Covariance dependent kernels, a Q-affine GARCH for multi-asset option pricing
Introduction
The class of Generalized Autoregressive Conditional Heteroscedasticity, or GARCH, models of Bollerslev (1986) and Engle (1982), and their applications to derivative pricing and risk management has evolved remarkably over the past three decades. Among various specifications of GARCH models, of particular interest is the class of affine GARCH models. The moment generating function in an affine GARCH model can be written in closed form with coefficients given by a set of recursive algebraic equations. Hence, calculation of European option prices and their hedge ratios can be performed in closed-form which is considerably faster and more accurate than Monte Carlo methods, see Badescu et al., 2015, Date and Islyaev, 2015, and Menn and Rachev (2005) for examples of non-Affine GARCH option pricing.
The first affine GARCH specification was proposed by Heston and Nandi (2000), henceforth the HN-GARCH model, as a univariate model with Gaussian innovations and a linear pricing kernel. Since then, a number of non-Gaussian affine models or extensions of the HN-GARCH model have been proposed by researchers. Christoffersen, Heston, and Jacobs (2006) successfully obtained a closed-form moment generating function with Inverse Gaussian shocks. Ornthanalai (2014) incorporated a combination of Gaussian innovations and Levy jumps in an affine setting, and Bégin, Dorion, and Gauthier (2020) presented a bivariate extension of this Levy jump GARCH model. A general non-Gaussian conditional affine GARCH model was introduced in Escobar-Anel, Rastegari, and Stentoft (2021).
The use of non-linear or variance-dependent pricing kernel in affine models was first introduced in Christoffersen, Heston, and Jacobs (2013) which extends the classical HN-GARCH model. The change of measure in this model is driven by an additional variance premium parameter accounting for a shift in variance of innovations from the physical to the risk-neutral model. It also maintains the affine structure of the model under the equivalent martingale measure. Badescu, Cui and Ortega (2019) formulated a general affinity condition for univariate GARCH models which guarantees both a closed-form moment generating function and a variance-dependent kernel.
A general specification of an affine multivariate factor GARCH model is presented in the recent work by Escobar-Anel, Rastegari, and Stentoft (2020). The model, henceforth the ERS-ARCH model, is based on decomposition of conditional covariance matrix such as spectral or Cholesky decomposition, and expresses the dynamics of the conditional covariance matrix as a linear combination of independent components driven by univariate affine GARCH models. Such factor GARCH models have been explored in Van der Weide (2002) and Vrontos, Dellaportas, and Politis (2003) in a non-affine setting. They exhibit certain advantages over other multivariate GARCH specifications, namely, guaranteed positive-definiteness of conditional covariance matrix, reduced number of parameters, straightforward estimation procedure, and meaningful economical interpretation.
The affine multivariate ERS-GARCH model is interestingly equipped with a covariance-dependent pricing kernel which maintains the closed-form structure of the moment generating function under risk-neutral process, thus generalizing the univariate model in Badescu, Cui et al. (2019) and Christoffersen et al. (2013). However, due to the affine structure of the model, the pricing kernel is somewhat restricted. For example, in the bivariate case the pricing kernel is driven by only two free parameters accounting for changes in the three elements of the conditional covariance matrix (i.e. two variances and a correlation).
Our main objective in this paper is to present a full-parameter covariance-dependent pricing kernel in an affine setting, and to study the impact of the additional flexibility of this pricing kernel on model fit and option pricing. The key observation here is that from a pricing point of view, having a closed-form moment generating function under the risk-neutral measure is desirable but is not necessary in the physical process. Hence, we extend the bivariate ESR GARCH model in Escobar-Anel et al. (2020) by relaxing the affinity condition on the physical process while ensuring that the model is affine under the risk-neutral dynamics. The new model admits a flexible covariance-dependent pricing kernel driven by at least three parameters to capture individual assets variance premia and the correlation premium.
