# SKEW and VIX relations?

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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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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