Covariance Dependent Kernels, a Q-Affine GARCH for multi-asset option pricing

Abstract

This paper introduces a class of multivariate GARCH models with sufficient flexibility to allow for pricing kernels dependent on variances and correlation. This extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The superiority of our model is captured by improvements in both the overall likelihood and the VIX-implied likelihoods, as well as a better fitting of marginal distributions with up to 15% less error on one-asset option prices. The new degree of freedom in the covariance-dependent kernel is also shown to significantly impact prices of two-asset correlation options leading to up to 53% differences for out-of-the-money claims with short maturity. The added flexibility is also noticeable in the shape of marginal and joint pricing kernels as demonstrated with our empirical estimates.

Keywords: Pricing, multi-asset options, GARCH models, Closed form solutions, Covariance dependent kernel, maximum likelihood estimation

JEL Classification: C15, G12, G13