optionmeasure

Option measure is a term used in financial mathematics to describe the probability measure under which option prices are computed in a no-arbitrage framework. More precisely, it is an equivalent martingale (or risk-neutral) measure Q defined on the same probability space as the real-world measure P, such that the discounted prices of tradable assets are martingales. Under Q, the present value of a payoff is obtained by taking the expected value of the payoff discounted at the risk-free rate.

The option measure is related to the state price density or pricing kernel, which links payoffs to today’s prices. In a complete market, there exists a unique equivalent martingale measure, and every contingent claim has a single arbitrage-free price given by the discounted expectation under Q. In incomplete markets, multiple such measures can exist, leading to a range of arbitrage-free prices; pricing may then rely on additional criteria such as replication, utility, or hedging considerations.

Practically, pricing an option with the option measure involves computing E^Q[ e^{-∫ r_t dt} X_T ], where X_T is the payoff and r_t is the short rate process. In models like Black-Scholes, the measure Q replaces the real drift with the risk-free rate, yielding closed-form prices for European options. Calibration of the option measure to observed market prices often uses liquid options to determine the market’s implied risk-neutral dynamics.

In summary, the option measure is the probabilistic framework used to price options consistently with no-arbitrage, reflecting how market prices encode the trade-off between risk and return through a chosen risk-neutral perspective.