Pricing of variance swap rates and investment decisions of variance swaps: Evidence from a three-factor model

Abstract

This paper proposes a tractable three-factor model with self-exciting jumps for the S&P 500 index and its variance. The statistical results show that this new model empirically outperforms the two-factor models widely studied in the literature in terms of fitting variance swap rates on the S&P 500 index across maturities ranging from one month to two years over the sample period from 2008 to 2020. We also provide closed-form solution for variance swap rates and analytically solve the optimal portfolio choice problem in variance swaps for an investor with a power utility function. Unlike the optimal investment with two variance swaps in two-factor models, the investor follows a “long-short-long” trading strategy involving three swap contracts. Hence, a third swap is not redundant in our three-factor model. In particular, our portfolio optimization exercises illustrate that ignoring a third variance swap in the investment problem may incur significant economic costs in the proposed model.