Differential Equations and Optimization in Financial Markets
For several decades, the prevailing theory of price dynamics has been based upon the efficient market hypothesis (EMH) which asserts that the asset price is determined solely through the information on the asset. A small deviation from the fundamental (or realistic) value is rapidly corrected through a first order differential equation that does not allow for oscillations and overshootings of the fundamental value. The evidence from both world markets and the newly developing field of experimental markets has cast doubt on the EMH in some situations. The large price bubbles and market crashes in recent years have been among the most dramatic examples.
Research work continues on modeling price dynamics by deriving systems of nonlinear differential equations that incorporate the tendency of investors to be influenced by the trend as well as the fundamentals of an asset. The equations also utilize natural conservation laws involving total asset and cash supply. The resulting system can then describe both momentum and liquidity in financial markets.
These equations are being utilized in conjunction with laboratory asset markets (see summary). This work is in collaboration with Vladimira Ilieva, David Porter and Vernon Smith.