Two Physics problems and the Birth of Quant Finance
April 19, 2012
The foundations of Quantitative Finance, the discipline which deals with the valuation and modeling of financial derivatives and financial assets, were laid on theories and models of theoretical and statistical physics. The twin concepts of "equilibrium" and "linearity" which underlie almost all of the theories and models of quant finance are both borrowed from physics, as is most of the mathematics. In essence, two problems in physics chiefly contributed to creating the entire edifice of quant finance.
Problem #1: Albert Einstein and the Random Walk (Diffusion) Problem
Diffusion is one of the fundamental physical processes by which material in nature moves. It is found in biology, chemistry, geology, engineering, and above all in physics. The process first originated in physics in the form of Brownian motion and was studied by none other the famous physicist Albert Einstein in 1905. Diffusion arises due to the constant thermal motion of atoms, molecules, and particles and causes material to move from areas of high concentration to low concentration. Therefore, the final outcome of diffusion would be a state of constant concentration across space. Take a bottle of deodorant or perfume and spray it heavily inside a small closed room. Eventually the smell spreads across the room and the entire room starts to smell nice. This is an example of diffusion. Take an iron rod and heat one end of it. Eventually, the other end becomes hot as well, because heat was transferred from the hot end to the cold end.
Actually, Albert Einstein had solved the Black-Scholes problem long ago. While studying Brownian motion of particles to complete his Ph.D. thesis, he realized that the random motion of the molecules at the microscopic level is ultimately responsible for the process of diffusion that occurs at the macroscopic level. Physicists had long studied the macroscopic phenomenon of diffusion and established the governing partial differential equation, PDE, (see below for more details on what is a PDE). However, it took Einstein’s genius to realize that the constant coefficient of diffusion in the governing PDE of the diffusion process is actually the same as the volatility parameter, , of the microscopic random process of the molecules.
Note: Fischer Black and Myron Scholes, while formulating the option pricing problem in early 1970s, followed more or less the same rationale and thought process as Einstein. The math and the physics was exactly the same. Only the context was different and certain broad (and sweeping) assumptions were made about the economy and the investors in relating the problem in physics to that in financial economics.
Problem #2: Richard Feynman’s "Path Integrals" and Feynman-Kac Approach
In 1948, Richard Feynman, another eminent physicist of the twentieth century made a simple and stunning discovery. He found out that the Schrodinger’s partial differential equation (another example of a PDE) in quantum mechanics could be solved by some kind of averaging over the paths. This led to the reformulation of the entire quantum mechanical theory in physics in terms of what is known as "path integrals". Mark Kac, a mathematician and a colleague of Feynman at Cornell, immediately realized that the concept of "averaging of paths" could be applied to the solution of heat equation with boundary conditions and other kinds of diffusion equations in physics. In short, a diffusive partial differential equation can be solved as an expectation, under a certain probability measure, of a function that contains a Brownian motion. This expectation approach has come to be known as the Feynman-Kac solution.
Note: The expectations approach, developed by latter day quants and still used extensively by practicing quants in the banks and the academic theorists, to solve the pricing problem for vanilla and exotic options is exactly like Feynman and Kac’s approach. In fact, even quants call it the Feynman-Kac approach. Everywhere, in finance textbooks, research papers and other technical documents, you’ll see the expectation operator, E written as and inside the brackets you’d see an option payoff. This is nothing but Feynman-Kac approach.
Reference: The Book of Greeks, Rahul Bhattacharya, CFE School Publication (2012)
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