Using relative returns to accommodate fat-tailed innovations in processes and option pricing

Abstract

The canonical process used to describe financial time series is based on a logarithmic random walk. Adding a fat-tail distribution for the innovations in this framework creates fundamental inconsistencies, essentially related to a diverging integral. The problems are related to (1) the (infinite) values of statistical quantities for processes with fat-tail innovations, (2) the robustness of computations when dealing with large events (genuine or noise), and (3) the pricing of options with heteroskedasticity and fat tails. Instead of logarithmic processes, we advocate using geometric processes and relative returns so that these problems do not occur. The mathematical properties of geometric processes are explored for setups of increasing complexity. For empirical time series and for processes, the skews are evaluated using a robust estimator and are compared for both return definitions. Finally, the European option pricing framework is modified to use geometric processes for the underlying, allowing the incorporation of natural fat-tail innovations.

Keywords:

• Geometric processes
• Relative returns
• Non-Gaussian distributions
• Non-Gaussian option pricing
• ARCH

JEL Classification::

• C2
• C22
• G1
• G12

Notes

†In general, the prices are not positive, say, for example, for a long/short portfolio, or for a swap.

†A fat-tailed distribution decays with a power law: for α > 0, Pr(X > x) ∼ x ^{−(1 + α)}.

‡The continuum limit is the limit process one obtains by letting the time period go to zero while scaling the process parameters appropriately.