W-Shaped Implied Volatility Curves and the Gaussian Mixture Model

35 Pages Posted: 2 Nov 2021 Last revised: 28 Jun 2022

See all articles by Paul Glasserman

Paul Glasserman

Columbia University - Columbia Business School

Dan Pirjol

Stevens Institute of Technology

Date Written: October 27, 2021


The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most 2(N-1) times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N=3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima.

Keywords: Convexity, Risk-neutral density, Option pricing

JEL Classification: C02, C63, G13

Suggested Citation

Glasserman, Paul and Pirjol, Dan, W-Shaped Implied Volatility Curves and the Gaussian Mixture Model (October 27, 2021). Available at SSRN: https://ssrn.com/abstract=3951426 or http://dx.doi.org/10.2139/ssrn.3951426

Paul Glasserman

Columbia University - Columbia Business School ( email )

3022 Broadway
403 Uris Hall
New York, NY 10027
United States
212-854-4102 (Phone)
212-316-9180 (Fax)

Dan Pirjol (Contact Author)

Stevens Institute of Technology ( email )

Hoboken, NJ 07030
United States

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