Martingale Process

A martingale is a sequence of random variables where the conditional expectation of the next value, given all past values, is equal to the current value. In financial terms, this represents a fair game where the expected future price of an asset is its current price.

Martingales are central to the theory of derivative pricing, as they allow for the use of risk-neutral valuation. Under a risk-neutral measure, the discounted price of a derivative is a martingale, which simplifies the process of finding its fair value.

This concept is deeply integrated into the mathematics of options and futures, providing a rigorous way to ensure that prices are consistent and arbitrage-free. For traders, it reinforces the idea that in an efficient market, you cannot profit from information that is already known.

It is a foundational concept in advanced quantitative finance and is essential for understanding how derivative prices are derived from underlying assets.