What is the swap measure and how is it used for swaptions?

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AcadiFi · Accepted Answer

The swap measure (annuity measure) uses the swap annuity A(t) = sum of delta_i * P(t, T_i+1) as numeraire. Under this measure Q^A, the FORWARD SWAP RATE S(t) becomes a martingale.

The forward swap rate is defined as S(t) = (P(t,T_0) - P(t,T_N)) / A(t) — the fixed rate that makes a forward-starting swap have zero value today. Under Q^A, modeling S(t) directly with lognormal dynamics leads to Black's formula for swaptions:

Payer swaption price = A(0) * [S(0) * N(d1) - K * N(d2)]

where d1, d2 follow Black-76 with the swap-rate implied vol. This is exactly how traders quote swaptions — as Black vols — so the swap measure aligns theory with market practice.

Comparison:
- Forward measure: underlying is a forward PRICE; numeraire is one bond
- Swap measure: underlying is a forward RATE; numeraire is the swap annuity

At an institutional desk like Chancery Swaps Group, swaption books are managed using swap-measure Black vols for each expiry-tenor grid cell. Vol surface calibration, hedging, and risk are all done in swap-measure space.

Inconsistency warning: the LMM uses forward measures for caps; the swap market model uses swap measures for swaptions. The two can't both be exactly lognormal simultaneously — this creates the classical CAP-SWAPTION arbitrage puzzle in interest rate modeling. Correlation assumptions bridge them.

What is the swap measure and how is it used for swaptions?

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