Classical Option Pricing and Some Steps Further

This paper takes the trade dataset of the value C and the volume V of executed transactions and regards relations C=pV as the only definition of the implemented price p. Any other price definitions, price models and forecasts form agents price expectations. Expectations force agents perform transactions and thus impact the price p dynamics. This paper considers the classical Black-Scholes-Merton (BSM) model for the underline asset price determined by the trade dataset and takes into account agents expectations. We show that the BSM model implicitly uses assumption that the value and the volume of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. The impact of agents expectations can further increase the dimension of the BSM model. Agents expectations may depend on the option price data and that can lead to nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations.