In a recent working paper, cited as Thul and Zhang (2014) below, we propose a novel jump-diffusion model whose jump sizes follow an asymmetrically displaced double gamma (AD-DG) distribution. In this blog post I discuss some of the feedback that we received during seminar and conference presentations. The collection is not exhaustive and will be extended from time to time.

Stochastic Volatility

While the model proposes a very flexible parametrization of the jump size distribution, it fails to incorporate other well-documented properties of asset returns such as, for example, heteroscedasticity.

We agree that the AD-DG jump-diffusion model does not capture all stylized empirical facts. This is however a deliberate modelling choice that was made in order to preserve a closed-form solution for plain vanilla options. In this blog post, I discuss the corresponding trade-off in more detail and present extension of the asset dynamics to include a Heston (1993)-type square-root stochastic volatility component.

Calibration under the Risk-Neutral Probability Measure

In addition to presented estimation under the physical probability measure, it would be interesting to analyze the model’s goodness of fit to the market prices of listed options.

Due to the stationarity of the logarithmic return process, the AD-DG jump-diffusion model is not able to fit the term structure of implied volatilities observed in most real-world option markets. The parameters could however be calibrated to the implied volatility smiles of any fixed maturity. Alternatively, we could incorporate a stochastic volatility component and analyze, whether the improvement in the goodness-of-fit achieved by the additional flexibility introduced to the jump size distribution is significant. While we currently work on addressing the former point, the latter lies outside the scope of this paper and might be the topic of future research.

Time-Dependent Parameters

Can the model dynamics be augmented to include time-dependent but non-random parameters while preserving a closed-form solution for the prices of European plain vanilla options?

Such an extension is indeed possible. In particular, the mean return, the risk-free interest rate, the diffusion coefficient and the jump intensity can be made time-dependent without sacrificing analytical tractability. We deliberately excluded this generalization from the working paper as it does not add significant insights while at the same time complicating the notation. I plan on briefly outlining it in a future blog post.

References

Heston, Steven L. (1993) “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, pp. 327-343

Thul, Matthias and Ally Q. Zhang (2014) “Analytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model,” Working Paper, University of New South Wales and Swiss Finance Institute