## Abstract

The Homoscedastic Gamma (HG) model characterizes the distribution of returns by its mean, variance and an independent skewness parameter. The HG model preserves the parsimony and the closed form of the Black–Scholes–Merton (BSM) while introducing the implied volatility (IV) and skewness *surface*. Varying the skewness parameter of the HG model can restore the symmetry of IV curves. Practitioner’s variants of the HG model improve pricing (in-sample and out-of-sample) and hedging performances relative to practitioners’ BSM models, with as many or less parameters. The pattern of improvements in Delta-Hedged gains across strike prices accord with predictions from the HG model. These results imply that expanding around the Gaussian density does not offer sufficient flexibility to match the skewness implicit in options. Consistent with the model, we also find that conditioning on implied skewness increases the predictive power of the volatility spread for excess returns.

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## Notes

- 1.
The BSM option price formula is a function of the strike price, stock price, interest rate, maturity of the option and of anticipated volatility but only the latter is unobservable.

- 2.
- 3.
Note also that closed-form option prices typically result from a first-order approximation. This may not be relevant in practice for option pricing but the impact of this approximation on estimates of implied skewness has not been discussed.

- 4.
Note, also, that this approach requires approximations of integrals over the moneyness domain. Although Dennis and Mayhew (2000) consider the impact of sampling error under the null of the BSM model, the accuracy of skewness estimates are unknown in the presence of measurement errors or in a non-gaussian setup.

- 5.
Bakshi and Madan conclude that historical skewness do not play an important role in the determination of the volatility spread but they do not consider risk-neutral skewness.

- 6.
One can show that an SDF exists such that the returns distribution belongs to the HG family under both measures with

*both*the variance and the skewness parameter shifted. However, this SDF is not in general within the exponential-affine class and the link between moments is not transparent. - 7.
In the limit, as skewness becomes zero, stock returns follow the usual square-root process.

- 8.
This follows directly from the fact that the Gamma distribution is summable.

- 9.
We use the standard notation for the regularized gamma functions,

*P*(*a*,*z*) and*Q*(*a*,*z*), possibly at the cost of some confusion with the usual notations for the historical and risk-neutral probability measures*P*and*Q*. - 10.
Note that we have \(P(a,z)+Q(a,z)=1\), which is a convenient property when computing derivatives (see below).

- 11.
This highlights the importance of using a model that can handle maturity differences. In particular, models based on density approximation are not robust to this composition effect.

- 12.
- 13.
The curve is not strictly flat and this may be due to the impact of kurtosis, or to a composition effect. We discuss these possibilities below.

- 14.
This contrasts with the theoretical results of Zhang and Xiang (2005). They argue that in the Gaussian case and up to a first-order approximation \(\sigma _0(\beta ,\kappa )\) is linear in the risk-neutral volatility, \(\gamma _1(\beta ,\kappa )\) is linear in skewness, and \(\gamma _2(\beta ,\kappa )\) is linear in kurtosis. However, they assume that the skewness and excess kurtosis of the underlying distribution can be chosen independently while in fact there is a tight link between the two for any given correctly specified density. Moreover, they linearize around the case where \(\sigma =0\) and this may lead to a poor approximation.

- 15.
Note that merely imposing \(\gamma _1(\alpha ,\kappa )=0\) does not identify an estimator of \(\alpha \) with skewness.

- 16.
We differ from Zhang and Xiang (2005) who linearize the restrictions around \(\sigma =0\). Arguably, linearizing around the HG distribution is likely to provide a better approximation than linearizing around the deterministic case.

- 17.
For smoothed model, we estimate parameters that are held constant through the sample in a first pass.

- 18.
This abstracts from the hedging errors due to discrete adjustments. See Galai (1983) for details.

- 19.
This term is a function of both skewness and volatility but the first term of its Taylor expansion is the usual correction in the Gaussian case, \(\frac{1}{2}\sigma ^2\).

- 20.
Precisely, our measures of risk-neutral moments pertain only to the distribution of returns at a horizons of 12 months or less. Nonetheless, if these moments exhibit persistence, their predictive power will extend to longer horizons as is indeed the case

- 21.
Note that our results contrast with Bakshi and Kapadia (2003). They consider regressions of BSM Delta-Hedged gains on skewness (and kurtosis) and find that it plays a small roll relative to volatility. These regression pooled Delta-Hedged gains across moneyness and are limited to near-the-money options. Looking at Fig. 5d, we expect that they produce low coefficients.

- 22.
Bakshi and Madan (2006) find that measure of

*historical*skewness plays a relatively small role in the determination of volatility spread. They did not consider measures of skewness based on options prices.

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## Additional information

An earlier version of this paper was circulated and presented at various seminars and conferences under the title “The Equity Premium and the Volatility Spread: The Role of Risk-Neutral Skewness”. We thank Peter Christoffersen, Redouane Elkamhi, René Garcia, Scott Hendry, Steve Heston, Teodora Paligorova and Jun Yang for their comments. We thank seminar participants at the Bank of Canada, Duke University, CIRANO 2009 Financial Econometrics Conference, Econometric Society 2009 NASM, EFA 2009 and the Fifth International Conference MAF 2012.

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### Cite this article

Feunou, B., Fontaine, JS. & Tédongap, R. Implied volatility and skewness surface.
*Rev Deriv Res* **20, **167–202 (2017). https://doi.org/10.1007/s11147-016-9127-x

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### Keywords

- SP500 options
- Implied skewness
- Implied volatility
- Volatility spread
- Delta-hedged gains

### JEL Classification

- G12
- G13