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In the paper "Optimal Delta Hedging for Options" link the author shows that the minimum variance delta is a function of change in implied volatility. If you use equation on page 9 i.e. $E[\Delta \sigma ]=(\frac{a+b \delta_{bs} + c \delta^2_{bs}}{\sqrt T}) \frac{\Delta S}{S}$ you will get what you want. The parameters a, b, c are fit with OLS ...


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I'm going to go through a coded example to show how you might attempt this, using the python port of the QuantLib library. It will all seem a little mechanical, but hopefully it is instructive. There is quite a bit of setup code required (specifying the interest rates, spot etc., and also a utility function that I use for plotting surfaces), I've pushed this ...


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You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and simulate from these using a uniform draw. That's the brute force approach. The variance gamma process is typically represented as a difference of gamma processes or a ...


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One can use the Euler-Maruyama discretization scheme for CIR, 'fixed' for $v$ positivity, to get: $$ v(t+\epsilon) -v(t)\approx \kappa (\bar{v} -v(t)^+)\epsilon + \omega \sqrt{v(t)^+} (W_v(t+\epsilon) - W_v(t)). $$ So, one approximation of the Brownian increment, when $v(t)$ and $v(t+\epsilon)$ are given, is: $$ W_v(t+\epsilon) - W_v(t) \approx \frac{v(t+\...


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Edit: This is probably incorrect. The Quadratic Exponential scheme is the best one I have seen as it converges in distribution and is pretty fast, so nice choice there! When $\eta$ is constant you can simplify the integral $$ \int_t^{t+\varepsilon}\eta dW(u)=\eta\int_t^{t+\varepsilon}dW(u)=\eta\left(W(t+\varepsilon)-W(t)\right) $$ In the QE scheme you either ...


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