A non-linear Langevin equation was constructed in order to approximate the behavior of the S&P 500 Index intraday price movements. Price changes were assumed to be a function of supply/demand offsets that resulted from new information. Based on this observation, I constructed an equation that describe the rate of return for the market data, which depended on (1) the sensitivity of the market to supply/demand offsets, (2) the liquidity of the market, (3) the memory effects of recent returns, and (4) the memory effects of the volatility of recent returns. MATLAB was used to find appropriate coefficients for these terms in the Langevin equation. I found that the Langevin model was capable of very accurately modeling the volatility and liquidity of the market, however, it was more difficult to model memory effects. Overall, I found that data sets with larger geometric mean values had a commensurately larger alpha and beta value, which suggests that the returns experienced in the short term are due to the dependence of future market prices on past market prices. This model approximates market returns over short periods of time—approximately one day—and should not be expected to remain valid over long time periods, given that underlying market conditions that are assumed constant in this Langevin model would be expected to change.
Benoit, Mitchell, "Stochastic Modeling of US Equity Returns" (2016). Honors College. 368.