Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2002-05-03
Quantitative Finance Vol 2 (2002) 415-431
Physics
Condensed Matter
Statistical Mechanics
Published version, revised
Scientific paper
Option pricing formulas are derived from a non-Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker-Planck equation which maximizes the Tsallis nonextensive entropy of index $q$. A generalized form of the Black-Scholes differential equation is found, and we derive a martingale measure which leads to closed form solutions for European call options. The standard Black-Scholes pricing equations are recovered as a special case ($q = 1$). The distribution of stock returns is well-modelled with $q$ circa 1.5. Using that value of $q$ in the option pricing model we reproduce the volatility smile. The partial derivatives (or Greeks) of the model are also calculated. Empirical results are demonstrated for options on Japanese Yen futures. Using just one value of $\sigma$ across strikes we closely reproduce market prices, for expiration times ranging from weeks to several months.
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