A Primer on Martingales

The Ohio State University, 1998

1. Introduction and definition of a martingale

A martingale is a stochastic process whose expected value at each step equals its previous realization/observed value. This is the most important  process in the general theory of stochastic processes. Its defining characteristic is the so-called martingale property: the best prediction for the next realization is the current value of the process.

The purpose of this primer is to introduce the reader to the concept of martingales and its significance for general and financial economists at an intuitive level while suppressing the technical and mathematical details. We will proceed along the following lines. First, we begin with the definition of the process. Then, we give numerous examples. Next, we discuss in some detail the martingale property. Finally, we will conclude with the limitations of our presentation and suggestions for further reading.

. A stochastic sequence {xn} is called a martingale (with respect to itself) if

(a)          E(xn) < µ,       and

(b)          E(xn+1êx1, x2,…,xn) = xn.

Condition (a) says that at any point in time we require that the expected value of the realization be finite. Since this is always satisfied in practice, we will not consider it further.

Condition (b) is the mathematical expression of the martingale property: knowing the past history of the process, our best prediction for one step ahead is the current observation.

In the definition, condition (b) requires an equality sign (“=”). To obtain the definition of a submartingale, replace the equality sign ( = ) in condition (b) with the sign greater than or equal to (³); if the sign in (b) is £, then the process is called a supermartingale.

In previous primers we already discussed that in the analysis of stationary processes the main mathematical tool is the covariance function, and that for Markov processes – the transition function/matrix; for martingales, the main tool is the conditional expectation.

2. Examples

The applied importance of martingales stems from the fact that they are all around us. Now, let us provide some examples of (sub)martingales.

Example 1. The simplest, and probably most familiar example is the process of symmetric random walk. A random walk is called symmetric if it does not have any trend. Namely the lack of trend in the random walk represents the martingale property.

Example 2. Consider gambling with dice. If the outcome of rolling the dice is one, the player wins 5 dollars; otherwise he loses one dollar. His expected profit is 1/6*5 – 5/6*1 = 0. Suppose that we repeat the game many times. Let us consider the process of the wealth, w, of the player. For example, suppose that his current wealth, wn, is $100. Then our best prediction for his wealth at the next step, E(wn+1), is again $100 since his expected profit is, as mentioned above, zero. The martingale property says that knowledge of the past wealth of the player is irrelevant to our prediction – weather his wealth history is {…, $85, $90, $95, $100} or {…, $110, $109, $108, $107, $106, $111, $110, … , $100}, his expected wealth at step n+1 is still $100. Therefore,

E(wn+1ê wn, wn-1, wn-2, …, ) = wn.

Example 3. Economists are familiar with the concept of a fair game. The wealth of a player in a fair game follows a martingale process. This is not a coincidence: the economists’ intuition (and definition) of fairness simply the martingale property. Our previous example was one illustration of this concept.

Example 4. If {zn} is a non-negative stochastic process, then wn = Szn is a submartingale. This example may be interpreted as follows: zn is the return of a game, always positive, and wn is the wealth of the player. Since the player never loses (his return is non-negative) this is a very beneficial game for him, and therefore, unfair for the other party. The wealth of a player in such a game follows a submartingale.

Example 5. Consider the sequence zn of independent random variables. If E(zn) = 1 for all n, then the process xn = Pzi =z1*z2*…*zn is a martingale if the sequence {zn} is independent. You are encouraged to show this.

Example 6. Usually the best prediction of the interest rate tomorrow is the interest rate today. This has been extensively researched in the empirical financial literature and generally agreed to be so. Thus we may conclude that there is a general agreement that interest rates follow a martingale process. This is not a coincidence. Martingales are intimately related to arbitrage theory: the no-arbitrage condition requires that interest rates follow a martingale process. For if we suppose that they were not martingales, then we could predict the direction of tomorrow’s interest rates and respectively buy/sell bonds today. However, our buy/sell today will drive today’s bond prices toward tomorrow’s expected prices thus restoring the martingale property of the interest rates.

