Forward measure / T-forward measure
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Probability measure mathbbQ^T under which the T-maturity zero-coupon bond P(t,T) is the numeraire. Under mathbbQ^T, the forward rate F(t; T_1, T_2) for T_1 = T becomes a martingale, and any time-T payoff prices as V_0 = P(0,T),mathbbE^mathbbQ^T[payoff] — the discount factor pulls out of the expectation, making caplet and FRA pricing immediate.
Forward measure / T-forward measure in plain English
Pricing a future payment means answering 'what's a promise of money later worth today?' — normally you'd have to juggle both an uncertain payoff AND the uncertain interest rate used to discount it, which is a tangled double-headache. The T-forward measure is a clever accounting trick: you switch to measuring everything in units of one specific zero-coupon bond (a bond that pays exactly one dollar at time T), and suddenly the messy discounting factor slides cleanly out front of your calculation. It's like pricing every item in a store in 'gift-card dollars' instead of cash so that one annoying conversion step vanishes. Under this lens, the relevant forward interest rate behaves as a fair-bet average (a martingale), which makes pricing common rate contracts like caplets and FRAs remarkably clean.
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