Generalizing the Black and Scholes Equation Assuming Differentiable Noise

Abstract

This article develops probability equations for an asset value through time, assuming continuous correlated differentiable Gaussian distributed noise. Ito’s (1944) stochastic integral and a generalized Novikov’s (1919) theorem are used. As an example, the mathematical model is used to generalize the Black and Scholes’ (1973) equation for pricing financial instruments. The article connects the Kolmogorov (1931) probability equation to arbitrage and to how financial instruments are priced, where more generally, the mathematical model based on differentiable noise may improve or be an alternative for forecasts. The article contrasts with much of the literature which assumes continuous nondifferentiable uncorrelated Gaussian distributed white noise.