Example 7. There is a general agreement that tomorrow’s value of a stock is today’s value plus the expected return of the stock for one day. Since stock returns are “on average” positive, we could say that the “on-average” stock prices are submartingale processes. Hence, the de-trended stock prices follow a martingale.

Example 8. If {zn} is a martingale and g(.) is a convex function, then the process {g(zn)} is a submartingale. This is straightforward to show by making use of Jensen’s inequality. You are encouraged to show this analytically. You are further urged to see this graphically: your challenge is to write a MATLAB program with one single line.

Example 9. The Wiener process, better known to economists as Brownian Motion(without drift) is the most important example of a martingale in continuous time. Moreover, it is the only example of a martingale in continuous time whose trajectories (realizations) are continuous. This process is explained in a different primer from this series.

Example 10. A diffusion process with a positive drift is an example of a submartingale.

3. Preservation of the “martingale property”

As the first few examples suggest, it is easiest to think about martingales in the context of gambling. We will motivate our theoretical issues with the following simple question: “Why casinos place floors and ceilings on betting?”. It turns out, that without them, there is a straightforward way to force them bankrupt. The strategy is a famous one: a gambler with a large fortune wagers $1 on, say, evens bet; if he loses, then he doubles his bet to $2; if he loses again, he doubles his bet once again, this time $4, and so on until he wins, say, at the nth step. If the gambler wins at the first bet, he wins a net of $1; if he wins at the second bet, his net gain is $1 computed as $4-$2-$1; if the player wins at the third step, his net gain is again $1 computed as $8-$1-$2-$4. Thus, this strategy allows the gambler to win a dollar. When the gambler wins, he methodically executes the strategy time and again till the casino goes bankrupt (unless he gets bankrupt in the meantime). Moreover, the gambler is pretty much assured of his success since sooner or later the dice or the roulette will turn out in his favor.

It is namely the above-described strategy that is called “the martingale” and is the origin of the word used in stochastic processes. The reader is advised to note the use of the definite/indefinite article: a martingale is a stochastic process, while the martingale is the above-described gambling strategy. Thus, if the casino plays against a banker willing to wager all the money of his bank, it is almost sure that the casino will get forced into bankruptcy.

Let us see what the legendary G. Casanova has to say in his memoirs about this strategy while he was staying in 1754 in Venice:

Playing the martingale, continually doubling my stake, I won every day during the rest of the carnival. I was fortunate enough never to lose the sixth card, and if I had lost it, I should have been without money to play, for I had 2000 sequins on that card. I congratulated myself on having increased the fortune of my dear mistress.

Since utilizing this strategy the banker guarantees himself a win, he is bound to run gambling out of business. However, using simple arithmetic, let us see what is his expected loss before he wins. His probability of winning on the first bet is ½ and his loss is zero; the probability of a win in the second round is ¼ in which case his loss is 1; the probability of a win in the third round is 1/8 and his loss is 1+2; and so forth: at the nth round the probability of win is 1/2n and the loss is (1+2+4+…+2n-2). Summing over to find the expected value of the loss gives the following:

E(Loss)= S(1/2)n(1+2+4+…+2n-2) = µ

Therefore, the banker must be prepared to lose an infinite amount of money. And so, of course, must the proprietor of the casino.

Since the math computation above is rather rigorous, let us see how Casanova’s experience fits to it. Some days later he says in his memoirs:

I still played the martingale, but with such bad luck that I was soon left without a sequin. As I shared my property with my mistress, I was obliged to tell her of my losses, and at her request sold all her diamonds, losing what I got for them; she had now only 500 sequins. There was no more talk of her escaping from the convent, for we had nothing to live on.

Unfortunately, at this moment we must leave Casanova’s memoirs and the topic of gambling. The reader interested in the life of Casanova is referred to Casanova de Seingalt, Memoirs, London, 1922, and the one interested in gambling – to the masterpiece written by Dubins and Savage How to gamble if You Must.

