Martingales

a Martingale M_t is a time-dependent random process with the property that, given everything we know at time s, its expected value at a future time t > s, is equal to its value at s: E [M_t | F_t] = M_s , t > s

Of course, this property depends on the stochastic process driving M --> it depends on the probability measure.

So a process is a martingale with respect to a particular probability measure.

Why are martingales useful?

Suppose that M_tis the value of a ptf of tradeable assets measured with respect to some numeraire B_{t -->}M_t= A_t / B_t

where A_t is the tradeable asset constructed from the stock price and B_t is the rolling money market account.

If M_t is a Martingale, then for a given expiry time T we have: E [M_T | F₀] = M₀

If there is one or more paths for which M_T > M₀ then there must also be one or more paths with M_T < M₀otherwise the Martingale condition could not be true. Therefore, this particular ptf does not represent an arbitrage opportunity because its value as measured by the numeraire can go up and down.

If we have found a probability measure for which all tradeable assets are Martingale with respect to the numeraire Bt.

Then a linear combination of the assets must also be Martingale, and so it is impossible to set up a portfolio of the assets that contains an arbitrage opportunity.

Our aim is to find an arbitrage-free price of a derivative contract.

To begin with, we find a measure so that the tradeable Asset A_t = S_t is Martingale with respect to the numeraire.

Let say the price of the derivative at time t is C(S_t, t). As long as we can find a function C(S_t, t) that has the correct payout at expiry T, and which is a Martingale with respect to the numeraire, then we will know we have found a price such that it is impossible to set up a ptf of the option, stock and bonds to create an arbitrage.

If this is possible, then the Martingale condition gives us a simple formula for the value of the contract now:

Martingale pricing equation