Abstract
In this paper, we propose a new multivariate mean-reverting model incorporating state-of-the art 4/2 stochastic volatility and a convenient principal component stochastic volatility (PCSV) decomposition for the stochastic covariance. We find a quasi closed-form characteristic function and propose analytic approximations, which aid in the pricing of derivatives and calculation of risk measures. Parameters are estimated on three bivariate series, using a two-stage methodology involving method of moments and least squares. Moreover, a scaling factor is added for extra degrees of freedom to match data features. As an application, we consider investment strategies for a portfolio with two risky assets and a risk-free cash account. We calculate value-at-risk (VaR) values at a 95% risk level using both simulation-based and distribution-based methods. A comparison of these VaR values supports the effectiveness of our approximations and the potential for higher dimensions.
Highlights
IntroductionPrincipal component analysis (PCA) is used to reduce dimensionality in the explanation of a vector of asset returns; see, for instance, Alexander (2001)
We studied in detail a multivariate mean-reverting 4/2 stochastic volatility model based on principal component analysis (PCA), which is inspired in the general framework of Cheng et al (2019)
Estimation of multidimensional processes is rare in the literature, and our work demonstrates that many, but not all, of the parameters are statistically significant, confirming stylized facts of commodity prices and volatility indexes such as stochastic correlation and spill-over effects
Summary
Principal component analysis (PCA) is used to reduce dimensionality in the explanation of a vector of asset returns; see, for instance, Alexander (2001). The methodology has been used in continuoustime stochastic processes for financial applications; see Escobar et al (2010) and Escobar and Olivares (2013) for its usage in collateralized debt obligations (CDO) and exotic financial derivatives pricing, as well as Escobar and Gschnaidtner (2018) and more generally. When modeling financial or any complex data, one can focus on capturing the stylized facts reported in the literature. Examples of a refined fact of financial data are the smiles and smirks of the implied volatility surface. In Christoffersen et al (2009), the authors proposed a PCA-inspired stochastic covariance (SC) model using the popular Heston stochastic volatility model Heston (1993) as the underlying component
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