  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.