To assess the impact and need of the proposed full-parameter covariance-dependent kernel, we perform a joint estimation on daily returns and volatility index values for three pairs of market indices. Our empirical results confirm statistical significance of all parameters in the pricing kernel, which offers the very first evidence of correlation dependency in the kernel, i.e. in addition to the well documented variance dependance in the kernel. It also shows a clear improvement in returns likelihood and volatility index likelihood compared to the restricted ERS-GARCH model. Next, the effect on the marginal pricing kernels over periods of one day, one week and one month are examined and shown to be different from those given by existing models especially in the tails. Using single asset call and put options, we demonstrate that the new model has much better option pricing performance with up to 38% improvement in the implied volatility error. Finally, we consider the joint pricing kernel and show that this differs significantly from existing models. Comparing the price of two-asset correlation options calculated from our proposed model we observe a relative difference of up to 53% for out-of-the money short-maturity options when compared to the restricted ERS-GARCH model.
We summarize our contributions as follows:
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We introduce a multivariate model of log returns which admits both a variance and a correlation dependent kernel. In particular, a general pricing kernel is created; this pricing kernel is driven by, at least, three free parameters accounting for shifts from the historical () measure to the risk-neutral () measure in the conditional covariance matrix. The model is an extension of the ERS-GARCH model of Escobar-Anel et al. (2020).
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Conditions ensuring a proper change from historical to risk-neutral measures are provided and though the model may be non-affine under the historical measure a special subclass allows for affine dynamics. The role of the new parameters in creating differences between the and covariances is explained.
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A joint calibration on three bivariate pairs of indexes and their respective volatility indexes, chosen from S&P500 (SPX and VIX), Dow Jones (DJX and VXD), NASDAQ 100 (NDX and VXN) and Russell 2000 (RUT and RVX) is performed. The calibration provides the very first empirical evidence of correlation-dependent kernels, and it improves, compared to the embedded ERS-GARCH model, on the likelihood of returns, the likelihood of volatility indexes, and the new parameters are, in most cases, statistically significant.
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The added flexibility of our model is further justified empirically when comparing the implied marginal and joint pricing kernel to the embedded models. This improvement results in significantly reduced errors in univariate option fitting. Moreover, when compared to the restricted ERS-GARCH model significant differences of up to 53% are found when pricing a popular two-assets correlation option.
The rest of the paper is organized as follows. Section 2 presents the general model, provides viable changes of measures along with a specific setting that ensures affine risk-neutral dynamics, and discusses the interpretation of the new parameter. Section 3 applies the model empirically providing parameter estimates from joint calibration of three bivariate series. The implications on the marginal and joint pricing kernels are studied here and the effect on fitting single asset options empirically and of pricing multi-asset options are discussed. Section 4 concludes.
Section snippets
Multivariate GARCH models with a covariance-dependent pricing kernel
In this section, we first present our multivariate GARCH model. We then provide general conditions for a viable changes of measure, as well as specific conditions that ensure an affine risk-neutral structure. Finally, an in-depth analysis of the implications of our model for variance-dependent and correlation-dependent kernels is presented.
Empirical analysis
In this section we consider an empirical application of our model with the two-parameter specification for . We first demonstrate how the parameters of the model can be estimated by maximizing the likelihood of returns and volatility indices. We then examine the effect that the new parameter has on the marginal pricing kernel and illustrate the improved performance in terms of pricing options on one asset. Finally, we examine the effect on the bivariate pricing kernel and document the
Conclusions
This paper introduces a class of multivariate GARCH models which admits both a variance and a correlation dependent kernel. In particular, the pricing kernel we propose is driven by, at least, three free parameters accounting for shifts from the historical () measure to the risk-neutral () measure in the conditional covariance matrix. The model is an extension of the ERS-GARCH model of Escobar-Anel et al. (2020) and extends the existing literature by explicitly modeling correlation dependent
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