The concepts of Path Integral in Quantitative Finance

Question

I realize that path integral techniques can be applied to quantitative finance such as option value calculation. But I don't quite understand how this is done.

Is it possible to explain this to me in qualitative and quantitative terms?

Answer 1

The fundamental equation which serves as the basis for the path-integral formulation of finance and many physical problems is the Chapman-Kolmogorov equation.

$$p(X_f|X_i)=\int p(X_f|X_k)p(X_k|X_i) dX_k $$

This is analogous to the following equation for amplitudes in quantum mechanics

$$\langle X_f|X_i \rangle=\int \langle X_f|X_k\rangle\langle X_k|X_i\rangle dX_k $$

That's right, it's the same form, but the interpretation of the basic entities changes. In the former, they are probability densities and thus real and positive, in the latter they are probability amplitudes and thus complex.

The class of physical problems that can be tackled with the first type of equation are called Markov processes, their characteristic is that the state of the system depends only on its previous state. Despite its seeming limitedness, this comprises many phenomena since any process with a long but finite memory can be mapped onto a Markov process provided the state space is enlarged appropriately. On the other hand, the second equation is pretty natural and general in quantum mechanics. It is basically stating that the unity operator can always be decomposed into a, possibly overcomplete, sum of pure states

$$\mathbb{I}=\int |X_k\rangle\langle X_k| dX_k \; .$$

Now, constructing a path integral is done by slashing up the path from $X_i$ to $X_k$ into ever smaller components. Let's suppose that the endpoints are fixed, then we might assume that to go from one endpoint to another, the system has to go through paths $(X_i,X_1(t_1),X_2(t_2),\ldots,X_n(t_n),X_f)$. This leads to the following integral

$$p(X_f|X_i)=\int\cdots\int \prod_{k=0}^n p(X_{k+1}(t_{k+1})|X_k(t_k)) \prod_{k=1}^n dX_k(t_k) $$

where I put $X_0(t_0)=X_i$ and $X_{n+1}(t_{n+1})=X_f$. The tricky part is now to see if the limit can be defined meaningfully. This can be very problematic, especially in the quantum case. Ironically, the cases that are used for finance and statistical mechanics are often much more well-behaved. This is again related to one integral being over complex numbers and the other over real numbers, but it's not the only reason. Up till now, I have not been specific about the kind of system I want to study, this will play an important role as well.

So, let's take an option which is a financial security of which the price is dependent on the price of the underlying stock and time. So we can write $O(X,t)$ for the price of the option and we'll assume the underlying stock follows a geometric brownian motion:

$$\frac{dX}{X}=\mu dt + \sigma dW$$

where $W$ represents a Wiener process with increments $dW$ having mean zero and variance $dt$. Also assume that the pay-off of the option at the expiration time $T$ is:

$$O(X_T,T)=F(X_T)$$

with $F$ a given function of the terminal stock price.

Then, Fisher Black and Myron Scholes have shown that the option, under the 'no arbitrage' assumption, satisfies the following PDE

$$\frac{\partial O}{\partial t} + \frac{1}{2}\sigma^2X^2\frac{\partial^2 O}{\partial X^2} + r X \frac{\partial O}{\partial X} - rO = 0$$

in which $r$ is the risk free interest rate. If instead of the geometric brownian motion variable $X$, I reformulate this into $x=\ln X$ which is an arithmetic brownian motion variable, I can reformulate the equation as:

$$\frac{\partial O}{\partial t} + \frac{1}{2}\sigma^2\frac{\partial^2 O}{\partial x^2} + (r-\frac{\sigma^2}{2}) \frac{\partial O}{\partial x} - rO = 0$$

This is nothing else but a special case of the PDE's that can be solved by using the Feynman-Kac formula, which includes also the Fokker-Planck equation and the Smoluchowski equation, both related to the description of diffusion processes in physics. In the diffusion problem, O is to be interpreted as a distribution of velocities of the particle (Fokker-Planck) or of the positions of the particle (Smoluchowski). That's how we relate to what I introduced above. Also note that the Schrödinger equation in quantum mechanics is very similar in form, except you'll get complex coefficients.

The Feynman-Kac formula tells us that the solution to the PDE is:

$$O(X,t) = e^{-r(T-t)}\mathbb{E}\left[ F(X_T)|X(t)=X \right]$$

It is this expectation value that will now be represented as a pathintegral:

$$O(X,t) = e^{-r(T-t)}\int_{-\infty}^{+\infty}\left(\int_{x(t)=x}^{x(T)=x_T} F(e^{x_T}) e^{A_{BS}(x(t'))} \mathcal{D}x(t')\right) dx_T$$

where

$$A_{BS}(x(t'))=\int_t^{T} \frac{1}{2\sigma^2}\left(\frac{dx(t')}{dt'}-\mu\right)^2$$

is the action functional.

The reason this path integral can be built is the same explained before, here it is possible to split the conditional expectation ever further in smaller intervals:

$$\begin{array}{rcl}\mathbb{E}\left[ F(e^{x_T})|x(t)=x \right] & = & \int_{-\infty}^{+\infty} F(e^{x_T}) p(x_T|x(t)=x) dx_T \\ & = & \int_{-\infty}^{+\infty} F(e^{x_T}) \int_{\tilde{x}(t)=x}^{\tilde{x}(T)=x_T} p(x_T|\tilde{x}(\tilde{t})) p(\tilde{x}(\tilde{t})|x(t)=x) d\tilde{x}(\tilde{t}) dx_T \end{array}$$

Each of the conditional probabilities satisfying the PDE for the arithmetic brownian motion as noticed above.

I'll stop here for now, but I refer to the article by Linetsky titled The Path Integral Approach to Financial Modeling and Options Pricing for further details.

This book provides a much more in-depth reference: Physics and Finance by Volker Ziemann