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.
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 = 1
1 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_n
latex W_0 = 0
latex W_n
latex 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} = 1
latex W_n = -(1 + 2 + \ldots + 2^{n-1}) = -2^n +1
latex W_n
latex \mathbb{E}[W_{n+1} | W_n] = W_n
latex W_n = 1
latex P(W_{n+1} = 1 | W_n = 1) = 1
latex \mathbb{E}[W_{n+1} | W_n = 1] = 1 = W_n
latex W_n = -2^n +1
latex 2^n
latex (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_n
latex \mathbb{E} [W_{n+1} | W_n] = W_n
latex X_n
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}
latex M_n := \frac{X_n}{n+2}$ is a martingale.
[…] Introduction to Martingales […]