Edgeworth Expansions Link Implied Volatility Skew to Return Cumulants

# Relationship between Implied Volatility Curve Derivatives and the Underlying's Moments


# Relationship between Implied Volatility Curve Derivatives and the Underlying's Moments












Very probably this question has been posed before, so if someone can pose the link to the relevant question, it would be appreciated.

What is the relationship between the implied volatility skew and the skewness and kurtosis of underlying, or more generally the relationship between implied volatility curve derivatives (slope, curvature,...) with respect to the strike and the moments of the underlying? Does Edgeworth expansion provides the answer?

After posting this question, I found David Backus, Silverio Foresi, and Liuren Wu: Accounting for Biases in Black-Scholes dealing with just this topic and using precisely the Edgeworth/Gram-Charlier expansion. Other references on similar topics would be most welcome.

## Answer by Frido (score 5)

https://quant.stackexchange.com/a/81200

I'm quite sure you've figured it all out by now, but maybe for the benefit of others it's useful to write one or two things about this.

First of all, the seminal paper applying Edgeworth expansion to option pricing is by Jarrow & Rudd, Approximate option valuation for arbitrary stochastic processes, 1982. To the best of my knowledge (nearly) all subsequent papers on moment expansion techniques for option pricing cite Jarrow & Rudd.

The idea of an Edgeworth expansion is to write the characteristic function (CF) of a random variable $Z$ with unknown distribution as the product of the CF of a known reference distribution and an error term. Specifically $$ \phi(t) = \phi_0(t) e^{\sum_{n=1}^\infty \frac{(it)^n}{n!} \gamma_n}, \quad \gamma_n := \kappa_n - \kappa_{0,n} $$ where $\kappa_n$ is the $n$-th cumulant of the unknown distribution and $\kappa_{0,n}$ that of the reference distribution.

Since the exponent is an analytic function, the above expression can be rewritten as $$ \phi(t) = \phi_0(t) \sum_{n=0}^\infty \frac{(it)^n}{n!} E_n $$ where the first few $E_n$ are \begin{gather} E_0 = 1 \\ E_1 = \gamma_1 \\ E_2 = \gamma_2 + \gamma_1^2 \\ E_3 = \gamma_1^3 + 3\gamma_1\gamma_2 + \gamma_3 \\ E_4 = \gamma_1^4 + 3\gamma_2^2 + 4 \gamma_1\gamma_3 + 6\gamma_1^2\gamma_2 + \gamma_4 \end{gather}

Now if one takes the inverse Fourier transform of $\phi(t)$ then the probability density $p(z)$ of $Z$ can be expressed as $$ p(z) = \sum_{n=0}^\infty \frac{ (-1)^n }{n!} E_n p^{(n)}_0(z) $$ with $p^{(n)}_0(z) := d^n p_0(z)/dz^n$. It is common to truncate the series, for instance: $$ p(z) \approx \sum_{n=0}^4 \frac{ (-1)^n }{n!} E_n p^{(n)}_0(z) $$ so that only the first four moments/cumulants of a distribution are considered.

It is at this stage that the approach of Jarrow & Rudd differs from Corrado & Su and Backus, Foresi & Wu. Jarrow & Rudd take $Z = S_T$ where $S_T$ is the terminal stock price, and for the reference distribution $p_0(z)$ they logically choose the lognormal distribution with some constant volatility $\sigma$, e.g. the ATM implied volatility.

Corrado & Su and Backus et al. on the other hand choose $$ Z := \frac{ \log S_T/S_0 + \frac12 \sigma^2 T}{ \sigma \sqrt{T}} $$ and for the reference distribution they take the standard normal density $$ p_0(z) = \frac{1}{\sqrt{2\pi}} e^{ -\frac12 z^2} $$ where again $\sigma$ is a constant such as the ATM IV.

What Jarrow & Rudd, Corrado & Su and Backus et al. do in common is that they assume that the first two cumulants of the reference distribution and the unknown distribution are equal. In the case of Corrado & Su and Backus et al. this leads to the following simple expression for $p(z)$: $$ p(z) \approx p_0(z) \left( 1 + \frac{ \kappa_3}{3!} He_3(z) + \frac{ \kappa_4}{3!} He_4(z) \right) $$ where $He_n(z)$ are the probabilist's Hermite polynomials. Backus et al. then continue to derive, after further approximations, that the IV is given by $$ IV(K) \approx \sigma \left( 1 - \frac{\kappa_3}{3!} d_1 + \frac{ \kappa_4}{4!} (d_1^2 - 1) \right) $$ with $$ d_1 = \frac{\log S_0/K + \frac12 \sigma^2T}{\sigma\sqrt{T}} $$ The expression for $IV(K)$ above is nice because it related the IV to cumulants of the distribution. In particular, it can be derived that $$ \left.\frac{\partial IV }{\partial \log K}\right|_{K = S_0} \approx \frac{\kappa_3}{3!} $$

A remark:

Jarrow and Rudd, Corrado & Su and Backus et al. all assume that the first two cumulants of the distribution and the reference distribution are equal. This is in fact not necessary. Indeed I think this matching of the first two moments leads to the 'martingality problem' that seems to occur with the Edgeworth/Gram-Charlier expansion.

Not just a bit of self-promotion, but I have applied the Gram-Charlier expansion to pricing options on realized volatility in stochastic volatility models where I do not assume that the first two cumulants are equal. The results are not bad; in fact the results are not so good / much worse if I do assume equality of the first two cumulants.

Finally then, an aside to close: I am surprised that Backus et al. do not cite Corrado & Su (or vice versa) as their work are essentially the same.