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.
