Derivative pricing with non-linear Fokker–Planck dynamics

We examine how the Black–Scholes derivative pricing formula is modified when the underlying security obeys non-extensive statistics and Fokker–Planck dynamics. An unusual feature of such securities is that the volatility in the underlying Ito–Langevin equation depends implicitly on the actual market rate of return. This complicates most approaches to valuation. Here we show that progress is possible using variations of the Cox–Ross valuation technique.