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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:  ...

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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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