# LO 11.8: Calculate the value o f a constant maturity Treasury swap, given an

LO 11.8: Calculate the value o f a constant maturity Treasury swap, given an interest rate tree and the risk-neutral probabilities.
In addition to valuing options with binomial interest rate trees, we can also value other derivatives such as swaps. The following example calculates the price of a constant maturity Treasury (CMT) swap. A CM T swap is an agreement to swap a floating rate for a Treasury rate such as the 10-year rate.
Example: CM T swap
Assume that you want to value a constant maturity Treasury (CMT) swap. The swap pays the following every six months until maturity:
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$1,000,000 2 X (y V C M T Tc m t ls a semiannually compounded yield, of a predetermined maturity, at the time of payment (y( is equivalent to 6-month spot rates). Assume there is a 76% risk-neutral probability of an increase in the 6-month spot rate and a 60% risk-neutral probability of an increase in the 1-year spot rate. Fill in the missing data in the binomial tree, and calculate the value of the swap. Figure 5: Incomplete Binomial Tree for CM T Swap Int. rate = 7.25% Swap price = ? 0.76 Int. rate = 6.75% Swap price = ? Int. rate = 7.50% Swap price = ? 0.60 Int. rate = 7.00% Swap price = ? Int. rate = 6.50% Swap price = ? 0.40 Today Six Months One Year 2018 Kaplan, Inc. Page 139 Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7 Answer: In six months, the top node and bottom node payoffs are, respectively: payoff^ = ———— x (7.25% 7.00%) =$1,250
$1,000,000 payoff^ = ————-x (6.75% 7.00%) =$1,250
$1,000,000 rr rr 2 2 ^ ^ Similarly in one year, the top, middle, and bottom payoffs are, respectively: r r$ 1, 000,000
/ r r i r A f t /
payoff2ju = ————-x (7.50% 7.00%) = $2,500 ^ ^ payoff2jM = ————-x (7.00% 7.00%) =$0
/ r r / w w w
r
r
$1,000,000 2 rr$ 1,000,000
payoff2 L = ————-x (6.50% 7.00%) = $2,500 2 The possible prices in six months are given by the expected discounted value of the 1-year payoffs under the risk-neutral probabilities, plus the 6-month payoffs ($1,250 and
$1,250). Hence, the 6-month values for the top and bottom node are, respectively: l,u Vi V 1>L ($2,500×0.6)+ ($0x0.4) 1+ 0.0725 +$1,250 = $2,697.53 ($ 0 x 0 .6 )+ (-$2,500×0.4) + 0.0675 1 -$1,250 = -$2,217.35 Today the price is$1,466.63, calculated as follows:
w ^ V Q — —————————ri'”r\~v———————- $1,466.63 ($2,697.53 x 0.76) + ($2,217.35 x 0.24) . 1+ 0.07 Page 140 2018 Kaplan, Inc. Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7 Figure 6 shows the binomial tree with all values included. Figure 6: Completed Binomial Tree for CM T Swap 0.60 Int. rate = 7.50% Swap price =$2,500.00
Int. rate = 7.00%
Swap price = $1,466.63 0.76 0.24 Int. rate = 7.25% Swap price =$2,697.53
Int. rate = 6.75%
Swap price = -$2,217.35 Int. rate = 7.00% Swap price =$0
Int. rate = 6.50%
Swap price = -\$2,500.00
0.40
Today
Six Months
One Year
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