Martingales and Why Pricing Theory Depends on Them

A martingale's conditional expectation of the future equals today's value, and that single property is why discounted asset prices cannot drift.

A martingale is a stochastic process whose conditional expectation of its value at any future time, given all the information available today, equals its value today. Pricing theory depends on martingales because the discounted price of a traded asset must be a martingale under the risk neutral measure, or an arbitrage exists that a trading desk could exploit for a riskless profit.

Formally, a process X adapted to a filtration F is a martingale under a probability measure Q if the expectation of X at time t given F at time s equals X at time s, for every s less than t, and if X is integrable at every time. The filtration F represents the information available to an observer at each point in time, typically the full history of prices up to that moment. The defining equation says precisely that, conditional on everything known today, your best forecast of the process at any future date is its current value, with no drift up and no drift down.

This is stronger than saying the unconditional expectation is constant over time. A process can have a constant unconditional mean while still being predictable from current information, and predictability of that kind is exactly what the martingale property rules out.

A sub-martingale has conditional expectation of the future value greater than or equal to today's value, meaning the process tends to drift upward. A super-martingale drifts downward in the same conditional sense. Interviewers ask for this distinction because discounted asset prices carrying a positive risk premium are sub-martingales under the real world measure, and only become true martingales once both the measure and the discounting are changed correctly, a distinction covered fully in Risk Neutral Pricing Explained From First Principles.

The connection between no arbitrage and the martingale property runs through replication. If a derivative's payoff can be replicated exactly by trading the underlying asset and a risk-free bond, the derivative's price today must equal the cost of setting up that replicating portfolio, or a trader could buy the cheaper of the two and sell the more expensive one for a riskless profit. The fundamental theorem of asset pricing formalizes this: a market has no arbitrage if and only if there exists a measure, equivalent to the real world measure, under which every discounted traded asset price is a martingale.