Dissecting skewness under affine jump-diffusions
-
Fang Zhen
and
Jin E. Zhang
Abstract
This paper derives the theoretical skewness in a five-factor affine jump-diffusion model with stochastic variance and jump intensity, and jumps in prices and variances. Numerical analysis shows that all of the uncertainties in this model affect skewness. The information regarding jumps in prices is mainly reflected in the short-term skewness. The skewness for other maturities carries the information that is highly correlated with variance. Furthermore, the theoretical VIX and skewness under a simplified five-factor model are used to fit the market risk-neutral volatility and skewness sequentially. The fitting performances are better than traditional double-jump models.
Funding source: Central University of Finance and Economics
Award Identifier / Grant number: 023063022002/010
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 71771199
Funding statement: Fang Zhen has been supported by the Research Fund from the Central University of Finance and Economics (Funder Id: http://dx.doi.org/10.13039/501100002942, Project No. 023063022002/010) and the Program for Innovation Research in Central University of Finance and Economics. Jin E. Zhang has been supported by an establishment grant from the University of Otago and the National Natural Science Foundation of China grant (Funder Id: http://dx.doi.org/10.13039/501100001809, Project No. 71771199).
A Appendix
A.1 Third central moment
The asset log-return is modelled by
where
Its conditional expectation at time t is then given by
Thus, adopting the notations in Zhang et al. (2017), we decompose the de-meaned return into four parts as follows:
where
Converting the variance process described by Equation (43) into the integral form gives
The conditional expectation of vs at time t (s > t) is given by
where
Conducting interchange of the order of integration, we decompose YT into three parts:
where
Note that
where
Note that, in our setup, there are seven risk resources, which are the diffusive risk XT in the asset return process, the Heston-type diffusive risks
where
Zhang et al. (2017) explicitly derives the Heston-type third moment as follows:
where the third and co-third moments of XT and
and the weights are given by
Next, we derive the third moment stemming from the risk in the stochastic long-term variance as follows:
Before deriving TCMM, we need the following results.
The correlations between
where
See Appendix A.1.1. □
Using Ito’s Lemma and the martingale property of Xu,
Plugging Equation (79) into the above equation, we obtain
where
Using Ito’s Lemma and the martingale property of
Plugging Equation (78) into the above equation, we obtain
where
Next, we obtain the third moment resulting from the jumps in prices and variance as well as the uncertainty in the instantaneous jump intensity. Making using of the properties of compound Poisson processes and the previous results on the Heston-type third moment, we have
where the third moment of
the co-third moment of
the co-third moment of
the third moment of
and the expectations of functions of jump sizes are
Finally, we obtain the third moment coming from the uncertainty in the stochastic long-term jump intensity as follows:
where the co-third moment of
and the third moment of
A.1.1 Proof of lemma 1
Using Ito’s Lemma and the martingale property of
Solving the above ordinary differential equation (ODE) gives
Differentiating the above result with respect to s yields Equation (78).
Similarly, using Ito’s Lemma and the martingale property of
Solving the above ODE gives
Differentiating the above result with respect to s yields Equation (79).
A.2 Second central moment
Adopting the same notations as those used for computing the third moment in the Appendix A.1, the second central moment of the continuously compounded return is given by
where
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Supplementary Material
The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2018-0086).
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Articles in the same Issue
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- Uncertainty and Forecasts of U.S. Recessions
- Dissecting skewness under affine jump-diffusions
- The term structure of Eurozone peripheral bond yields: an asymmetric regime-switching equilibrium correction approach
- Unconventional monetary policy reaction functions: evidence from the US
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Articles in the same Issue
- Research Articles
- Uncertainty and Forecasts of U.S. Recessions
- Dissecting skewness under affine jump-diffusions
- The term structure of Eurozone peripheral bond yields: an asymmetric regime-switching equilibrium correction approach
- Unconventional monetary policy reaction functions: evidence from the US
- The nonlinear effects of uncertainty shocks
- Bayesian analysis of periodic asymmetric power GARCH models
