Martingale

Explained: martingale submartingale supermartingale

The word "martingale" has long had associations with . Historically, a martingale was a strategy of doubling your stake every time you lose in order to recoup the loss. If you lose a dollar, two. If you lose again, four. Continuing with the strategy, you are almost guaranteed to finish with a dollar in total , but there is a remote chance that you will instead lose all you money. Today, "martingale" has a very different meaning. It is used in mathematics to describe a form of stochastic process that is akin to a fair .

Most people have an intuitive sense for what constitutes a fair . If you go to a and play , that is not a fair . On average the house wins money, so on average you lose money. A fair is one at which, on average, you break even.

Let be a stochastic process, and let denote a realization of at time t (see the notation conventions documentation). The process is a martingale if, at any time t, the expected value of all future values equals the current value . Formally, in the notation of conditional probabilities, this means

[1]

for all t and all .

If you are on a game of chance, the game is fair if and only if your wealth follows a martingale through playing. A martingale always has zero drift. For this reason, quantities such as equity or commodity prices are generally not modeled with martingales. Equity prices are generally assumed to drift upward with time while some commodity prices rise and fall in a seasonal manner. Nonetheless, martingales play a central role in financial engineering. Brownian motion and random walks are martingales, and martingales appear in the fundamental theorem of asset pricing.

If the second equality in [1] is replaced with , the process is a submartingale—a process with positive drift is a submartingale. If the second equality in [1] is replaced with , the process is a supermartingale—a process with negative drift, is a supermartingale. Note that neither inequality is strict, so martingales are included in both definitions.

Exercises
Here is a question to tease your intuition. At the start of this article, I gave the old definition of a martingale as a strategy of doubling your each time you lose. If a is playing a fair game and pursues such a martingale strategy, will his wealth over time follow a martingale (according to today's definition of martingale)? [solution]
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Brownian motion A simple continuous stochastic process that is widely used in physics and finance for modeling random behavior that evolves over time. expected value A parameter describing the "center of gravity" of a distribution. fundamental theorem of asset pricing A theorem that relates the existence of an equivalent martingale measure to the no-arbitrage condition and completeness of markets. mean reversion A tendency for a stochastic process to remain near, or return over time to a long-run average. option pricing theory An introductory article. random walk A discrete stochastic process whose increments form a white noise. time series and stochastic processes An introductory article.
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