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6

To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi$$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$... 3 About the integration problem: Your integrand is highly oscillatory, and the adaptive quadrature of Matlab doesn't handle such integrands very well. In general, I would recommend Mathematica when Matlab's standard procedures don't perform well. In this case, a Levin-type method would perform much better. The reason that quadv produces NaN values is because ... 3 The model is similar to the Barndorﬀ-Nielsen - Shephard model. But this model is much more general. On the other hand in this paper by Heston it is exactly your form that is used. Already Scott in 1987 considered a model of your form (see this) Finally in this thesis you find the names Hull-White model (of course there is the interest rate model too) and ... 2 Let dS_t = \mu_tS_tdt + \sigma_tS_tdW_t be the underlying GBM (Geometric Brownian Motion)-like dynamics as in the question. Let B_t a Brownian motion such that d[B,W]_t = \rho dt, \rho\in[-1,1]. CIR (Cox-Ingersoll-Ross) for \sigma_t^2 (when combined with GBM-like underlying dynamics, it is the popular Heston SV model)$$d\sigma_t^2 = ...

2

Historical returns are not to be used 'untreated' for the calculation of option prices. The expectation that you will be using in Monte Carlo will take the form $$C(K,T) = E^Q\{D(T)\ \max[0, S_T-K, 0]\}$$ where $T$ is the maturity, $K$ is the strike price, $S$ is the stock price and $D$ is the discount factor. But the expectation is taken under the 'risk ...

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I would suggest that you use a more 'modern' method to recover option prices from characteristic functions. The approach of this papers (for practical calculations of option prices) is somewhat outdated. The backbone of affine models (such as SVJJ) is the characteristic function $\psi(u)$ of the log-price distribution, which is known in closed form. The ...

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GBM is defined as $$S_t = S_{t-1}\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)dt + \sigma dW_t\right)$$ So, in your notation, assuming your daily parameters: $$S_{new} = S_{previous}\cdot\exp\left( \left({drift} - \frac{{volatility}^2}{2} \right)days + volatility \,\sqrt{days}\,N(0,1)\right)$$ So your formula was incorrect. The youtube you quote is ...

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To calibrate BS you compute volatility $\sigma$, to calibrate SABR you compute implied $\alpha$, the volvol and $\beta$, the skewness. These parameters does not play the same role. So you can't really use the parameters of one models to calibrate another. But you can build equivalent parmaters, i.e. compute an equivalent vol under SABR to use BS pricing ...

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I think a sketch of the proof would look like this Let's say you start from $$dS_t = S_t \odot (\mu_t dt + \sigma_t dW_t)$$ where $S$ is an vector valued process of your $n$ risky assets prices, $W$ a standard $k$-dimensionnal brownian motion under the historic probability, $\sigma_t$ an $n \times k$ matrix valued process and $\odot$ is the Hadamard ...

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