Classical Option Pricing and Some Steps Further

This paper modifies single assumption in the base of classical option pricing model and derives further extensions for the Black-Scholes-Merton equation. We regard the price as the ratio of the cost and the volume of market transaction and apply classical assumptions on stochastic Brownian motion not to the price but to the cost and the volume. This simple replacement leads to 2-dimensional BSM-like equation with two constant volatilities. We argue that decisions on the cost and the volume of market transactions are made under agents expectations. Random perturbations of expectations impact the market transactions and through them induce stochastic behavior of the underlying price. We derive BSM-like equation driven by Brownian motion of agents expectations. Agents expectations can be based on option trading data. We show how such expectations can lead to nonlinear BSM-like equations. Further we show that the Heston stochastic volatility option pricing model can be applied to our approximations and as example derive 3-dimensional BSM-like equation that describes option pricing with stochastic cost volatility and constant volume volatility. Diversity of BSM-like equations with 2 – 5 or more dimensions emphasizes complexity of option pricing problem. Such variety states the problem of reasonable balance between the accuracy of asset and option price description and the complexity of the equations under consideration. We hope that some of BSM-like equations derived in this paper may be useful for further development of assets and option market modeling.