Hermite Expansion for Transition Densities of Irreducible Diffusions with An Application to Option Pricing
49 Pages Posted: 2 Jul 2019
Date Written: July 2, 2019
A diffusion is said to be reducible if there exists a one-to-one transformation of the diffusion into a new one whose diffusion matrix is the identity matrix, otherwise it is irreducible. Most multivariate diffusions such as the stochastic volatility models are irreducible. As pointed out by Ait-Sahalia (2008), the straight Hermite expansion of Ait-Sahalia (2002) will not in general converge for irreducible diffusions. In this paper we manage to develop the Hermite expansion for transition densities of irreducible diffusions, which converges as the time interval shrinks to zero. By introducing a quasi-Lamperti transform unitizing the process’ diffusion matrix at the initial time, we can expand the transition density of the transformed process using Hermite polynomials as the orthogonal basis. Then we derive explicit recursive formulas for the expansion coefficients using the Itˆo-Taylor expansion method, and prove the small-time convergence of the expansion. Moreover, we show that the derived Hermite expansion unifies some existing methods including the expansions of Li (2013) and Yang et al. (2019). In addition, we demonstrate the advantage of Hermite expansion by deriving explicit recursive expansion formulas for European option prices under irreducible diffusions. Numerical experiments illustrate the accuracy and effectiveness of our approach.
Keywords: Hermite expansion, Irreducible diffusions, Transition densities, European option pricing, Stochastic volatility models
JEL Classification: C13, C32, G13, C63
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