Stock Market Models

Mathematical Modeling · Axiom Academy

REAL WORLD Stock Market Models How discrete mathematical models attempt to capture the behavior of financial markets - and why they sometimes fail spectacularly The Ultimate Modeling Challenge For centuries, mathematicians, economists, and now computer scientists have tried to model one of humanity's most complex systems: the stock market. Why? Because if you can predict price movements - even slightly better than random - you can make enormous profits. These prices change thousands of times per second during trading hours The stock market is the perfect arena for discrete models . Prices exist at specific moments in time: the opening price, the closing price, prices at each tick. Between these discrete observations, we can only interpolate. This makes difference equations and discrete dynamical systems natural tools for analysis. A Brief History of Market Modeling The quest to mathematically model stock prices has a surprisingly long and fascinating history: Key Milestones in Financial Modeling The simplest discrete model for stock prices is the random walk . It suggests that price changes are essentially unpredictable - like flipping a coin. Let S_t be the stock price at time t . The random walk model says: where _t is a random "shock" at each time step. Often we assume _t follows a normal distribution with mean (the drift) and variance ^2 . In practice, we often model returns rather than prices: Define the log-return as r_t = (S_ t /S_ t-1 ) . The model becomes:

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