Modeling the variance risk premium of equity indices: the role of dependence and contagion (Q2813080)

By applying several deep results from mathematical finance and stochastic analysis, the authors of this paper study in detail the variance risk premium (VRP) of a value weighted stock index and its driving stochastic dynamics.NEWLINENEWLINESuppose there are two trading entities, bank \(B\) and its counterparty \(C\), say. Similarly to \(B\)'s demanded addition of a credit valuation adjustment (CVA) to a default-free price of \(B\)'s offered portfolio of derivatives to \(C\) to cover for \(B\)'s risk of a potentially large loss due to a default of \(C\) before the maturity of the trade between \(B\) and \(C\), a VRP could be viewed as a premium (respectively ``model risk adjustment'') that investors demand for holding (possibly correlated) assets the volatility of which varies stochastically in time and is exposed to stochastic shocks.NEWLINENEWLINETherefore, following [\textit{V. Todorov}, ``Variance risk-premium dynamics: the role of jumps'', Rev. Financ. Stud. 23, No. 1, 345--383 (2010; \url{doi:10.1093/rfs/hhp035})], the authors' definition of a VRP is based on the stochastic model of the value weighted stock index which is driven by a stochastic volatility component and jumps in the stock price components of the index.NEWLINENEWLINEBy applying the standard probabilistic Itô machinery to describe incomplete financial markets and their induced information by semimartingales with jumps and including a few simplifying assumptions (such as a constant weight index vector \(\mathbf{w}\) over time), the then so derived stochastic dynamics of the model of the value weighted index is given as NEWLINENEWLINE\[NEWLINEdY_t = {\mathbf{w}}^{\top}{{\mathbf{a}}_t}\,dt + {\mathbf{w}}^{\top} \Sigma^{\frac{1}{2}}_t \,d{{\mathbf{W}}_{t}} + {\mathbf{w}}^{\top}\,d{{\mathbf{J}}_{t}} + \int_{\mathbb{R}} (e^x - (1 + x)){\mathbf{w}}^{\top}{{\mathbf{M}}(dt, dx)},NEWLINE\]NEWLINE NEWLINEwhere the stochastic process \(Y \equiv \{Y_t : t \geq 0 \}\) models the log-returns of the sum of the stock prices of the \(D\) stock components of the index, \(\mathbf{w}\) is a \(D\)-dimensional vector of weights which is assumed to be constant over time, \(\mathbf{W}\) is a \(D\)-dimensional Brownian motion, \(\mathbf{J}\) is a \(D\)-dimensional pure-jump process and \({\mathbf{M}}\) the \(D\)-dimensional vector, consisting of the \(D\) jump measure components of \(\mathbf{J}\). \(\Sigma\) describes a stochastic process with values in the cone of all symmetric positive semidefinite matrices, representing a random change of the covariance matrices of the \(D\) stock prices over time. The \(D\)-dimensional vector process \({\mathbf{a}}\) models a transformed \(D\)-dimensional drift vector which itself depends on the stochastic covariance matrix-valued process \(\Sigma\). The details of the whole construction of the process \(Y\) are listed in Assumption~1--8.NEWLINENEWLINE\(\Sigma\) is consecutively described by three individual multivariate models which are analysed in detail. The first two (already established and now revisited) models account for dependency between the single stock components in the value weighted stock index and hence are classified by the authors as dynamic correlation models. The first dynamic correlation model is built on a multivariate non-Gaussian Ornstein-Uhlenbeck (OU) process (Section~3.1.1), and the second one is induced by a Wishart process (Section~3.1.2), where the latter is shown to lead to a straight-forward generalisation of a popular stochastic volatility model which has been frequently used in both, academic research and in the financial industry (since 1993) -- the Heston model. Already the Heston model, actually introduced to tackle the famous -- and still unsolved -- ``smile problem'' in a Black-Merton-Scholes world with constant volatility (generalising the local volatility model of Dupire) lead to an vibrating research field on its own in both, academia and the financial industry.NEWLINENEWLINEIn addition to the two known multivariate dynamic correlation models for \(\Sigma\) the authors introduce and investigate in detail a third multivariate model, where \(\Sigma\) now is modelled in time by \(D \times D\)-diagonal matrices of non-Gaussian OU processes (each one of them given by a Lévy process driven SDE) and the components of the driving \(D\)-dimensional Brownian motion \(\mathbf{W}\) are correlated. The theoretical features of this third model are in line with empirical findings in e.