The path integral has many applications:

**Mathematical Finance**:

In mathematical finance one is faced with the problem of finding the price for an "option."

An *option* is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or sell a specified asset, the *underlying*, on or before a specified future date, the option's *expiration date*, at a given price, the *strike price*. For example, an option may give the buyer the right but not the obligation to buy a stock at some future date at a price set when the contract is settled.

One method of finding the price of such an option involves path integrals. The price of the underlying asset varies with time between when the contract is settled and the expiration date. The set of all possible paths of the underlying in this time interval is the space over which the path integral is evaluated. The integral over all such paths is taken to determine the average pay off the seller will make to the buyer for the settled strike
price. This average price is then *discounted*, adjusted for for interest, to arrive at the current value of the option.

**Statistical Mechanics:**

In statistical mechanics the path integral is used in more-or-less the same manner as it is used in quantum field theory. The main difference being a factor of $i$.

One has a given physical system at a given temperature $T$ with an internal energy $U(\phi)$ dependent upon the configuration $\phi$ of the system. The probability that the system is in a given configuration $\phi$ is proportional to

$e^{-U(\phi)/k_B T}$,

where $k_B$ is a constant called the Boltzmann constant. The path integral is then used to determine the average value of any quantity $A(\phi)$ of physical interest

$\left< A \right> := Z^{-1} \int D \phi A(\phi) e^{-U(\phi)/k_B T}$,

where the integral is taken over all configurations and $Z$, the *partition function*, is used to properly normalize the answer.

**Physically Correct Rendering:**

*Rendering* is a process of generating an image from a model through execution of a computer program.

The model contains various lights and surfaces. The properties of a given surface are described by a material. A material describes how light interacts with the surface. The surface may be mirrored, matte, diffuse or any other number of things. To determine the color of a given pixel in the produced image one must trace all possible paths form the lights of the model to the surface point in question. The path integral is used to implement this process through various techniques such as path tracing, photon mapping, and Metropolis light transport.

**Topological Quantum Field Theory:**

In topological quantum field theory the path integral is used in the exact same manner as it is used in quantum field theory.

Basically, anywhere one uses Monte Carlo methods one is using the path integral.