Closed-form pricing formulas for variance swaps in the Heston model with stochastic long-run mean of variance

• Published in: Computational & applied mathematics Vol. 41; no. 6
• Main Authors: Yoon, Youngin, Seo, Jun-Ho, Kim, Jeong-Hoon
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Brownian motion; Closed form solutions; Computational mathematics; Computational Mathematics and Numerical Analysis; COVID-19; Exact solutions; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Pricing; Stochastic models; Volatility; Long-run mean of variance; 91G60; Heston model; Variance swap; He–Chen model; 91G20; rDMR model
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-022-01939-7

The Heston model is a popular stochastic volatility model in mathematical finance and it has been extended or modified in several ways by researchers to overcome the shortcomings of the model in the context of pricing derivatives. However, the extended models usually do not lead to a closed-form formula for the derivative prices. This paper is focused on a stochastic extension of the constant long-run mean of variance in the Heston model for the pricing of variance swaps. The extension is given by a positive function perturbed by an amplitude-modulated Brownian motion or Ito integral. We obtain two closed-form formulas for the fair strike prices of a variance swap under two corresponding underlying models. The formulas are explicitly given by elementary functions without any integral terms involved. Further, the two models show better performance than the Heston model when the market implied volatility has a concave-down pattern as shown in an unstable market circumstance caused by the COVID-19 pandemic.