Non-Gaussian GARCH option pricing models and their diffusion limits

Abstract

This paper investigates the weak convergence of general non-Gaussian GARCH models together with an application to the pricing of European style options determined using an extended Girsanov principle and a conditional Esscher transform as the pricing kernel candidates. Applying these changes of measure to asymmetric GARCH models sampled at increasing frequencies, we obtain two risk neutral families of processes which converge to different bivariate diffusions, which are no longer standard Hull-White stochastic volatility models. Regardless of the innovations used, the GARCH implied diffusion limit based on the Esscher transform can be obtained by applying the minimal martingale measure under the physical measure. However, we further show that for skewed GARCH driving noise, the risk neutral diffusion limit of the extended Girsanov principle exhibits a non-zero market price of volatility risk which is proportional to the market price of the equity risk, where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution. Our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes.

Valuation of VIX and target volatility options with affine GARCH models

“…Moreover, since the valuation is typically performed in the risk‐neutral world, we need to identify a pricing kernel which preserves the affine property of our models after the change of measure. In the literature, there are different methods proposed for specifying the pricing kernel (see, e.g., Badescu, Elliott, & Ortega, 2015, 2017) within a GARCH setting. Following Christoffersen, Heston, and Jacobs (2013; see also Bormetti, Corsi, & Majewski, 2016; Corsi, Fusari, & LaVecchia, 2013; Majewski, Bormetti, & Corsi, 2015), we use a variance‐dependent stochastic discount factor whose Radon–Nikodym derivative is given by The parameters and in (1) represent the market prices of equity and variance risk, respectively.…”

Discriminating Between GARCH Models for Option Pricing by Their Ability to Compute Accurate VIX Measures

“…Let us start with marginal effects: the impact of the choice of a non-affine GARCH structure accounting for the leverage effect is small with a 2.2% improvement in favor of the GJR model. In the same way, in the case of the NIG-NGARCH model estimated using returns and VIX information, the Esscher and the extended Girsanov principle SDF give rise to almost identical results with a difference of 1.4% for the benefit of the exponential-affine parameterization (see also Badescu et al (2011) and Badescu et al (2015) that deliver the same conclusion). Finally, using an estimation strategy based on options and returns information only improves by around 1% the IVRMSE with respect to its VIX-Returns counterpart (however, this improvement is around 10.5% when using returns only) as already observed in Chorro & Fanirisoa (2019).…”

“…Therefore, Gaussian hypothesis for the conditional distribution of log-returns has to be relaxed and a myriad of possible choices may be used to take into account all the mass in the tails and the asymmetry (Chorro et al (2015) Chapter 2). Among them, the Generalized Hyperbolic (Chorro et al (2012), Badescu et al (2011)) family and its Normal Inverse Gaussian (NIG) subclass (Stentoft (2008), Badescu et al (2015)), the Inverse Gaussian (IG) distribution (Christoffersen et al (2006a)), or the mixture of Gaussian (Badescu et al (2008)) clearly improve forecasting performances of related GARCH models.…”

Non-parametric news impact curve: a variational approach

“…Our model is written in equation (1) in discrete time, with the time t in a subset of integers, like in most of the literature about ARCH models. However, some papers deal with the continuous limit of ARCH models [35] or even of GARCH models [3,9]. In such a continuous framework, our model would be the limit, when τ → 0, of…”