Risk Latte - Working Miracles with the Differential Operator - II

Black-Scholes: A Linear and an Equilibrium Model

Rahul Bhattacharya
28th August 2014

Of late, while everybody has been throwing rocks at the Gaussian distribution that is used in the world of financial derivatives, very few have paused to ponder over the real reason for the failure of Black-Scholes model and all Black-Scholes type models that are derived from a diffusion-like partial differential equation in physics. The entire discipline of quantitative finance is predicated on the twin notions of "equilibrium" and "linearity". That is how it is in most of physics. And that is where the real problem lies.

Black-Scholes is essentially a heat equation and it is premised upon key concepts of "equilibrium" and "linearity". The general behavior of the solution to heat equation is such that as time passes, heat energy stops moving around, i.e. as time tends to infinity the process of heat transfer "hits diminishing returns". The first derivative of heat energy with respect to time tends to zero. This leads to the steady state equilibrium solution whereby the diffusion equation transforms into a Laplace's equation which in one dimension (and one spatial dimension suffices to understand the valuation of financial derivatives) is simply an ordinary differential equation, the solution of which is just a straight line. In other words, our world, while in equilibrium, is linear. Just input the time to maturity of an at the money call option as 50 years or 100 years in your Excel spreadsheet and see what happens to the theta. It becomes vanishingly small. This is when, on a piece of paper, the call option becomes just a straight line function of the underlying asset. But talk to any seasoned option trader and he will tell you that this is all nonsense. No matter whether it is 1 month in maturity or 100 years, a call option will always contaminate the space around it and create curvature.

Financial crises occur every other decade because markets are neither in equilibrium nor linear. Quantum mechanics is mostly linear but Quant Finance is not.

Reference:

  1. Bhattacharya, Rahul, The Book of Greeks, CFE School Publishing, Hong Kong, 2014
  2. Bhattacharya, Rahul, Quantitative Finance: The Long Road Through Physics, CFE School Publishing, Hong Kong, 2014.

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