lecture_slides-TermStructure_Binomial_#1

We assume delivery takes place just after a coupon

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Unformatted text preview: on at t = 3 4 Payoff at t = 3 is max(0, 88 − Z3,. ) Underlying ZCB Matures at t = 4 0 ¨¨ 4.92 ¨ ¨ ¨ 8.73 ¨ ¨ 10.78 ¨ ¨ t=0 ¨ ¨¨ ¨¨ ¨ 0.65 ¨ ¨ ¨ 0 ¨¨ 0 ¨ ¨ ¨¨ ¨¨ 0 3.57 ¨ 0¨ ¨ ¨ t=1 t=2 t=3 e.g. 4.92 = max 88 − 83.08 , e.g. 8.73 = max 88 − 79.27 , 1 1 + .0938 1 1 + .075 1 1 ×0 + ×0 2 2 1 1 × 4.92 + × 0.65 2 2 . . Turns out it’s optimal early-exercise everywhere – not a very realistic example. 5 Financial Engineering & Risk Management Fixed Income Derivatives: Bond Forwards M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research Columbia University Our Sample 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 2 Pricing a Forward on a Coupon-Bearing Bond Delivery at t = 4 of a 2-year 10% coupon-bearing bond. We assume delivery takes place just after a coupon has been paid. In the pricing lattice we use backwards induction to compute the t = 4 ex-coupon price of the bond. 6 Let G0 be the forward price at t = 0 and let Z4 be the ex-coupon bond price at t = 4. Then risk-neutral pricing implies 0 = EQ 0 6 Z4 − G0 B4 where B4 is the value of the cash-account at t = 4. Rearranging terms and using the fact that G0 is known at date t = 0 we obtain EQ [Z 6 /B4 ] G0 = 0Q 4 . (10) E0 [1/B4 ] Recall that EQ [1/B4 ] is time t = 0 price of a ZCB maturing at t = 4 but 0 with a face value $1 – have already calculated this to be .7722. 3 Pricing a Forward on a Coupon-Bearing Bond 110 102.98 ¨ ¨ ¨¨ 110 ¨¨ ¨¨ ¨107.19¨¨ 110 91.66 ¨¨ ¨ ¨¨ ¨ ¨ ¨ 98.44¨¨110.46¨¨ 110 ¨ ¨ ¨ ¨¨ ¨¨¨ ¨¨ ¨ ¨¨110 .96 ¨ ¨ 103.83¨ 112¨¨ ¨¨ ¨ ¨¨ ¨¨ ¨¨¨ ¨ ¨114.84¨¨ 110 ¨ 108.00 ¨¨ ¨¨ ¨ ¨ ¨¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨111.16¨¨116.24¨¨¨110 ¨ ¨ ¨ ¨ ¨ ¨ t=0 t=1 t=2 t=3 t=4 t=5 t=6 6 First find ex-coupon price, Z4 , of the bond at time t = 4: e.g. 98.44 = 1 1 + .1055 1 1 × 107.19 + × 110.46 . 2 2 4 Pricing a Forward on a Coupon-Bearing Bond 91.66 ¨ ¨¨ 85.08 ¨ 98.44 ¨ ¨¨ ¨ ¨¨93.27¨¨¨ .83 81.53 103 ¨ ¨¨ ¨¨¨ ¨¨¨ ¨ 79.99 ¨ 90.45 ¨ 99.85 ¨ 108.00 ¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨¨89.24¨¨¨97.67¨¨ 104.99¨¨¨ .16 79.83 111 ¨ t=0 t=1 t=2 t=3 t=4 6 Now work backwards in lattice to compute EQ [Z4 /B4 ] = 79.83. 0 Can now use (13) to obtain G0 = 79.83 = 103.38. 0.7722 5 Financial Engineering & Risk Management Fixed Income Derivatives: Bond Futures M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research Columbia University Pricing Futures Contracts Let Fk be the date k price of a futures contract that expires after n periods. Let Sk denote the time k price of the security underlying the futures contract. Then Fn = Sn , i...
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This note was uploaded on 01/09/2014 for the course FIN 347 taught by Professor Martinhaugh during the Fall '13 term at Bingham University.

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