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TL;DR

A stochastic process \( \{M_t\} \) where \( E[M_t | \mathcal{F}_s] = M_s \) for all \( s \leq t \). Intuitively, the best prediction of the future value is the current value.

By Valenke Exam Prep Team·Last updated 2026-06-03

Martingale

A stochastic process \( \{M_t\} \) where \( E[M_t | \mathcal{F}_s] = M_s \) for all \( s \leq t \). Intuitively, the best prediction of the future value is the current value.

Why it matters for interviews

Martingale theory is the mathematical backbone of fair pricing. Under the risk-neutral measure, discounted asset prices are martingales. The optional stopping theorem provides powerful tools for solving gambling and trading problems.

Definition and Mathematical Foundation

A stochastic process \( \{M_t\} \) where \( E[M_t | \mathcal{F}_s] = M_s \) for all \( s \leq t \). Intuitively, the best prediction of the future value is the current value.

Application in Quantitative Finance

Martingale theory is the mathematical backbone of fair pricing. Under the risk-neutral measure, discounted asset prices are martingales. The optional stopping theorem provides powerful tools for solving gambling and trading problems.

Related Terms

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Frequently Asked Questions

What is the optional stopping theorem?▼
For a martingale \( M_t \) and a bounded stopping time \( \tau \), \( E[M_\tau] = E[M_0] \). This is used to solve gambler's ruin, barrier hitting probabilities, and optimal stopping problems.
How do martingales relate to options pricing?▼
The First Fundamental Theorem of Asset Pricing states that a market is arbitrage-free if and only if there exists an equivalent martingale measure under which discounted prices are martingales.
What is a submartingale?▼
A process where \( E[M_t | \mathcal{F}_s] \geq M_s \). It has an upward drift on average. The square of a martingale is a submartingale, which is useful in variance calculations.