Closed-form variance swap prices under general affine GARCH models and their continuous-time limits

• Published in: Annals of operations research Vol. 282; no. 1-2; pp. 27 - 57
• Main Authors: Badescu, Alexandru, Cui, Zhenyu, Ortega, Juan-Pablo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2019; Springer; Springer Nature B.V
• Subjects: Asymptotic properties; Autoregressive processes; Business and Management; Closed form solutions; Combinatorics; Continuity (mathematics); Convergence; Discrete-time systems; Exact solutions; Gaussian process; Mathematical models; Operations research; Operations Research/Decision Theory; Prices; S.I.: Application of O. R. to Financial Markets; Stochastic models; Studies; Theory of Computation; Canada; Diffusion limits; Affine GARCH models; C58; G13; Variance dependent pricing kernels; G17; Variance swaps; Realized variance
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-018-2941-9

Fully explicit closed-form expressions are developed for the fair strike prices of discrete-time variance swaps under general affine GARCH type models that have been risk-neutralized with a family of variance dependent pricing kernels. The methodology relies on solving differential recursions for the coefficients of the joint cumulant generating function of the log price and the conditional variance processes. An alternative derivation is provided in the case of Gaussian innovations. Using standard assumptions on the asymptotic behavior of the GARCH parameters as the sampling frequency increases, the diffusion limit of a Gaussian GARCH model is derived and the convergence of the variance swap prices to its continuous-time limit is further investigated. Numerical examples on the term structure of the variance swap rates and on the convergence results are also presented.