Louis Bachelier’s paper Théorie de la Spéculation, published in 1900, was the first notable attempt to apply mathematical methods to the prediction of stock prices and the behavior of financial markets. Bachelier used a random walk model; the same model was later used by Albert Einstein to explain the phenomenon of Brownian motion. Bachelier’s approach was groundbreaking, but his original model is now widely agreed to be too simplistic. Various methods and approaches from theoretical physics have been co-opted by economists in an attempt to build more accurate models of price movements in financial markets. This paper surveys the benefits and drawbacks of some of these approaches.
Theoretical physics and financial modeling seem, at first glance, to be strange bedfellows. Theoretical physics is devoted to building models of various aspects of the natural world, in a tradition dating back to Galileo Galilei, the first Western scientist to use mathematics to describe the regular behavior of the pendulum and other recurring patterns in the world around him. Financial modeling studies and attempts to predict the behavior of financial markets, which are driven by the hopes, fears, beliefs, and decisions of numerous individual investors. There is, however, a deep connection between the two disciplines: they both attempt to model and understand the behavior of complex systems by looking for macroscopic rules, rather than trying to predict the behavior of individual components. In the case of theoretical physics, calculating the behavior and interactions of each separate atom and molecule in a gas, fluid, or solid object may be possible in principle but is not feasible in practice. Instead, physicists develop equations of motion or equations of state, which describe connections between the overall behavior of a system and its macroscopic properties such as temperature and pressure. Similarly, models of financial markets do not try to predict the motivations and decisions of individual investors; instead, they attempt to model patterns in the behavior of the most easily observed attribute of markets, their prices.
For most financial assets, it is not feasible to build an exact model that will predict the future market price of the asset with no uncertainty – indeed, if this were possible then the modeled asset would, by definition, become a risk-free asset. However, the goal of market price modeling is more realistic and more achievable than this. Its goal is to build a probabilistic or “stochastic” model of market prices, which can be used to predict the probability distribution of future prices. Stochastic market pricing models are used by cash market traders to decide when to buy or sell a given asset; by derivatives traders to determine the “fair” value of options and other financial derivatives; by risk managers to quantify the risk of loss on a portfolio of assets; and by portfolio managers to determine the construction of a portfolio that gives the optimum balance of risk and return.
The same ideas used in theoretical physics to build quantitative models of physical phenomena are often found in market price models. Sometimes this is the result of parallel but independent thinking; in other cases, a conscious effort is made to take a concept from physics, such as phase transition, and apply it to financial markets. This cross-fertilization of ideas has given rise to a discipline called “econophysics.” The econophysics paradigm is not universally accepted, and has been criticized for lacking theoretical underpinnings [Gallegati et al. (2006)]. However, there are parallel lines of development of mathematical models in both theoretical physics and in economics, from simple random walk models, through more sophisticated linear models, to highly nonlinear models.