## Nickeled & Dimed

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## Quantum Decision Theory Series: #4 Path Integral Formulation and Finance

By Sudip Patra

Earlier in this series we have briefly seen how quantum like modelling can be done in finance/ decision theory in general, and how such a modelling is close to quantum information theory. However, it is also the fact that there is different formulation of quantum theory in general. The beauty is that all of these formulations provide same answers for the mother nature.

One stunning and crazy formulation is what is known as the path integral formulation, first developed in details by Feynman and Gibbs in early 40s and 50s, and then highly developed since. Today, the whole of quantum field theory, or particle physics, or string theory, or quantum gravity, or quantum cosmology etc are all based on the path integral approach. Interestingly the genius, Feynman, while developing the same in his PhD thesis (under Wheeler, another Legend), humbly noted that path integral technique was just another clever method to arrive at the same results in quantum physics! Well at times geniuses are humble (contrast them with know it all of today!).

So what’s the idea? Well in classical physics it is well known for ages that equations of motions (Euler–Lagrange equations which can also be written as Newtonian equations) can be obtained by the so called extremising action principle, i.e. to say everybody (like a red ball) has a well-defined path to follow when in motion, and that path can be derived by setting the change of the action integral=0, where action integral is the integral over the so called Lagrange function. Such a principle gives rise to observed equations of motions.

Well Feynman thought of applying/extending the basic idea into a crazy formulation in quantum physics. Where any particle, like an electron, actually travels from one point in space time to another following huge number of ways all at the same time, but we only see a definite path, since most of ‘contribution’ of different paths which are at different phases cancels out. Here contribution means contribution to the probability of an electron to start from A and getting at B. We here remember that quantum mechanics is inherently the science of measuring probabilities/probability amplitudes as they are known, and hence, the path integral so measured also gives a probability amplitude of the particle to evolve from space time point A to space time point B.

This idea actually made it simpler for physicist to measure highly complicated things in advanced quantum theory and its myriads of applications.

Fast forward to finance now. Since the discovery of the fact that asset prices behaves like a geometric Brownian motion, the world of finance has changed. This discovery has not only changed the academic research, but also the trading strategies, and today there are numerous publications on the same field. The whole branch of stochastic finance is based on the same idea.

During the late half of 90s and early 2000s some mathematicians and physicists (Andrei Khrennikov, Bellal Baquiee to name a few pioneers) observed that Brownian motion in financial markets can be mathematically analyzed as path integrals. There are surprising mathematical similarities between the two theories. Hence based on the path integral formulation now asset pricing equations have been derived. Option pricing theory can be given a new shape altogether with the path integral formulation.

Physical science and finance is getting ever more closer, we still do not know the underlying reason though.

Professor Sudip Patra is an Assistant Professor of Management Practice at O.P. Jindal Global University . His research interest encompass dividend signaling theory under information asymmetry, game theory for applied corporate finance, econometric modeling, and allied areas.

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