Introduction to Martingales

This is a rather important topics for anyone interested in doing Finance.
Lets look at their definition first.
A Martingale is a random process $\{X_n: 0 \le n \le \infty \}$ with respect to the information filtration $F_n$ and the probability distribution $P$ , if
$\mathbb{E}^P [|X_n|] < \infty$ for all $n \ge 0$
$\mathbb{E}^P[X_{n+m}|F_n] = X_n$ for all $n, m \ge 0$

Martingales are used widely and one example is to model fair games, thus it has a rich history in modelling of gambling problems. If you google Martingale, you will get an image related to a Horse, because it started with Horse-betting.

Martingales. Source: NYU

We define a submartigale by replacing the above condition 2 with
$\mathbb{E}^P [X_{n+M}| F_n] \ge X_n$ for all $n, m \ge 0$
and a supermartingale with
$\mathbb{E}^P [X_{n+M}| F_n] \le X_n$ for all $n, m \ge 0$ .
Take note that a martingale is both a submartingale and a supermartingale. Submartingale in layman terms, refers to the player expecting more as time progresses, and vice versa for supermartingale.

Let us try to construct a Martingale from a Random Walk now.
Let $S_n := \sum_{i=1}^n X_i$ be a random walk where the $X_i$ ’s are IID with mean $\mu$ .
Let $M_n := S_n -n \mu$ . Then $M_n$ is a martingale because:
$\mathbb{E}_n [M_{n+m}]$
$= \mathbb{E}_n [\sum_{i=1}^{n+m} X_i - (n+m) \mu]$
$= \mathbb{E}_n [\sum_{i=1}^{n+m} X_i] - (n+m) \mu$ since expectation distributes linearly
$= \sum_{i=1}^n X_i + \mathbb{E}_n [\sum_{i=n+1}^{n+m} X_i] - (n+m) \mu$
$= \sum_{i=1}^n X_i + m \mu - (n+m) \mu = M_n$

So how will a martingale betting strategy be like?
Here, we let $X_1, X_2, \ldots$ be IID random variables with $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$ . We can imagine $X_i$ to represent the result of a coin-flipping game where,
– player win $1 if the coin comes up heads, that is,$ latex X_i = 1 $- player lose$ 1 if the coin comes up tails, that is, $X_i = -1$

Consider further now a doubling strategy where we keep doubling the bet until we eventually win. Once we win, we stop and our initial bet is $1. The first thing we note is that the size of bet on the$ latex n^{th} $play is$ latex 2^{n-1} $assuming we are still playing at time n. And we can let$ latex W_n $denote total winnings after n coin tosses, assuming$ latex W_0 = 0 $. Then$ latex W_n $is a martingale! To see this, let us prove that$ latex W_n \in \{ 1, -2^n +1 \} $for all n. Suppose we win for first time on$ latex n^{th} $bet. Then$ latex W_n = -(1 + 2 + \ldots + 2^{n-2}) + 2^{n-1}latex = -(2^{n-1} – 1) + 2^{n-1} = 1 $If we have not yet won after n bets then,$ latex W_n = -(1 + 2 + \ldots + 2^{n-1}) = -2^n +1 $Finally, to show$ latex W_n $is a martingale, we just need to show$ latex \mathbb{E}[W_{n+1} | W_n] = W_n $which can be easily prove using iterated expectations. For case 1,$ latex W_n = 1 $, then$ latex P(W_{n+1} = 1 | W_n = 1) = 1 $so$ latex \mathbb{E}[W_{n+1} | W_n = 1] = 1 = W_n $For case 2,$ latex W_n = -2^n +1 $, meaning we bet$ latex 2^n $on$ latex (n+1)^{th} $toss so$ latex W_{n+1} \in \{ 1 , -2^{n+1} + 1 \} $. Since$ latex P(W_{n+1} = 1) | W_n = -2^n +1) = \frac{1}{2} $, and$ latex P(W_{n+1} = -2^{n+1}+1 | W_n = -2^n +1) = \frac{1}{2} $, then$ latex \mathbb{E}[W_{n+1} | W_n = -2^n +1] = \frac{1}{2} \times 1 + \frac{1}{2} \times (-2^{n+1} +1) = -2^n + 1 = W_n $Thus, we showed that$ latex \mathbb{E} [W_{n+1} | W_n] = W_n $To bring what we learnt a further step, lets look at Polya's Urn briefly. Consider an urn which contains red balls and green balls. Initially there is just one green ball and one red ball in the urn. At each time step, a ball is chosen randomly from the urn: If ball is red, the its returned to the urn with an additional red ball. If ball is green, then its returned to the urn with an additional green ball. Let$ latex X_n $denote the number of red balls in the urn after n draws. Then$ latex P(X_{n+1} = k+1 | X_n = k) = \frac{k}{n + 2}latex P(X_{n+1} = k | X_n = k) = \frac{n + 2 – k}{n + 2} $We can show that$ latex M_n := \frac{X_n}{n+2}$ is a martingale.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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