Presenter: Thomas Kloster, AU

Title: An orthogonal expansions approach to joint SPX and VIX calibration in the SVJJ model

Abstract: We consider the problem of jointly calibrating a selection of models from the SVJJ framework of Duffie, Pan and Singleton (2000) to SPX and VIX option smiles. Conventionally, the SVJJ model is specified with exponential jumps in the variance process. Under this assumption, the Laplace transform of the underlying becomes available in closed form, which leads to efficient option pricing formulas via transform inversion methods. We show that by considering alternative distributions for the jumps in variance, one may substantially improve the performance of the joint calibration. Specifically, we find that a simple extension to gamma distributed jumps leads to a much better fit to both SPX and VIX option smiles. The downside of this improvement, however,  is a severe increase in computational time, due to the relevant transform no longer being explicitly available. We then show how to amend this loss of tractability by constructing novel approximations to option prices, based on orthogonal polynomial expansions. Contrary to the classical method of selecting a simple reference density with known closed form, we expand the methodology to include all densities with an explicit and tractable Laplace transform, thereby offering increased flexibility. In particular, we exploit the analytical tractability of the standard SVJJ model with exponential jumps to build approximations having its density as leading term. Through a series of joint calibration experiments, we find that our option price expansions are very robust and yield essentially the same results as using exact transform inversion formulas, but taking only a small fraction of the computational time.

PhD presentation as part of mandatory 1st or 3rd year presentation. The time is extended to one hour; the presenter has 40 minutes for the presentation, 5-10 minutes for the discussant, and 5-10 minutes for questions.

Organizers: Stefan Hirth and Anders Merrild Posselt