lecture_slides-TermStructure_Binomial_#1

Haugh g iyengar department of industrial engineering

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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 Coupon-Bearing 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 Short-Rate 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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