W-shaped implied volatility curves in a variance-gamma mixture model

In liquid option markets, W-shaped implied volatility curves have occasionally been observed. We introduce a single-period market model based on a mixture of two variance-gamma distributions – representing the diverging views of two groups of investors – in which such shapes can be reproduced. This is in contrast to lognormal models, where at least three different distributions have to be mixed in order to produce a W-shape, as shown by Glasserman and Pirjol [Quantitative Finance, 23(4), 2023]. Next to analytic proofs for the appearance of W-shapes curves, we also provide an explicit option pricing formula for our model. The methodology we develop can possibly be extended and applied to other mixture models constructed from (semi-)heavy-tailed distributions, offering a systematic way of capturing complex implied volatility patterns.