Willow Tree Algorithms for Pricing VIX Derivatives Under Stochastic Volatility Models

29 Pages Posted: 1 Nov 2019 Last revised: 19 Jan 2020

See all articles by Changfu Ma

Changfu Ma

affiliation not provided to SSRN

Wei Xu

Tong Ji University - Mathematical Department

Yue Kuen Kwok

Hong Kong University of Science & Technology - Department of Mathematics

Date Written: October 23, 2019

Abstract

VIX futures and option are the most popular contracts traded in the Chicago Board Options Exchange. The bid-ask spreads of traded VIX derivatives remain to be wide, possibly due to lack of reliable pricing models. In this paper, we consider pricing VIX derivatives under the consistent model approach, which considers joint modeling of the dynamics of S&P index and its instantaneous variance. Under the affine jump-diffusion formulation with stochastic volatility, analytic integral formulas can be derived to price VIX futures and option. However, these integral formulas invariably involve Fourier inversion integrals with cumbersome hyper-geometric functions, thus posing various challenges in numerical evaluation. We propose a unified numerical approach based on the willow tree algorithms to price VIX derivatives under various common types of joint process of the S&P index and its instantaneous variance. Given the analytic form of the characteristic function of the instantaneous variance of the S&P index process in the Fourier domain, we apply the fast Fourier transform algorithm to obtain the transition density function numerically in the real domain. We then construct the willow tree that approximates the dynamics of the instantaneous variance process up to the fourth order moment. Our comprehensive numerical tests performed on the willow tree algorithms demonstrate high level of numerical accuracy, run time efficiency and reliability for pricing VIX futures and both European and American options under the affine model and 3/2-model. We also examine the implied volatility smirks and the term structures of the implied skewness of the model prices of VIX options.

Keywords: VIX derivatives, willow tree algorithm, affine jump-diffusion model, 3/2-model

JEL Classification: G13

Suggested Citation

Ma, Changfu and Xu, Wei and Kwok, Yue Kuen, Willow Tree Algorithms for Pricing VIX Derivatives Under Stochastic Volatility Models (October 23, 2019). Available at SSRN: https://ssrn.com/abstract=3474165

Changfu Ma

affiliation not provided to SSRN

Wei Xu

Tong Ji University - Mathematical Department ( email )

1239 Siping Road
Shanghai
China

Yue Kuen Kwok (Contact Author)

Hong Kong University of Science & Technology - Department of Mathematics ( email )

Clearwater Bay
Kowloon, 999999
Hong Kong

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