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The Heston Model is given by:

$$ dS_t = \mu S_t dt + \sqrt{v_t}S_tdB_{1t}$$ $$ dv_t = \kappa(\theta - v_t)dt + \xi \sqrt{v_t}dB_{2t}$$.

The parameters are:

$\theta$ is the long term variance

$\kappa$ is the rate at which $v_t$ reverts to $\theta$

$\xi$ is the volatility of the volatility

Can anyone explain what these 3 parameters mean in simple terms? Also, how are these parameters calculated in practice?

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  • volatility of the volatility controls convexity of the skew/smile => more vol of vol generates more convex function ( = more smile)

  • mean revertion and correlation between brownian motions both control ATM skew.

  • long term variance controls overall level of skew (moves whole skew graph higher)

In practice these parameters are calibrated to market quotes of vanilla options using Heston's semi analytic formula and numerical optimizer.

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