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Lorig, Matthew; Pagliarani, Stefano; Pascucci, Andrea

A family of density expansions for Lévy-type processes. (English) Zbl 1329.60122

Ann. Appl. Probab. 25, No. 1, 235-267 (2015).
The authors consider scalar Lévy-type processes with sufficiently smooth coefficients which include local volatility models with or without jumps of various kinds extensively investigated in the past. In particular, earlier results and methods of S. Pagliarani et al. [SIAM J. Financ. Math. 4, 265–296 (2013; Zbl 1285.60084)] that were obtained for state-independent Lévy measures and no-default situations are extended and generalized here to include state-dependent Lévy measures and the local default intensity case. No small diffusion or small jump size/intensity assumptions as in [E. Benhamou et al., Finance Stoch. 13, No. 4, 563–589 (2009; Zbl 1195.91153)] are made in this paper.
The authors derive a family of asymptotic expansions (with respect to an arbitrary basis) for the transition densities of Lévy-type processes as well as European-style derivative prices and defaultable bond prices. Pseudo-differential calculus techniques are used to provide explicit formulae for the Fourier transform of every term in the transition density and option pricing expansions. For models with Gaussian-type jumps, respective inverse Fourier transforms are explicitly computed, and pointwise error estimates for the transition densities are provided. Numerical examples that illustrate the versatility and effectiveness of the methods include CEV-like processes and processes with Gaussian or Variance-Gamma Lévy measure.
Reviewer: Martynas Manstavičius (Vilnius)

Cited in 16 Documents

MSC:

60G51 Processes with independent increments; Lévy processes
35S05 Pseudodifferential operators as generalizations of partial differential operators
35R09 Integro-partial differential equations
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories

Keywords:

Lévy-type processes; asymptotic expansions; local volatility; pseudo-differential calculus; defaultable asset

Citations:

Zbl 1285.60084; Zbl 1195.91153

Software:

Mathematica
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Full Text: DOI arXiv Euclid

References:

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