# Derivation of non-classical stochastic price dynamics equations

## TL;DR

This paper derives a new stochastic price dynamics equation based on supply and demand, revealing that variance depends on market regimes and challenging traditional models that decouple return and volatility, with implications for risk and options pricing.

## Contribution

It introduces a non-classical stochastic differential equation for asset prices derived from supply and demand considerations, linking variance to market regimes and providing a new framework for risk assessment.

## Key findings

- Variance peaks during large price changes
- Market calm near peaks can be misleading for risk assessment
- Traditional models decouple return and volatility, unlike this approach

## 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.