Applications of Fractional Calculus in Option Pricing

Dear Colleagues,

Since the famous Black–Scholes model, several robust generalizations have been introduced, including stochastic volatility models, regime-switching models, or models driven by jump-diffusion and pure jump processes. In the context of pure jump models, particularly important was the pioneering work by Peter Carr and Liuren Wu, who generalized the Black–Scholes setup by considering the totally asymmetric alpha stable Lévy process known as the finite moment log-stable process (FMLS). It was shown, notably in the works of Cartea and Del Castillo Negrete, that Carr and Wu’s FMLS model corresponds to the generalized diffusion equation with the fractional differential operator in the spatial coordinates: this was a motivation for further investigations of fractional diffusion and fractional calculus in option pricing, with the introduction of fractional operators in both the spatial and temporal coordinates. (Anti-)differential fractional operators also arose in the context of option pricing via fractional Brownian motion, which has its origins in the work of Necula, and found very promising recent applications with rough volatility models and the works of Bergomi, Gatheral, and Rosenbaum, to name a few. This Special Issue is therefore dedicated to applications of fractional calculus in option pricing. The topics include:

• Option pricing under fractional diffusion equation, stable and tempered stable laws;
• Applications of fractional PDEs in option pricing, fractional Black–Scholes equation;
• Option pricing under models featuring fractional Brownian motion: rough volatility, rough Bergomi model, etc.;
• Subordinated option pricing models and their connection to fractional diffusion;
• Volatility clustering, memory phenomena in fractional option pricing;
• Comparison of fractional option pricing models with other option pricing models, such as stochastic volatility models and jump processes.

Dr. Jan Korbel
Dr. Jean-Philippe Aguilar
Guest Editors

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• option pricing
• Black–Scholes model
• fractional calculus
• fractional diffusion
• long-term memory
• Lévy stable processes
• jump processes
• fractional Brownian motion
• subordinated models
• Bergomi model
• rough volatility models