Derivation of non-classical stochastic price dynamics equations

Abstract

We analyze the relative price change of assets starting from basic supply/demand considerations subject to arbitrary motivations. The resulting stochastic differential equation has coefficients that are functions of supply and demand. We derive these rigorously. The variance in the relative price change is then also dependent on the supply and demand, and is closely connected to the expected return. An important consequence for risk assessment and options pricing is the implication that variance is highest when the magnitude of price change is greatest, and lowest near market extrema. This occurs even if supply and demand are not dependent on price trend. The stochastic equation differs from the standard equation in mathematical finance in which the expected return and variance are decoupled. The methodology has implications for the basic framework for risk assessment, suggesting that volatility should be measured in the context of regimes of price change. The model we propose shows how investors are often misled by the apparent calm of markets near a market peak. Risk assessment methods utilizing volatility can be improved using this formulation.