Murmuring about a martingale
In the field of financial mathematics, for 50 or so years we have essentially been restating and coming up with different ways to answer the same question; if I know a stock price today, can I know the stock price tomorrow?
Naturally, you would expect to hear a resounding “No!” from the audience you asked. Yet in financial mathematics, we do this all the time. In fact, most models including that of the famous Black Scholes Merton model imply it (although you’d need to run through the derivations to see where).
Mathematically, if there exists a random variable that is real (below infinity) and fluctuates over time, we call this variable a martingale. By definition, for a martingale we expect that tomorrow’s value will be equal to today’s value. Applied to the stock market, we can theoretically say that by knowing today’s stock price, we now know tomorrow’s price.
The key word here is ‘expect’, as in; we believe it’s most likely that tomorrow’s value will be equal to that of today’s. This is an issue because there is an infinite array of factors that can influence the value of a stock beyond those that are random in nature that the definition of a martingale cannot incorporate.
Without the martingale assumption however, many financial models either cannot be derived or produce strange results. Yet their application by the financial industry on the markets is a double edged sword; in times of normality, the models work reasonably well to price risk and allow commerce to function in an orderly manner, yet as left-tailed events enter the fray, chaos can ensue – just ask Long-Term Capital Management about this one…
So let’s look at two different perspectives; i) the more traditional discounted cash flow model and ii) a more mathematical expression that states that stock prices rise over time at a known rate, but are also subject to random fluctuations at each time step. (I’ll leave out the equations so as to not overcomplicate the matter, but for those interested, Google ‘Geometric Brownian motion’).
Both models are essentially trying to explain how a financial instrument can be expected to act in the future, yet both approach the issue from different angles. And both are based on assumptions (however good or poor).
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