Useful math ideas to describe to a general audience
mathematical game theory
At first glance, mathematical game theory (Nash equilibrium etc) seems to offer a powerful tool for producing optimal strategies to win games (or maximize the expected winnings of games). This is certainly true, but it must be understood in the proper context.
- Some games have correct game theoretical solutions, which, for example, can maximize expected winnings in the long run for games with a random component, but not how to win an individual trial. So, for example,
- if you are betting on a coin toss, you can expect heads to come up half the time, but that tells you absolutely nothing about the next toss
- if you are playing many games of rock paper scissors, you can expect to break even if you play randomly each of the strategies one third of the time. But this tells you nothing about how to win a single game.
- if you are playing many hands of poker, a good understanding of pot odds will help you to maximize your expected winnings.
But it is wrong to apply these strategies in different situations. If you or your opponents are playing a single trial, and you have very different values, then it makes sense to play a different strategy. (If you are going to be evicted if you can't make rent, it might make sense to play double or nothing to win rent). The fact that different players have different values is part of what creates a market.
- many games may have a theoretical solution, but one that is of no practical use. For example, game theory can tell use that there is a winning strategy for Hex for the first player, but in practice we don't know it, and both players do equally well
Black and Scholes, Pricing Stock Options
Many products in the marketplace have a price which is determined by supply and demand. There is no mathematical formula for the price of potash, instead the price is determined by a compromise between buyers and sellers.
So why is there a mathematical formula for the price of certain stock options? And why is it difficult to price other options? It's important to begin from the premise that most prices are determined by the market. But if a product is simply a package of other freely traded products, its price should be determined entirely by the underlying products. No one would dispute that if a pound of sugar cost one dollar, and a pound of flour costs 50 cents, then a pound of both should cost a dollar fifty. At least this should be true if we are talking about thousands of pounds of each.
There are plenty of derivative products for sale on the marketplace which are priced according to market forces. But other can clearly be priced by breaking them down into underlying assets. If the package is price wrongly, for example is a pound of sugar and flour costs only $1.25, arbitrage will correct the price: someone will see an opportunity to buy a lot of sugar and flour packages, and then sell them separately to make at $.25 profit on each package.
While it isn't obvious that financial derivatives can be broken down into individual components, sometimes, but not always, this can be done. The breakthrough observation made by Black and Scholes was that an American Option, could be hedged (i.e. broken up into pieces) by a clever technique which involves buying and selling fractions of the a combination of stock and bond in response to the change in price, so that at the end, the value of the option is the same as the value of the combination. This meant, at least in principle, that if an option was wrongly priced, an profit could be made based on the arbitrage procedure.
The economically sound idea behind derivates is to allow people to buy insurance against risk. If I am a farmer, and I have to pay my bills today, I might sell my corn now, for a fixed price, before the harvest, rather than later, after the harvest, when I am exposed to the risk that the price might have gone down. Or if I am a company that exports to the U.S. but pays bills in Canadian dollars, I might want to hedge against currency fluctuations which might be unfavorable for me.
Unfortunately, there are all kinds of complicated derivatives out there, which are so complicated, that it's hard for a lay person to understand how they are designed to protect against risk. Rather, they might appear to be dangerous gambles. Usually, the more complicated the derivative, the higher the price.
The mathematics of option pricing is limited: in some cases because there is no theoretical price for the options (it just can't be hedged), and in other cases because the potential hedging procedure is simply too complicated to compute.
As a mathematician, it is important for me to ask the question: is this work being used to properly price options being used to stabilize the market, or simply as technical justification for charging high prices for products that are so complicated that the potential buyers might not understand them.
What does Merton's Optimal Porfolio Theory have to say about planning for retirement
Merton won the Nobel prize for Economics for his optimal portfolio theory.
A topic that I find much more interesting, and more practically useful for people (rather than the market makers) is optimal portfolio theory.
Here we are faced with a simple setup which many people face in planning for their retirement.
Suppose you have a defined set of contributions to a retirement fund, along with an approximate idea of future spending after retirement. The goal is to invest the bulk, and the future contributions to maximize the overall value in retirement. The choices the investor faces are between a risky but higher yielding stock, and a safe but lower yield bond.
What makes this problem interesting, is that the person needs to express their risk aversion/greed with a value function in order to come up with an optimal strategy.
Before going into the math, let us discuss a typical optimal solution.
What are simple solutions:
- buy only the bond. The problem with this is that, on average, the stock pays better
- buy only the sock. The problem with this is that, while in general the stock pays better, when you are ready to retire, it could go be low.
- buy half of each. This just averages the two previous solutions. We want something better.
- rebalancing
Here the idea is to rebalance to portfolio so that, at the end of each year (or quarter, or month) half the portfolio is in the stock, and half in in the bond. This way, we are selling the stock when it's high and buying the bond, and doing the reverse when the stock is low.
This solution is better than the last two, but it still doesn't take into account the time horizon. When we are about to retire, we want to avoid the possible fluctuations of the stock. So the better solution is the simple strategy. Take 110 minus your age. Invest that percent in stocks, and the remainder in the bond. Rebalance each year. This way, if you are 30 years old you are 80 percent in stocks, but by the time you are 60 you are 50 percent in stock, and 50 percent in bonds. When you are young, you can take advantage of more of the long terms gains of the stock, but you gradually move into bonds as you get closer to retirement.