At the career fair a few weeks ago I went to the stand of a quantitative hedge fund. They had a brochure which discussed their firm, but also an interesting math problem which they would reward the best answer with a $5,000 prize. The math problem was essentially as follows:

A gold trader is traveling down a river and will stop at 10 towns. At the last town he must sell all of his gold for $1 per unit. At each town he can buy or sell as much gold as he wants. Farther away from the final stop, the gold price will be lower than $1, but the closer to the final town, the gold price will converge to $1. The gold prices are nonetheless random variables and also depend on amount purchased. He starts with $1000

The problem is then to “devise and justify a trading strategy” to maximize expected profit.

My first idea for a trading strategy is very simple. Basically, I would just buy all the gold in the first town, since it has the largest probability of being the lowest price, and then I would hold it all until the final town where I would sell it all.

This obviously is not very comprehensive. It would make more sense to assume that certain functions model the random variable of prices and then make decisions based on each price that is given. For instance, if I have a continuous random variable from 0 to 1 with all probabilities equal for the first town, from .1 to 1 for the 2^{nd} town, from .3 to 1 for the 3d town, etc., then if the price in the first town falls at 80 cents, you would wait until the second town to purchase gold, etc.

They suggested a probability function as: Log p(t) = (1-t/10) + (log p(t-1) + d(t) + c[x(t)]), 2<=t<=10, where t is the town they are at, x(t) is the amount of gold purchased and p(1)=.95, and d(t) is normally distributed with mean zero and standard deviation 10% and c is a small nonnegative parameter.

I would think though that the most efficient strategy would be something rather simple.

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Woah. You know why I get behind on all the comment on this blog? It’s because you guys are always trading these tricky finance problems and then I want to think about them and then I don’t get my own research done. 🙂

By:

Prof. Kateon October 22, 2008at 3:34 pm

I agree… The most efficient strategy does seem as though it would be far more simple. Also, it seems like it may be far more effort than it’s worth. Sometimes in the finance world, people spend so much time concerned with improving a single investment, that they loose sight of the bigger picture. In the time that it took someone to derive the best formula for spending their money, they could have made several investments of a smaller size. Not only could this improve your cash flow, but it gives you a better strategy probability wise.

They always say that it is better to not put all of your money into one venture…

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