\,g. [\textit{J. Driessen}, \textit{P. J. Maenhout} and \textit{G. Vilkov}, ``The price of correlation risk: evidence from equity options'', J. Finance 64, No. 3, 1377--1406 (2009; \url{doi:10.1111/j.1540-6261.2009.01467.x})]. Since all off-diagonal elements of the diagonal matrix \(\Sigma\) are \(0\), the authors view the third model as a constant correlation model (Section~3.2).NEWLINENEWLINEIt should be explicitly noted that the stochastic dynamics of both, \(Y\) and \(\Sigma\) is driven by the physical ``real-world'' measure \(\mathbb{P}\) and not by a risk-neutral equivalent (local) martingale measure (\(\mathbb{Q}\), say).NEWLINENEWLINEGiven the log-returns \(\{Y_t : t \geq 0 \}\) of the sum of the prices of the \(D\) stock components of the index, an arbitrary filtration \(\{{\mathcal{F}}_t : t \geq 0\}\) satisfying the usual conditions and some change of the real-world physical measure \(\mathbb{P}\) to a so-called ``structure preserving'' risk-neutral equivalent martingale measure \(\mathbb{Q}\), the VRP is then defined as NEWLINENEWLINE\[NEWLINEVRP_{t, h} : = {\mathbb{E}}^{\mathbb{P}}\big[[Y]_{t+h} - [Y]_{t} \bigm| {\mathcal{F}}_t\big] - {\mathbb{E}}^{\mathbb{Q}}\big[[Y]_{t+h} - [Y]_{t} \bigm| {\mathcal{F}}_t\big],NEWLINE\]NEWLINE NEWLINEwhere \([Y] \equiv \{[Y]_t : t \geq 0 \} = \{[Y, Y]_t : t \geq 0 \}\) denotes as usual the quadratic variation of the process \(Y\) (and \(h, t \geq 0\)). A VRP therefore describes the spread between the physical and the risk neutral volatility. A justification of this definition is given in Section~5.2 of [\textit{O. E. Barndorff-Nielsen} and \textit{A. E. D. Veraart}, ``Stochastic volatility of volatility and variance risk premia'', J. Financ. Econom. 11, No. 1, 1--46 (2013; \url{doi:10.1093/jjfinec/nbs008})].NEWLINENEWLINEUnder such a structure preserving probability measure \(\mathbb{Q}\) both, \(Y\) and \(\Sigma\) preserve their respective probabilistic properties and follow a stochastic dynamics driven by the same classes of processes although different values for the parameters are allowed.NEWLINENEWLINEIn the case of dependence between the \(D\) stock prices in the index, explicit formulas and examples of VRPs are provided under both multivariate dynamic correlation models for \(\Sigma\). The authors reveal that in their model of \(Y\) and the related VRP both dynamic correlation models, the Wishart model and the multivariate non-Gaussian OU model for \(\Sigma\) actually do no longer fit with the empirical results in [Driessen et al., loc. cit.]. However, these empirical results were build on historical ``real-world'' data from 1993 until 2003 only -- and hence were delivered several years before the contagion tail effect of the financial crisis 2007/2008 changed the shape of market data significantly.NEWLINENEWLINETo picture the impact of this shape, another source of jumps is used to model the stochastic dynamics of \(Y\). To this end, the \(D\)-dimensional Lévy process model for \(\mathbf{J}\) is substituted by a \(D\)-dimensional Hawkes process model so that for any component of the \(D\)-dimensional vector, consisting of the \(D\) jump measures \(M^{(1)}(ds, dx),\) \(M^{(2)}(ds, dx), \ldots, M^{(D)}(ds, dx)\) of the pure jump process vector \(\mathbf{J}\), the respective (intensity of a) Hawkes process now is implemented in its compensator.NEWLINENEWLINEHawkes processes are counting processes that allow for self- and mutual excitation, due to a peculiar form of their intensities and their specific stochastic dynamics. When a component of the \(D\)-dimensional Hawkes process exhibits a jump, the corresponding intensity increases (self-excitation). This implies an increase also in the other elements of the intensity vector, causing a boost in the probability of subsequent jumps in the other components (mutual excitation). Modelling \(\mathbf{J}\) in this way by a \(D\)-dimensional Hawkes intensity process accounts for the contagion effect between the stock components, which has a significant impact during periods of financial distress and bankruptcy.NEWLINENEWLINEThe paper concludes with an appendix where explicit calculations and proofs are given including the proof of the existence of a structure preserving risk-neutral equivalent martingale measure in the case of the VRP, built on Hawkes processes.NEWLINENEWLINEDue to some technical subtleties regarding an exclusion of arbitrage in incomplete financial markets with jumps it would be perhaps very useful for the reader of this interesting paper to see more clearly which stochastic dynamics is driven under a (structure preserving) risk-neutral (local) martingale measure and which one under the real-world probability measure, and how they are precisely related. In particular, this might also explain the important observation of the authors on top of Equation~13 in the paper.