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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: $$ ...


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To check your results, you might try "The Heston Model: A Practical Approach with Matlab Code" by Nimalin Moodley, http://math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdf , in particular the www.ingber.com open source C++ code for Adaptive Simulated Annealing (+ SWIG to wrap/parse it to the language you are using)


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You need to obtain a $4 \times 4$ correlation matrix. As you effectively observe, you have four random processes driving the system, with $i \in 1,2$ $$ \frac{dS_i}{S_i} = \mu_i dt + \sqrt{v_i} dW_{Si} \\ dv_i = \kappa(\bar{v}_{i}-v_i) dt + \xi \sqrt{v_i} dW_{vi} $$ Each of the $W_{ji},j\in\{S,v\},i\in 1,2$ is a brownian motion correlated with the others, ...


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It depends on the used optimization algorithm, esp. whether they act locally or globally. Just to give you some ideas: Local (deterministic) algorithms (e.g. gradient methods): a good initial guess is crucial. (Global) stochastic algorithms (e.g. simulated annealing): the initial guess is irrelevant. You can find more here: Heston’s Stochastic ...


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Different optimizations could help. Parallel computing makes even worse if each computation is fast enough due to overhead. Thus it may be better to use profiler to get what can be improved. Usually it helps to send larger problems to parallel computation cores. Matlab is very good at matrix operations and it could be better to treat different draws of MC ...



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