Abstract
We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs …) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.
Highlights
Many financial models are based on the idea that the instantaneous variations of market prices can be accurately described by some stochastic dynamics, resulting in the so-called class of exponential market models; this class includes, among others, the Black–Scholes model [1], the Heston model [2] or models based on Variance Gamma [3] or Normal inverse Gaussian [4] processes
When the stochastic dynamics is given by a Lévy process, one speaks of an exponential Lévy model
Evaluating the Mellin integral by means of residue summation, we have obtained a collection of practical pricing formulas for several exotic payoffs
Summary
Many financial models are based on the idea that the instantaneous variations (or log-returns) of market prices can be accurately described by some stochastic dynamics, resulting in the so-called class of exponential market models; this class includes, among others, the Black–Scholes model [1], the Heston model [2] (which features a supplementary dynamics for the volatility) or models based on Variance Gamma [3] or Normal inverse Gaussian [4] processes. From the point of view of financial modeling, the presence of a left fat tail in the FMLS model corresponds to the occurrence of negative jumps in the distribution of returns, i.e., to brutal drops in market prices, observed for instance in stressed market conditions or in case of a firm’s default (such events, sometimes called “black swans”, are described in detail in [8]) This model is very closely related to fractional calculus because its probability densities satisfy a space fractional diffusion equation, involving a Riesz–Feller derivative whose order governs the tail index of the distribution; when this fractional derivative coincides with the usual second derivative, the model degenerates into the Black–Scholes model [1], whose densities are given by the usual Gaussian kernel.
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