Stochastic Mathematics

One of the most common question people ask me is "What are you researching for your PhD?" So common, in fact, I would have to put it third behind "When are you finished?" and "How long have you been doing it now?"

So, here is a very simple explanation of what I am researching.

The financial world has contracts known as options. An option contract bestows upon the holder the right, but not the obligation, to buy or sell an asset for a given price (stipulated in the contract) on a given date in the future.

So, if the markets move favourably between now and then, the holder can exercise his option and make money. If the markets move unfavourably, the holder will not exercise the contract (as he would then lose money).

Following me so far? Good.

Now, since the other party in the contract is taking all the risk, she receives an upfront payment to write the contract. This value is known as the price of the option. The problem is, that if the option is priced wrongly, it is possible to make risk-free money from the market inaccuracy. Getting option pricing correct is important.

Options are popular because of the principle of leverage.

Leverage is very simple. Because the price of options is typically much less than that of the underlying assets, you can essentially multiply your investments. So, as an example, you could buy Microsoft stock for $150, let it grow to $200 on the 31 October, making $50 on the transaction.

Alternatively you could buy an option to buy Microsoft stock on 31 October at $150 for $10 (as an example price). You then exercise the option, buy it for $150 and immediately sell it on the market for $200. The end result is that you have made the same amount of money for a much smaller investment of capital.

In the above example, I have sacrificied a few technicalities at the altar of simplicity, but the overall concept is preserved.

Given the above, it is obvious why leveraging is so attractive to the markets. Options facilitate this leveraging.

In the financial world, there are two main types of options, European options and American options. The nomenclature has nothing to do with geography, and I am not certain why they are named such.

European options only allow the holder to exercise the option on the date of expiry stipulated in the contract. American options allow the holder to exercise early, if she wishes.

While this difference does not seem profound, mathematically the consequences are huge.

Until the 1970s, option-pricing was a black art, with no fully scientific method for performing these calculations. Then, two mathematicians solved the problem. Their approach was very novel. They removed all quantities that could not be quantified and assumed that the markets moved completely randomly. Their work was a revelation, but was not without its problems.

It relies heavily on a branch of mathematics called Stochastic Mathematics, which deals with function involving random numbers. While not particularly difficult, it is quite intimidating at first, despite being based an very simple and clever ideas.

The solution for simple European options was solved almost immediately, producing a relatively simple formula, easily programmed into the handheld computers that traders use on the exchange floor. Even the more exotic European options, which do not have an analytic solution, were straightforward to price using numerical techniques such as MonteCarlo integration.

Unfortunately, there is no analytic solution to the American option price problem. All results must be calculated numerically using a computer. Even then, it is hard to capture the early exercise features of American options in the models.

Enter your humble narrator.

My work takes a two-pronged assault on the problem. Using two completely distinct techniques (finite-difference and MonteCarlo integration), I am developing different algorithms to price American options. While the finite-difference approach has the benefit that implementing it for European and American options is almost identical (typically one line of code needs to be added for the early exercise feature), it has some major drawbacks - I will spare you the details.

Naturally, the MonteCarlo approach does not have these drawbacks, but has problems elsewhere - one of the most important being that the early exercise features can be very awkward to implement effectively.

Thankfully, I am almost finished. As I type this I await a program completion. I decided to write this entry since I have been waiting for a while. Hopefully, it will finish soon.

Here is hoping that the numbers are better than previous attempts.