Solving Path Integral Problem in Quantitative Finance using Computer

Question

I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again.

We know that for option value calculation, path integral is one way to solve it. But the solution I get from the Black-Scholes formula (derived from the above question):

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

is very cryptic and simply unusable on a computer.

My question is, how can we program this solution? Or more generally, how can we devise computer algorithms to solve path integral problem in quantitative finance?

Answer 1

There are many numerical approaches to solving stochastic integrals such as the above. Assuming that there is no closed form slight-of-hand, the easiest approach is the Monte Carlo approach. I would recommend referring to Glasserman's excellent "Monte Carlo Methods in Financial Engineering"

If you are not familiar with MC, think of it as evaluating millions of possible paths in N dimensional space (the space of your random variable x time) and computing the expectation from a probability weighted average.

Making MC work for you involves:

• modeling your distribution accurately
• being able to randomly sample your distribution over the simulation in such as way as to have uniformly sampled on its cumulative probability function
• having a good random N dimensional number generator with period > total # of samples
• various tricks to reduce the required sample space

Answer 2

You can use Monte Carlo methods to generate paths.

Answer 3

It seems to me that you are making the problem more complicated than it is in fact. What is the process $X_t$ and what is the motivation to find this expectation as a path integral? If you would like to find the value of integral on the trajectory of the diffusion process I think it is undefined.

Answer 4

I am not sure why you need to use ESKC relation to make an intermediary point appear since your payoff is on the final point. It renders the whole thing more complex than it needs to be.

Usually path integrals representations are then implemented as a MC simulation, using that the path integral action term gives a representation of small time transition, or if you can solve the path integral on paper, well then you get a more-or-less closed form for your transition then you work from that point forward.