Do you often struggle to make decisions? Do you spend too much time deliberating over a bunch of options, trying to decide which one is best? What if I told you that there is a mathematical strategy that can assist us with our decision making – does that sound useful? Over the summer, I was given the opportunity to research the mathematics of optimal stopping problems, a strategy that does just that! I learnt about the statistical tools used to model decision making scenarios and different mathematical methods to solve these problems.

What is an optimal stopping problem? It is the dilemma of deciding when the most ideal time is to make a particular choice such that the expected reward is maximised, i.e., what is the best time for me to do something so I can get the best out of it? The mathematics requires scenarios where there is a sequential flow of information, and the future is unknown. We have data coming in consecutively, but we don’t know what that data is going to look like.

A great example is the secretary problem. Imagine a massive line of people are outside your office waiting to be interviewed for a job. You must interview them one by one and have two options: you either reject the individual and continue the interviews, or you hire them and turn the rest away. The problem at hand is when do you stop interviewing people? When have you found the best candidate? If you hire someone, how do you know the next person won’t be better?  If you turn someone away, how will you know the next person will be better?

My research looks at these problems in financial applications. As the market is constantly changing, the data flow is more continuous opposed to the discrete (one-by-one) example described above, and my paper explores techniques to deal with such problems. Using the mathematical methods associated with optimal stopping problems, we can assist decision making in the study of real options. For instance, they can be used to analyse an investment and predict the ideal time to opt out and maximise profit.

A case study contained in my report is exercising a put option, which involves an investor locking in a price to sell a certain asset at a later date. The strategy is ideal if they believe the asset’s value will go down as they will make profit selling an over-priced asset. This led to a free boundary problem to find the optimal time to exercise the option, which can be quite tricky to solve.

I hope in the future to explore more complicated problems and allow for the presence of other parties who can also invest in the market. This could be achieved by applying game theoretic principles to model the strategic interaction between the parties.

Patrick Daley
Macquarie University

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