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Unformatted text preview: .e., at expiration the futures price and the underlying
security price must coincide.
Can compute the futures price at t = n − 1 by recalling that anytime we
enter a futures contract, the initial value of the contract is 0.
Therefore the futures price, Fn −1 , at date t = n − 1 must satisfy (why?)
0
B n −1 = EQ−1
n Fn − Fn −1
.
Bn 2 Pricing Futures Contracts
Since Bn and Fn −1 are both known at date t = n − 1, we therefore have
Fn −1 = EQ−1 [Fn ].
n
By the same argument, we obtain
Fk = EQ [Fk +1 ] for 0 ≤ k < n .
k
Can then use the law of iterated expectations to obtain
F0 = EQ [Fn ] .
0
Since Fn = Sn we have
F0 = EQ [Sn ]
0 (11) – holds regardless of whether or not underlying security pays coupons etc. In contrast corresponding forward price, G0 , satisﬁes
G0 = EQ [Sn /Bn ]
0
EQ [1/Bn ]
0 . (12) 3 A Futures Contract on a CouponBearing Bond
Futures contract written on same coupon bond as earlier forward contract
Underlying coupon bond matures at time t = 6
91.66
Futures expiration at t = 4
¨
95.05 ¨
¨ ¨¨ 98.44 ¨¨
¨¨
¨¨101.14¨¨¨103.83
98.09
¨
¨
¨
¨¨
¨
¨¨
¨¨
100.81 ¨ 103.52 ¨ 105.91 ¨ 108.00
¨
¨
¨¨
¨
¨¨
¨¨
¨
¨
¨¨
¨
¨
103.22 ¨ 105.64 ¨ 107.75 ¨ 109.58 ¨ 111.16
¨
¨
¨
¨
t=0 t=1 t=2 t=3 t=4 Note that the forward price, 103.38, and futures price, 103.22, are close
– but not equal! 4 Financial Engineering & Risk Management
Fixed Income Derivatives: Caplets and Floorlets M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research
Columbia University Pricing a Caplet
A caplet is similar to a European call option on the interest rate, rt .
Usually settled in arrears but they may also be settled in advance.
If maturity is τ and strike is c , then payoﬀ of a caplet (settled in arrears) at
time τ is
(rτ −1 − c )+
– so the caplet is a call option on the short rate prevailing at time τ − 1,
settled at time τ .
A ﬂoorlet is the same as a caplet except the payoﬀ is (c − rτ −1 )+ .
A cap consists of a sequence of caplets all of which have the same strike.
A ﬂoor consists of a sequence of ﬂoorlets all of which have the same strike. 2 Our ShortRate lattice
18.31% ¨
¨
¨¨ .18%
14.65%
¨ 13
¨¨
¨¨
¨
¨
11.72% ¨ 10.55% ¨
9.49%
¨
¨
¨
¨
¨¨
¨¨
¨
¨
¨
¨
9.38% ¨
8.44% ¨
7.59% ¨
6.83%
¨
¨¨
¨
¨¨
¨¨
¨
¨
¨¨.75% ¨¨ 6.08% ¨¨ 5.47% ¨¨¨.92%
7.5% ¨
6
4
¨
¨
¨
¨¨
¨¨
¨
¨
¨
¨
¨¨
¨
¨
¨ 5.4%¨¨ 4.86%¨¨ 4.37%¨¨ 3.94%¨¨ 3.54%
6%
¨
t=0 t=1 t=2 t=3 t=4 t=5 3 Pricing a Caplet
.138
Expiration at t = 6
Strike = 2% ¨
¨
¨¨.099
.103
¨
¨¨ ¨¨¨
¨ .076
.080 ¨
.068
¨
¨
¨
¨
¨¨
¨
¨
¨
¨
.064 ¨
.059 ¨
.053 ¨
.045
¨
¨
¨
¨
¨¨ ¨¨¨ ¨¨¨ ¨¨¨
¨ .047
.052
.041
.035
.028
¨
¨
¨
¨
¨
¨¨
¨¨
¨¨
¨¨ ¨¨¨
¨ .038¨¨ .032¨
¨ .021¨¨ .015
.042
.026
¨
¨
t=0 t=1 t=2 t=3 t=4 t=5 Note that it is easier to record the time t = 6 cash ﬂows at their time 5 predecessor
nodes, an...
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 Fall '13
 MartinHaugh

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