Given a rate tree together with risk-neutral probabilities, price a constant-maturity Treasury (CMT) swap.
The idea
A constant-maturity Treasury swap, or CMT swap, exchanges a fixed rate for a floating rate set at a Treasury par yield of fixed maturity, for example the two-year yield, rather than the short rate. To price it on a tree with risk-neutral probabilities, first work out the CMT rate at each node. Let N be the notional and K the fixed rate. For semiannual payments the payment at a node is Pay = N × (CMT − K) ÷ 2. Then discount backward node by node with the short rate and the risk-neutral probability p. A payment set at one date and paid six months later is discounted one more step at that node. The fixed leg can be valued from today's zero prices. Do not use forward CMT rates with today's discount factors. The expected rate and the discount factor are correlated, a convexity effect.
Formulas
Semiannual CMT swap payment to the CMT receiver
Pay = N × (CMT − K) ÷ 2
- N
- the notional
- CMT
- the constant-maturity Treasury par yield observed at the node
- K
- the fixed rate
Halve for semiannual accrual. If paid six months after the rate is set, discount it one more step at that node's own rate before rolling back.
A practice question
A constant-maturity Treasury (CMT) swap differs from a plain-vanilla swap because its floating leg:
- pays the realized average of the short rate between reset dates.Why not: That describes a compounded-overnight-style floating leg; a CMT leg references a fixed-maturity par yield.
- pays the forward rate observed today for each reset date, which fixes the leg.Why not: A CMT payment is set at the future date from the par yield then; it is not fixed at inception.
- exchanges fixed payments for payments tied to the Treasury's coupon, which is constant.Why not: A CMT leg references a market yield that varies, not a fixed coupon.
- pays a rate tied to the par yield of a fixed-maturity Treasury, such as the two-year yield, observed on each reset date, rather than the short rate. - correct
Because the floating rate is a par yield of a fixed maturity, it must be computed at each node of the tree from the bond prices there, and the payment is then discounted backward with the short-rate tree.
A harder one
On the same CMT tree, the present value of the expected CMT payments (before subtracting the fixed leg) is USD 4.4138 million for a notional of USD 100 million. The zero-coupon prices from the tree are 0.96119 for 1.0 year and 0.94241 for 1.5 years. A holder receives the CMT rate and pays 4.20% fixed on the same two payment dates (semiannual accrual). The swap's value to the holder is closest to:
- USD 0.21 millionWhy not: Discounts nothing in the fixed leg: subtracts 100 x 4.20%/2 x 2 = 4.20 undiscounted.
- USD 2.44 millionWhy not: Counts only the last fixed payment (discounted at the 1.5-year price) and omits the fixed payment at 1.0 year.
- USD 0.42 million - correct
- -USD 3.59 millionWhy not: Does not halve the fixed rate: charges the full 4.20% on each semiannual payment.
Fixed leg PV = 100 x (4.20%/2) x (0.96119 + 0.94241) = 3.9976. Value = 4.4138 - 3.9976 = 0.42 million.
The hardest tier
On the same tree, the present value of the CMT leg is 4.4138 per 100 of notional and the zero prices for 1.0 and 1.5 years are 0.96119 and 0.94241 (the six-month zero price is 0.98039). The swap's two payments are semiannual, at 1.0 and 1.5 years. The fixed rate that makes the swap worth zero at inception is closest to:
- 4.41%Why not: Values the fixed leg as an undiscounted annuity of two payments; the fixed payments must be discounted with the zero prices.
- 9.27%Why not: Forgets the factor 1/2 of semiannual accrual and so doubles the fixed rate.
- 3.06%Why not: Includes the six-month zero price in the fixed-leg annuity although no payment is made at six months.
- 4.64% - correct
Fixed rate K solves 4.4138 = 100 x K/2 x (0.96119 + 0.94241). K = 2 x 4.4138/(100 x 1.90360) = 4.64%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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