# Relationship between the Black-Scholes model and path integrals

This question was inspired by some interesting comments by Rod Vance on this answer.

Could you (Rod), or someone else, expand on these comments and give a brief summary of the connection between the famous model developed by Black, Scholes and Merton and the path integral formulation of QM/QFT? (related question: Relation between Black-Scholes equation and quantum mechanics). An answer should outline how interesting quantities can be calculated using path integrals (e.g. is there a diagrammatic technique analogous to Feynman diagrams?).

Note that I am not familiar with the BSM model but am quite comfortable with path integrals. I'm very curious to see how they can be applied in economics.

As an example of the application of path integrals to compute quantities of interest in finance, I will review, Path integral approach to Asian options in the Black-Scholes model by Devreese, et al.

An Asian option is an example of an exotic option, wherein the payoff is determined by an underlying price of the option, averaged over a pre-determined period of time. This is in constrast to for example, vanilla options like a European put or call option.

In the general case, we are interested in the log-return, $x_t = \log S_t / S_0$ where $S$ is the underlying asset. The path integral gives a propagator interpreted as a distribution of $x_t$, and is given by,

$$\mathcal K (x_T, T|0,0) = \int \mathcal D x \, \exp \left(-\int_0^T \mathcal L[x(t)]\, dt \right)$$

where we define a 'Lagrangian' from the Black-Scholes model,

$$\mathcal L = \frac{1}{2\sigma^2}\left[\dot{x}^2 - \left(\mu - \frac12 \sigma^2\right) \right]^2$$

where $\mu$ is the drift and $\sigma$ the volatility. Of interest in the paper is the case where the Asian option performs a geometric average to determine the payoff. When $\bar{S}_T$ is the geometric average, then $\bar{x}_T$ is the continuous average of $x(t)$. The path integral is then a weighted sum of paths, but split into subsets, according to $\bar{x}_T$, from which we have,

$$\mathcal K (x_T, T|0,0|\bar{x}_T) = \int \mathcal D x \, \delta \left( \bar{x}_T - \frac{1}{T}\int_0^T x(t) \, dt\right)\exp \left(-\int_0^T \mathcal L[x(t)]\, dt \right).$$

Re-writting the delta function shows $\mathcal K$ for an Asian option with geometric averaging is equivalent to the path integral for a particle in a constant field. From a stochastic calculus viewpoint, $\mathcal K$ is the density function of a Gaussian process for $\{x_T, \bar{x}_T\}$.

Something we'd like to be able to compute in finance is the price of the Asian option, $V_0$. If the payoff at time $T$ is $V_T$, then,

$$V_0 = e^{-rT} \int_{-\infty}^\infty dx_T \int_{-\infty}^\infty d\bar{x}_T \, V_T\mathcal K (x_T, T|0,0|\bar{x}_T)$$

where $r$ is the discount, risk-free interest rate. You can find out more from the paper, and many models applying path integrals to finance are based on the approach in,

• R.P. Feynman, H. Kleinert, Effective classical partition functions, Phys. Rev. A $\mathbf{34}$ (1986).

In addition, you may find the last chapter of,

• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets (World Scientific Publishing, 2009)

to be illuminating. Unfortunately I have yet to come across any literature that computes quantities in finance via a diagrammatic approach related to the path integral, as in quantum field theory.