options pricing language, the probability of a spot process being above a given level $K$ at time $T$ is the undiscounted price of a digital call option on that spot process. In the Heston model, there is an analytic expression for this in terms of Fourier transform. You can find this in various standard references, e.g. Alan Lewis's book "Option ...

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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Your adjusted scheme is correct. Basically, taking a maturity $T$, you can consider the forward price process $F_t^T = S_t e^{r(T-t)}$. You apply the Andersen scheme to $F_t^T$ and then note that \begin{align*} S_{t+\Delta} &= F_{t+\Delta}^T e^{-r(T-(t+\Delta))}\\ &=F_t^T \exp(\ \Box \ ) e^{-r(T-(t+\Delta))}\\ &=S_t e^{r(T-t)}\exp(\ \Box \ ) e^{-...

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COS method is an efficient way to recover the distribution function from the characteristic function in the Heston model. For other methods, you may refer to "Inverting Analytic Characteristic Functions and Financial Applications".

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