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Using Relative Returns to Accommodate Fat-Tailed Innovations in Processes and Option Pricing
Oct 13, 2009
Using relative returns to accommodate fat-tailed innovations in processes and option pricing
The canonical process used to described financial time series is based on a logarithmic random walk. Adding fat tail distributions for the innovations in this framework creates fundamental inconsistencies, essentially related to a diverging integral. We propose to use geometric processes and relative returns instead.
This change solves four problems related to
- the (infinite) values of statistical quantities for processes with fat-tail innovations,
- the robustness of computations when dealing with large events (genuine or noise),
- the negative skewness of returns in stocks, and
- the pricing of options with heteroskedasticity and fat-tails.
The mathematical properties of geometric processes is explored for set-ups of increasing complexity. The European option pricing framework is modified to use geometric processes for the underlying, allowing to incorporate naturally fat-tail innovations.
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