## Abstract

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.

Original language | English |
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Article number | 235 |

Journal | Computational and Applied Mathematics |

Volume | 41 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2022 Sept |

### Bibliographical note

Funding Information:We thank anonymous referees for having provided valuable comments on an earlier version of the manuscript. The research of J.-H. Kim was supported by the National Research Foundation of Korea NRF2021R1A2C1004080.

Publisher Copyright:

© 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

## All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics