Analytic solutions for variance swaps with double-mean-reverting volatility

See Woo Kim, Jeong Hoon Kim

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston's stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.

Original languageEnglish
Pages (from-to)130-144
Number of pages15
JournalChaos, Solitons and Fractals
Publication statusPublished - 2018 Sept

Bibliographical note

Funding Information:
We thank anonymous reviewers for comments on the manuscript which have made this paper improved significantly. The research of J.-H. Kim was supported by the National Research Foundation of Korea under Grant NRF-2017R1A2B4003226 .

Publisher Copyright:
© 2018 Elsevier Ltd

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics


Dive into the research topics of 'Analytic solutions for variance swaps with double-mean-reverting volatility'. Together they form a unique fingerprint.

Cite this