Path Integrals Series: 5. Examples and Overview

路径积分系列：5.例子和综述 苏剑林 · 2016-06-09 · Mathematics differential equation differential path integral finance

The path integral method brings a new perspective to solving certain classes of stochastic problems.

An Example: A Stock Price Model

Consider a risky asset (such as a stock), whose price at time $t$ is $S_t$, over the time interval $[0,T]$, where 0 denotes the initial time and $T$ denotes the maturity date. We treat $S_t$ as a continuous-time variable evolving over time, satisfying the following stochastic differential equation:

$$dS_t^0=rS_t^0 dt;\quad dS_t=S_t(\mu dt+\sigma dW_t).\tag{70}$$

where $\mu$ and $\sigma$ are two constants, and $W_t$ is a standard Brownian motion.

The equation for $S_t$ is a stochastic differential equation, and the general approach to solving it is via stochastic calculus. Stochastic calculus differs from ordinary calculus in that, when performing a first-order expansion, one cannot neglect the $dS_t^2$ term, because $dW_t^2=dt$. For example, let $S_t=e^{x_t}$, then $x_t=\ln S_t$

$$\begin{aligned}dx_t=&\ln(S_t+dS_t)-\ln S_t=\frac{dS_t}{S_t}-\frac{dS_t^2}{2S_t^2}\\ =&\frac{S_t(\mu dt+\sigma dW_t)}{S_t}-\frac{[S_t(\mu dt+\sigma dW_t)]^2}{2S_t^2}\\ =&\mu dt+\sigma dW_t-\frac{1}{2}\sigma^2 dW_t^2\quad(\text{remaining terms all below}dt\text{order})\\ =&\left(\mu-\frac{1}{2}\sigma^2\right) dt+\sigma dW_t\end{aligned} ,\tag{71}$$more

This converts it into the form $(48)$. Based on our earlier discussion, this is equivalent to an asymmetric random walk model, which allows us to perform numerical simulations; alternatively, based on $(12)$ we can write down the equivalent partial differential equation

$$\sigma\frac{\partial P}{\partial t}=\frac{\sigma^2}{2}\frac{\partial^2 P}{\partial x^2}+\left(\mu-\frac{1}{2}\sigma^2\right)\frac{\partial P}{\partial x},\tag{72}$$

or equivalently the path integral

$$\int_{x_a}^{x_b}\exp\left\{-\frac{1}{2\sigma}\int_{t_a}^{t_b}\left[\dot{x}-\left(\mu-\frac{1}{2}\sigma^2\right)\right]^2 dt\right\}\mathscr{D}x(t).\tag{73}$$

The path integral for this problem is quadratic, so it can be solved exactly. The answer is

$$\begin{aligned}&\exp\left[-\frac{1}{2\sigma}\frac{(x_b-x_a)^2}{t_b-t_a}-\frac{1}{2\sigma}\left(\mu-\frac{1}{2}\sigma^2\right)^2(t_b-t_a)\right.\\ &\qquad\qquad\qquad\left.+\frac{1}{\sigma}\left(\mu-\frac{1}{2}\sigma^2\right)(x_b-x_a)\right]\\ =&\exp\left\{-\frac{t_b-t_a}{2\sigma}\left[\frac{x_b-x_a}{t_b-t_a}-\left(\mu-\frac{1}{2}\sigma^2\right)\right]^2\right\} \end{aligned},\tag{74}$$

We can see that it depends only on the relative values $T=t_b-t_a$ and $\Delta x = x_b-x_a$:

$$\exp\left\{-\frac{T}{2\sigma}\left[\frac{\Delta x}{T}-\left(\mu-\frac{1}{2}\sigma^2\right)\right]^2\right\},\tag{75}$$

As emphasized before, this result is correct up to a normalization factor; after normalizing, the complete result is

$$P(\Delta x)=\frac{1}{\sqrt{2\pi \sigma T}}\exp\left\{-\frac{T}{2\sigma}\left[\frac{\Delta x}{T}-\left(\mu-\frac{1}{2}\sigma^2\right)\right]^2\right\}.\tag{76}$$

Note that this is the distribution of $x_t$, whereas what we want to analyze is the distribution of $S_t$. Through the change of variables $\Delta x=\ln S_b - \ln S_a = \ln(S_b/S_a)$, we obtain

$$P(S_b)=\frac{1}{S_b\sqrt{2\pi \sigma T}}\exp\left\{-\frac{T}{2\sigma}\left[\frac{\ln (S_b/S_a)}{T}-\left(\mu-\frac{1}{2}\sigma^2\right)\right]^2\right\}.\tag{77}$$

This is a log-normal distribution. It tells us that if the current value of the stock is $S_a$, then after a time $T$, the probability that its value becomes $S_b$ is $P(S_b)$.

Many financial problems can be described by stochastic differential equations, and stochastic differential equations can be converted into corresponding partial differential equations or path integrals. Since path integrals originate from quantum mechanics, in recent years the combination of the two has given rise to an emerging field—quantum finance, or quantum economics. But in essence, it is simply a matter of using the path integral method to transplant conclusions from quantum mechanics into the domain of finance; from a mathematical point of view, there isn't really anything substantively new. However, from a practical standpoint, it saves research time and cost, and is thus quite meaningful. A notable work in this area is Quantum Finance [10].

Summary of the Series

Through a number of illustrations and examples, this article has demonstrated the application of the path integral method to a broad class of stochastic problems.

However, the discussion here is far from complete. There remain many directions worth exploring regarding the application of path integral methods in this area:

1. For higher-order nonlinear stochastic field differential equations, find the corresponding path integrals, and construct analogues of the asymmetric random walk model for them;
2. Study the path integrals corresponding to stochastic partial differential equations;
3. Seek analogues of the asymmetric random walk model for stochastic partial differential equations.

I believe all of the above directions would be quite worthwhile to pursue.