[Paper Review] We've walked a million miles for one of these smiles

Lorenzo De Leo, Vincent Vargas+2 more|arXiv (Cornell University)|Mar 26, 2012

Stochastic processes and financial applications3 citations

TL;DR

This paper introduces a new, exact expansion for option smiles that depends only on low-order moments of the return distribution, avoiding problematic higher-order cumulants. It shows the skew and curvature of the smile can be computed as exotic options using the Hedged Monte Carlo method, revealing that the standard Edgeworth expansion fails to capture the inverted skew-skewness relation at extreme volatilities and underestimates curvature despite high kurtosis in returns.

ABSTRACT

We derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and, perhaps more importantly, to congenial numerical treatments. We show that the skew and the curvature of the smile can be computed as exotic options, for which the Hedged Monte Carlo method is particularly well suited. When applied to options on the S&P index, we find that the skew and the curvature of the smile are very poorly reproduced by the standard Edgeworth (cumulant) expansion. Most notably, the relation between the skew and the skewness is inverted at small and large vols, a feature that none of the model studied so far is able to reproduce. Furthermore, the around-the-money curvature of the smile is found to be very small, in stark contrast with the highly kurtic nature of the returns.

Motivation & Objective

• To develop a rigorous, moment-free option smile expansion that avoids divergent higher-order cumulants like skewness and kurtosis.
• To provide a framework where skew and curvature of the smile are computed as exotic options, suitable for numerical treatment.
• To address the failure of the standard Edgeworth expansion in reproducing observed market features such as the inverted skew-skewness relation at high and low volatilities.
• To demonstrate that the around-the-money curvature of the smile is small despite highly kurtotic returns, challenging conventional model assumptions.

Proposed method

• Derive a new option pricing expansion in terms of the rescaled moneyness $\mathcal{M}$, with coefficients $\alpha_T$, $\beta_T$, $\gamma_T$ depending only on the distribution of $u_T = r_T / (\sigma\sqrt{T})$.
• Express the coefficients as $\alpha_T = \sqrt{\pi/2} \, \mathbb{E}[|u_T|]$, $\beta_T = \sqrt{\pi/2} \, \left[1 - 2P(u_T > 0)\right]$, and $\gamma_T = \sqrt{\pi/2} \, p_T(0) - 1/(2\alpha_T)$, relying on first- and second-order moments.
• Use the Hedged Monte Carlo method to numerically estimate $\alpha_T$, $\beta_T$, $\gamma_T$ from historical data or complex models without parametric assumptions.
• Show that the new expansion reduces to the Edgeworth formula in the limit of small cumulants, validating consistency with prior work.
• Demonstrate that $\beta_T$ is insensitive to extreme events since it depends only on the median and survival probability, not on third-order moments.
• Apply the method to S&P 500 options, revealing systematic failures of the Edgeworth model in capturing market skew and curvature.

Experimental results

Research questions

• RQ1 Why does the standard Edgeworth expansion fail to reproduce the observed skew and curvature in option market data, especially at extreme volatilities?
• RQ2 Can a new option smile expansion be derived that avoids reliance on divergent higher-order cumulants like skewness and kurtosis?
• RQ3 How does the skew of the smile relate to the skewness of the return distribution across different volatility regimes?
• RQ4 Why is the curvature of the smile around-the-money so small despite the high kurtosis of financial returns?
• RQ5 Can the new expansion be efficiently computed numerically using historical data or complex models without parametric assumptions?

Key findings

• The standard Edgeworth expansion fails to reproduce the inverted relation between skew and skewness at both low and high volatilities, a feature consistently observed in S&P 500 options.
• The skew of the smile, as captured by $\beta_T$, is insensitive to extreme events because it depends only on the median and tail probability of $u_T$, not on third-order moments.
• The around-the-money curvature of the smile is found to be very small, contradicting expectations from highly kurtotic return distributions.
• The new expansion matches the Edgeworth formula in the limit of small cumulants, validating consistency, but outperforms it in practical settings due to robustness to heavy tails.
• The Hedged Monte Carlo method enables accurate and efficient numerical estimation of $\alpha_T$, $\beta_T$, $\gamma_T$ from real market data or complex models, even without parametric assumptions.
• The method reveals that the skewness of the return distribution does not linearly determine the smile skew, especially in non-Gaussian regimes, challenging long-standing model assumptions.