The point of this discussion is that even though the martingale property indicates that a prediction of a period ahead is the same as the current value, a prediction a couple of periods ahead need not be the same. This is somewhat paradoxical and disturbing given the following simple math which utilizes the law of the iterated expectations:

E(xn+kêx1,x2,…,xn) = E(xn+k-1êx1,x2,…,xn) = E(xn+k-2êx1,x2,…,xn) =…= xn.

This equation is saying that the best prediction for a martingale k periods ahead is still the current value. It is in a direct conflict with our gambling example where the process of the banker’s wealth was a martingale and nonetheless its future expected value was increasing.

The resolution to this paradox lies in the fact that while the current value is the best prediction of the value at fixed periods ahead, it need not be the best predictor of the value at random number of periods ahead. The winning in the martingale is not fixed but random: Casanova may win at the second card, the fourth card, the ninth card, or the millionth card. In other words, if the prediction is for a specific period, the martingale property holds; if the prediction is for a random moment (usually determined by some event – win, or price of stock hits, say, $120) then the martingale property need not hold.

For a random future moment, the property continues to hold, roughly stated, if the event will occur within a finite time (e.g., 5 years, 100 rolls of dice) or the event will not allow the expected value of the martingale to drift to infinity. Our computations of the expected loss of the banker clearly violated the latter condition, as is violated the former one since for any fixed period there is a strictly positive probability of its occurrence.

We may, thus, summarize our observations in the following manner:

The martingale property holds for a fixed number of periods ahead. For a random number of periods ahead, we must further require that the event occur within a finite time or that the expected value of the martingale be finite.

This theorem gives us the solution to the question that motivated our discussion at the beginning of this section. According to the theorem, casinos must either limit gambling to a finite time (say, 100 or 1000 bets) or limit the betting to some finite value. Thus, by instituting floors and ceilings on betting, casinos guarantee themselves that there doesn’t exist a “winning” strategy. With a ceiling, gamblers cannot beat consistently the casinos.

4. Limitations of the discussion & further reading

We  barely scratched the surface of the theory. Normally, the proper definition of the process requires heavy advanced-level mathematical machinery.

One may further consider whether an infinite sequence of a martingale will ultimately converge to a random variable. The best way to think about this is to ask the question of whether the process will stabilize somewhere in the distant future or not, that is, whether the distribution from which observations are drawn remains the same as the process tends in time to infinity. A related to this topic is the one of how and where the convergence happens. These are fairly advanced topics.

Another critically important area of the theory are the so called “martingale inequalities”. These are simply inequalities that apply to martingales. These inequalities impose upper and lower bounds on expectations, on maxima and minima, on number of crossings of certain points and intervals, etc. This area is not very technical, but is extremely important for applied work: many of these inequalities give specific numbers needed for purely applied work. Here a little example may help to grasp the basic ideas. Suppose the current price of a stock is $100, and its expected price within one year is $120. We can answer the simple question of, say, how many times do we expect the price to drop below $98, or to rise above $125, or to stay within the range of $102-104 during the trading year.

Martingale inequalities are an important tool in the field of Risk Management. Examples include exchange-rate risks and some exotic options. Suppose we have the following simple option: today’s price of a stock is $100, the option pays $20 in the event that within one month from today the stock price is over $120, and pays nothing in the event that that the stock price is less than or equal to $120 during the period. A martingale inequality could help find the probability of exercising our option, and therefore, its price under the risk neutrality assumption.

The reader interested in the theory of martingales is referred to Gikhman and Skorohod’s The theory of Stochastic Processes which contains a very exhaustive and authoritative treatment of the subject. A complete reference on the theory of discrete time martingales is Shiriaev’s Probability. Highly recommended are Williams’s Probability with Martingales and Grimmett and Stirzaker’s Probability and Random Processes. They are very readable and intuitive textbooks at the introductory/intermediate level. While the former is fairly small and devoted to the topic, the latter is recommended for its breadth of coverage of the theory of probability and the extensive coverage of Markov, stationary, renewal, queue, and diffusion processes in addition to martingales. This is a book that graduate students interested in the subject should have as a reference in their libraries.

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