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1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want to buy it now. Instead you enter a forward contract with someone that says that at time $T$ you will pay the amount $K$ and get the asset in exchange. What ...

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There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable ...

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Some simple improvements: 1) Replace the Euler discretization approximation of the volatility to a Milstein discretization approximation. See e.g. these notes by Rouah. 2) 100 Paths is a very low number of paths, and leads to a big standard error in your estimate. So this should be increased by a factor of ~100. 3) You should use some form of variance ...

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change the discretization and use the QE-M approach: Andersen (2006) the bias is way smaller than the one of the simple Euler. further u can try to use control variates/anthitetic numbers to reduce the sample variance.

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Heston gives an expression for the characteristic function, from which option prices can be computed. Therefore it can be calibrated (statically) on a set of vanilla option prices with different strikes and maturities. Hence this produces risk neutral parameters that can be used to price other more exotic products. However, it is a pain to estimate the ...

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Yes! Try this and this. But if you don't know the black-scholes basics well consider to read the book "Paul Wilmott in Quantitative Finance" before to go to Stochastic Volatility models and models with jumps.

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The Feller condition applies without modification. That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level $v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ...

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There are by now a lot of papers on discretizations of Heston. One objective of them being to avoid negativity. As has already been said, the Heston SDE has no negative solutions, but a crude discretization does give negative variance with positive probability. If you want to do small steps, then using a log-normal approximation or the QE approximation ...

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