Martingales in Option Pricing Explained

Let Mₜ be a process and Fₜ the information available at time t. A martingale satisfies E[Mᵤ | Fₜ] = Mₜ for every later time u, with integrability conditions

The conditioning is essential. The forecast uses everything already known, not only an unconditional long-run average

The definition always names a probability measure and information filtration. A process can be a martingale under one measure but not another

Fair game does not mean the next value equals the current value. It means positive and negative conditional changes balance in expectation

Large jumps, wide distributions, and path-dependent uncertainty are compatible with the martingale property if the required expectations exist

Nor does martingale mean independent increments. Dependence may remain as long as the conditional-mean restriction holds

A tradable asset normally has carry, dividends, or funding effects, so its undiscounted price need not be a martingale under a pricing measure

Divide total-return value by the chosen numeraire, such as a money-market account. Under suitable no-arbitrage assumptions, the discounted process becomes a martingale

Changing numeraire changes the associated measure and which relative prices are martingales. The unit of account is part of the statement

For a replicable payoff H at time T, its discounted value today equals the pricing-measure conditional expectation of the discounted payoff

This is not averaging under historical frequencies. The measure is selected to keep discounted tradable prices consistent with no arbitrage

The formula condenses a dynamic replication argument. It does not remove assumptions about trading, model states, dividends, or funding

Choose the up and down pricing weights so the discounted underlying's expected next value equals its current discounted value

Apply those same weights to the option's next-node values and discount backward. Repeating the step gives the root price

The tree's risk-neutral probability is therefore a martingale weight. It is not a claim that the actual stock has zero expected excess return

A submartingale has conditional expected future value at least as large as the present; a supermartingale has it at most as large

These concepts appear with constraints, consumption, American exercise, and superhedging, where exact equality may be replaced by a one-sided bound

The prefix does not by itself forecast a tradable profit. Measure, discounting, admissibility, and integrability still control the interpretation

It is tempting to think any rule for stopping a fair game preserves its expectation. Optional stopping theorems require boundedness, integrability, or related conditions

Doubling strategies can violate admissibility or require unlimited capital. They do not create a practical arbitrage from the martingale definition

American-option stopping problems use carefully defined exercise times and value processes, not an unrestricted promise that timing cannot matter

Observed prices include spreads, stale quotes, constraints, and discrete trading. A model martingale is a statement inside specified idealizations

Strict local martingales, bubbles, unbounded claims, and measure-existence issues show why technical conditions cannot always be skipped

Before applying a martingale formula, state the process, measure, numeraire, filtration, payoff integrability, and available trading strategies