Smile from the Past: a General Option Pricing Framework with Multiple Volatility and Leverage Components

Abstract

In the current literature, the analytical tractability of discrete time option pricing models is guaranteed only for rather specific types of models and pricing kernels. We propose a very general and fully analytical option pricing framework, encompassing a wide class of discrete time models featuring multiple-component structure in both volatility and leverage, and a flexible pricing kernel with multiple risk premia. Although the proposed framework is general enough to include either GARCH-type volatility, Realized Volatility or a combination of the two, in this paper we focus on realized volatility option pricing models by extending the Heterogeneous Autoregressive Gamma (HARG) model of to incorporate heterogeneous leverage structures with multiple components, while preserving closed-form solutions for option prices. Applying our analytically tractable asymmetric HARG model to a large sample of S&P 500 index options, we demonstrate its superior ability to price out-of-the-money options compared to existing benchmarks.

VIX term structure and VIX futures pricing with realized volatility

“…In this study, we denote the original LHARG model of Majewski et al () as LHARG‐M, as it contains volatility components up to the monthly average. We extend the model to LHARG‐Q by including the quarterly average (63 trading days), and to LHARG‐Y by including both the quarterly and the yearly average (252 trading days).…”

“…Unlike the Heston–Nandi leverage function, is no longer guaranteed to be positive. However, Majewski et al () provided numerical evidence that the analytical results can effectively describe a regularized version of the model.…”

“…Most studies focus on the performance of the HAR model and its extensions into forecasting volatility or realized volatility under the physical measure, but Corsi, Fusari, and Vecchia () and Majewski, Bormetti, and Corsi () have shown that the HAR framework is also capable of matching the volatility information implied by option prices, that is, under the risk‐neutral measure. Corsi et al () extended the HAR model with a gamma innovation (the heterogeneous autoregressive gamma, HARG) and specified an exponentially affine pricing kernel.…”

“…Corsi et al () extended the HAR model with a gamma innovation (the heterogeneous autoregressive gamma, HARG) and specified an exponentially affine pricing kernel. Majewski et al () further developed the model by allowing more flexible leverage components (LHARG) and derived the analytical pricing formula for European options. In this study, we extend the LHARG model by including lagged quarterly and yearly realized variance into the RV dynamics and derive the analytical formulas for the VIX term structure in addition to the VIX futures.…”

Econometric analysis of financial derivatives: An overview

“…The purpose of the special issue is to highlight a number of areas of research in which novel econometric, financial econometric and empirical finance methods have contributed significantly to the econometric analysis of financial derivatives, specifically market-based estimation of stochastic volatility models (Aït-Sahalia, Amengual and Manresa (2015)), the fine structure of equity-index option dynamics (Andersen, Bondarenko, Todorov and Tauchen (2015)), leverage and feedback effects in multifactor Wishart stochastic volatility for option pricing (Asai and McAleer (2015)), option pricing with non-Gaussian scaling and infinite-state switching volatility (Baldovin, Caporin, Caraglio, Stella and Zamparo (2015)), stock return and cash flow predictability: the role of volatility risk (Bollerslev, Xu and Zhou (2015)), the long and the short of the risk-return trade-off (Bonomo, Garcia, Meddahi and Tedongap (2015)), What's beneath the surface? option pricing with multifrequency latent states (Calvet, Fearnley, Fisher and Leippold (2015)), bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in 4 commodity spot and futures markets (Cavaliere, Ørregaard Nielsen and Taylor (2015)), a stochastic dominance approach to financial risk management strategies (Chang, Jiménez-Martín, Maasoumi and Pérez-Amaral (2015)), empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction (Duong, and Swanson (2015)), non-linear dynamic model of the variance risk premium (Eraker and Wang (2015)), pricing with finite dimensional dependence (Gourieroux and Monfort (2015)), quanto option pricing in the presence of fat tails and asymmetric dependence (Kim, Lee, Mittnik and Park (2015)), smile from the past: a general option pricing framework with multiple volatility and leverage components (Majewski, Bormetti and Corsi (2015)), COMFORT: A common market factor non-Gaussian returns model (Paolella and Polak (2015)), divided governments and futures prices (Sojli and Tham (2015)), and model-based pricing for financial derivatives (Zhu and Ling (2015)). …”