Numeraires and Forward Measures Explained

Switching from the money market numeraire to the forward measure removes the discount factor from a fixed income pricing calculation.

A numeraire is the asset used as the unit of account when computing prices, and choosing a different numeraire together with its associated equivalent martingale measure can turn a difficult pricing problem into a straightforward one without changing the answer. The forward measure, which uses a zero-coupon bond maturing at the payoff date as the numeraire, is the standard example candidates are expected to know because it removes the need to model the discount factor as a separate random quantity.

Any traded asset with a strictly positive price can serve as a numeraire, and the numeraire theorem states that for each choice of numeraire there exists an equivalent probability measure under which every other asset's price, divided by the numeraire, is a martingale. The standard risk neutral measure uses the money market account, the value of one unit of currency continuously reinvested at the risk-free rate, as its numeraire. Under that measure, every discounted asset price is a martingale, the property developed in Martingales and Why Pricing Theory Depends on Them.

Switching numeraires does not change the price of anything. It changes which measure makes the pricing calculation easiest, and moving between numeraires is carried out using exactly the change-of-measure machinery from Girsanov's Theorem and the Change of Measure, with the Radon-Nikodym derivative determined by the ratio of the two numeraires.

Pricing a claim that pays off at a future date T under the money market numeraire requires taking an expectation of a payoff divided by a random discount factor, since interest rates themselves are stochastic in any realistic model. This coupling between the payoff and the discount factor makes the expectation difficult to compute directly. Switching to the T-forward measure, which uses the price of a zero-coupon bond maturing at T as the numeraire, removes this coupling. The price becomes the T-maturity bond price today, multiplied by the expectation of the payoff alone under the forward measure, with no discount factor left inside the expectation.

This simplification is the entire reason forward measures are used in fixed income pricing rather than as a mathematical curiosity. Interest rate derivatives such as caps, floors and swaptions are priced far more tractably under the forward measure than under the money market measure, since it separates the randomness of the payoff from the randomness of discounting.