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 .
This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the 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 .
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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
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