Sunday, April 13, 2008

Dynamics of Texas Hold 'em and Financial Markets

I recently started getting back into the wonderful poker game that is Texas Hold 'em. I used to play on and off, but only within the past few months have I begun to see the true beauty of the game. As a financial mathematics student, I already have an affinity for its many dynamics. My rudimentary thought is that poker is not unlike the financial markets. The inherent concepts and strategy of the game make it remarkably similar to trading in financial markets.

Making decisions in a poker game depend on several factors. Among others, this can include:

1. Current state - how much information you have at any given time about your table, the strength of your hand or current position.
2. Position - the position you are in and your current financial position.
3. Betting - the amount and timing of your bets into the pot.
4. Predictions - Your read on what other players are holding.
5. Implied odds/pot odds - the odds of your hand winning at any given time, and how much you can potentially win with respect to how much you bet.
6. Player psychology - making decisions based on how others generally react to certain stimuli, e.g. is on person likely to fold his hand if I raise, or should I slow-play this hand to maximize my potential profit?

All of this information forms an optimization problem that gives you an optimal decision rule at any given time. Based on your position at the poker table, at any given time, you have a certain amount of data. This data, combined with either your aggressive or passive stance determine your decision rules during the game. The rest of the game is more or less stochastic and depends on the cards dealt and the play of others. You then base your subsequent decisions using all of this information and this becomes an iterative process.

Does this not sound familiar? If you are trading in financial markets, your decisions are based on exactly these types of rules. Here's an analog to the rules listed above applied to investing:

1. Data - be it company financial statements, historical data and trends (technical analysis), insider knowledge, etc.
2. Position - Your current financial situation and deciding the optimal time to execute trades
3. Preferences - the type of security you want and the amount you'd like to invest
4. Market knowledge - determining what other agents in the market are doing
5. Probabilities - calculating if it is worth it to make a particular investment, and how to maximize the probability of profit
6. Market psychology - how markets react to certain stimuli, e.g., is there a herd mentality among agents?

People invest optimally by making decisions based on having this type of information. In poker, most players make decisions based on what is optimal to them at the time, and how optimal others are playing. I find it very interesting that agents at the poker table as well as in the markets are acting based on similar strategies. Of course, there is always a certain level of sub-optimal decision-making, and this is perhaps where arbitrage opportunities exist on a skill-based level. However, there is of course a certain level of stochasticity in the markets and during a poker game as well, which confirms an element of luck is involved to some extent in order to become profitable. Eventually, I would like to look more in depth into the dynamics of poker and financial markets. Interesting stuff, eh?

Friday, April 4, 2008

Pricing American-style Options Using the Least-Squares Monte Carlo Approach

Pricing American-style options has been an important issue in modern finance for a very long time. In theory, an American-style option allows the holder to exercise at any point up to its expiration. This differs from a European-style option in which the holder can only exercise at the expiration date. This basically means that an American option is harder to valuate and is also more valuable, in general. With the Black-Scholes framework, the PDE for a European option has a fixed boundary and can be solved analytically. In contrast, an American-style option has a moving boundary condition, which prevents it from being solved analytically (however, a solution may have been found recently - I'll look into this some more). Thus, one must use numerical methods to approximate values for American-style options.

The most common numerical methods for approximating American options are:
1. Finite difference schemes, and
2. Monte Carlo methods

Throughout this semester, I have looked closely at one method of approximating values for American-options using Monte Carlo methods. The technique was developed by Longstaff and Schwartz (2001) in their paper entitled "Valuing American Options by Simulation: A Simple Least Squares Approach".

To understand their approach, one must realize that the holder of an American option must decide optimally at what point to exercise. That is, at every discrete time step, he must ask himself: Should I exercise now, or continue to hold the option (in hopes of getting a better payoff in the future)? This notion is the very basis behind finding the optimal exercise strategy and consequent value of an American-option. The Least Squares Monte Carlo (LSM) approach determines an approximation to this optimal stopping strategy to value the option, and the empirical results have been very good.

The idea is simple: after randomly generating many stock price paths using Monte Carlo methods, use least squares regression to estimate the conditional payoff to the option holder from continuation. This will allow the holder to compare the payoff of his option from immediate exercise to the expected payoff (found using the regression) from continuation. Naturally, if his expected value from continuation is higher than immediate exercise, he will avoid immediate exercise and continue the life of the option. In this way, the algorithm works backward from the final time step, say N, and creates a regression with time step N-1. The optimal stopping strategy is determined, and then this process is repeated with a regression of time N-1 and N-2. This continues until the optimal stopping strategy is found all the way to time 1. One can then average over these values and the price of the American-option is determined.

The results of this approach (in comparison with those using finite difference schemes) have been very similar. We receive a result that converges to an approximate solution close to the "true" value of the option found using the finite difference schemes, however, it appears the LSM approach necessarily underestimates the value of the option very slightly. This is because we use a finite set of basis functions for the regression and we are finding an approximation to the optimal stopping rule, and not the optimal stopping rule itself.

I could go much more in depth into the math and theory behind the LSM approach and its results, but I'll spare the reader. The purpose of this post is to show that the LSM algorithm is indeed an efficient and relatively simple way to approximate the value of an American-style option. Coding the method is easier than coding finite difference schemes and results are almost as good. Using MATLAB, most trials took up to 20 or 30 minutes max using 100,000 stock price paths generated using Monte Carlo methods and 50 discrete time steps. My trials were completed on my Toshiba laptop (2 GHz Centrino, 1 GB RAM). One would most likely get a convergence result faster coding the LSM algorithm in C++ or FORTRAN.