Unifying pricing formula for several stochastic volatility models with jumps. (English) Zbl 1420.91444

Baustian, Falko; Mrázek, Milan; Pospíšil, Jan; Sobotka, Tomáš

Appl. Stoch. Models Bus. Ind. 33, No. 4, 422-442 (2017).

Summary: In this paper, we introduce a unifying approach to option pricing under continuous-time stochastic volatility models with jumps. For European style options, a new semi-closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro-differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way.
In particular, we focus on a log-normal and a log-uniform jump diffusion stochastic volatility model, originally introduced by D. S. Bates [“Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options”, Rev. Financ. Stud. 9, No. 1, 69–107 (1996; doi:10.1093/rfs/9.1.69)] and G. Yan and F. B. Hanson [“Option pricing for a stochastic-volatility jump-diffusion model with log-uniform jump-amplitudes”, in: Proceedings of the American control conference, ACC’06. Los Alamitos, CA: IEEE Computer Society. 6 p. (2006; doi:10.1109/ACC.2006.1657175)], respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out-of-the money contracts.

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
60J75	Jump processes (MSC2010)
45K05	Integro-partial differential equations

Keywords:

stochastic volatility models; option pricing; fundamental transform; PIDE; fractional volatility