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 with respect to the information filtration and the probability distribution , if
for all
for all

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
for all
and a supermartingale with
for all .
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 be a random walk where the ’s are IID with mean .
Let . Then is a martingale because:

since expectation distributes linearly

So how will a martingale betting strategy be like?
Here, we let be IID random variables with . We can imagine to represent the result of a coin-flipping game where,
– player win latex X_i = 11 if the coin comes up tails, that is,

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 latex n^{th}latex 2^{n-1}latex W_nlatex W_0 = 0latex W_nlatex W_n \in \{ 1, -2^n +1 \}latex n^{th}latex W_n = -(1 + 2 + \ldots + 2^{n-2}) + 2^{n-1}latex = -(2^{n-1} – 1) + 2^{n-1} = 1latex W_n = -(1 + 2 + \ldots + 2^{n-1}) = -2^n +1latex W_nlatex \mathbb{E}[W_{n+1} | W_n] = W_nlatex W_n = 1latex P(W_{n+1} = 1 | W_n = 1) = 1latex \mathbb{E}[W_{n+1} | W_n = 1] = 1 = W_nlatex W_n = -2^n +1latex 2^nlatex (n+1)^{th}latex W_{n+1} \in \{ 1 , -2^{n+1} + 1 \}latex P(W_{n+1} = 1) | W_n = -2^n +1) = \frac{1}{2}latex P(W_{n+1} = -2^{n+1}+1 | W_n = -2^n +1) = \frac{1}{2}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_nlatex \mathbb{E} [W_{n+1} | W_n] = W_nlatex X_nlatex 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}latex M_n := \frac{X_n}{n+2}\$ is a martingale.

Comments
pingbacks / trackbacks
Contact Us

CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

Not readable? Change text.

Start typing and press Enter to search