What is the purpose of the Feynman-Kac formula in stochastic calculus, and how is it used in the context of option pricing?

The Feynman-Kac formula is an important result in stochastic calculus that connects partial differential equations (PDEs) to stochastic processes. Specifically, it provides a link between the solution of a certain class of PDEs and the expected value of functionals of a stochastic process, primarily Brownian motion. This connection has a significant impact on various areas of applied mathematics, particularly in finance, where it is used extensively to price derivative securities, such as options.

The formula is named after Richard Feynman and Mark Kac, who independently discovered its significance in physics and mathematics, respectively.

In the context of option pricing, the Feynman-Kac formula plays a crucial role in deriving the Black-Scholes equation, which is the backbone of modern option pricing theory. The formula helps to understand the relation between option prices, the underlying asset price, and the risk-free interest rate.

Let’s first state the Feynman-Kac formula. Consider a function u(t, x) that satisfies the following PDE with terminal condition:

$$\frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2 x^2 \frac{\partial^2 u}{\partial x^2} + rx\frac{\partial u}{\partial x} - ru = 0, \quad t \in [0, T), \: x \in (0, \infty)$$

Terminal condition:
u(T, x) = g(x)

where σ is the asset price volatility, r is the risk-free interest rate, and g(x) is the terminal payoff of the option.

Then, the Feynman-Kac formula provides a link between the solution u(t, x) of this PDE and the expected value of a certain function of a stochastic process, as follows:

u(t, x) = E[e^{− r(T − t)}g(X_T)∣X_t=x]

Where X_t is a geometric Brownian motion (GBM) driven by the following stochastic differential equation (SDE):

dX_t = rX_tdt + σX_tdB_t

Here, B_t is a standard Brownian motion and E[ ⋅ ] denotes the expectation operator.

To see how the Feynman-Kac formula helps in option pricing, consider a European call option, where the payoff at the maturity date T is given by:

g(x) = max (x − K, 0)

Where K is the option strike price. We seek the price of this option at an earlier time t. The Feynman-Kac formula allows us to express this price as an expectation of the discounted payoff under the risk-neutral probability measure Q. That is,

u(t, x) = E^Q[e^{− r(T − t)}max(X_T−K,0)∣X_t=x]

Transforming this expectation into an integral and applying it to the Black-Scholes formula, we can obtain the famous option pricing formula:

C(t, x) = xe^{− q(T − t)}N(d₁) − Ke^{− r(T − t)}N(d₂)

Where N( ⋅ ) is the cumulative distribution function of the standard normal distribution, d₁ and d₂ are given as:

$$\begin{aligned} d_1 &= \frac{\log\left(\frac{x}{K}\right) + \left(r - q + \frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{T-t}}, \\ d_2 &= d_1 - \sigma\sqrt{T-t}\end{aligned}$$

And q is the dividend yield of the underlying asset.

The Feynman-Kac formula thus provides the foundation for the option pricing theory, enabling the establishment of a deep connection between PDEs and stochastic processes, which further leads to a wide range of applications in quantitative finance.