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My question is about the CBOE published index VIX and SKEW.

To start with, I consider working on the variance dynamics. I calibrate the market data (such as VIX and VIX futures) into the Heston model. After that it's not hard to derive the dynamics of VIX.

But how about SKEW, how could I relate the Heston Model to it? Or I shall employ further stochastic models for the higher moments of the underlying log return?

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2 Answers 2

In fact, even your VIX dynamics are not exact, since you can only obtain dynamics for an approximation of the actual VIX calculation (I presume you are just running the variance variable through a square root using Ito's rule).

SKEW is even less tractable here since its calculation roughly goes as the third moment of the return distribution. I doubt you can even get a closed form for that third moment, let alone the actual calculation (which, like VIX, depends on discrete strikes).

I will add that the Heston model is notoriously bad at making plausible forward volatility skews, so the results of your endeavor are likely to be of academic interest only.

If you still want to try, your best bet is to fourier transform the characteristic function (as is done to get the closed-form Heston option pricing equation). You can probably obtain a closed form expression for the fourier transform of the third moment of returns. Turning that into an actual approximation to SKEW will probably not have any analytic solution to the fourier integral so you will have to use FFT techniques to get numbers out of the whole process.

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I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as $$ \frac{dS_t}{S_t} = \sqrt{v_t}dW_t $$ $$ dv_t = -\kappa(v_t-\eta)dt + \theta \sqrt{v_t}dZ_t $$ with $d\langle W,Z\rangle_t = \rho dt$ and $v_0 = \sigma_0^2$. Using Itô's lemma we can derive $$ \log\left(\frac{S_t}{S_0}\right) = \int_0^t \sqrt{v_s}dW_s - \frac{1}{2} \int_0^t v_s ds $$ According to the CBOE white paper, the SKEW index is computed from $$ SKEW = 100 - 10 \mathbb{E}\left[ \left(\frac{R-\mu}{\sigma}\right)^3\right]$$ with $R$ being the 30-day log return of the S&P500 and $\mu$, $\sigma$ its mean and variance. You can -not without some work- rewrite the SKEW as a function of $v_t$ moments. Indeed you will have to use :

  • Itô's lemma with $f(x)=x^\alpha$ to get $\mathbb{E}(X_t^\alpha)$ with $X_t := \int_0^t \sqrt{v_s} dW_s$
  • Itô's lemma with $f(x,y)=xy$ to get mixed expectation of form $\mathbb{E}(X_t^\alpha v_t^\beta)$

Eventually you will only worry about finding $v_t$ moments, which can be obtained by using the classical $$v_t - \mathbb{E}(v_t) = \theta\int_0^t e^{-k(t-s)}\sqrt{v_s}dW_s $$ and the above.

In case you need fractional moments (as you are looking at the VIX as well), the following should be of interest.

Let X be a random variable with Laplace transform $\mathcal{L}$. Then if $n\in\mathbb{N}$ and $\alpha>0$ then $$ \mathbb{E}[X^{n-\alpha}] = \frac{(-1)^n}{\Gamma(\alpha)}\int_0^\infty \frac{\partial^{(n)}\mathcal{L}}{\partial \lambda}(\lambda)\lambda^{\alpha-1}d\lambda $$

This can be applied to find $\mathbb{E}[\sqrt{v_t}]$, $\mathbb{E}[v_t^{3/2}]$, etc. The Laplace transform of a CIR process has a closed-form of affine type and can be easily found in the litterature.

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