lecture_slides-TermStructure_Binomial_#1

# Lecture_slides-TermStructure_Binomial#1

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Unformatted text preview: (8) we obtain Zt = EQ t Bt t +s Zt +s Cj + B Bt +s j =t +1 j (9) Pricing using (9) ensures no-arbitrage – note that (6) is a special case of (9). 5 A Sample Short-Rate lattice 18.31% ¨ ¨¨ ¨ 13.18% 14.65% ¨¨ ¨ ¨ ¨ ¨¨ 9.49% 11.72% ¨ 10.55% ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨¨ ¨ ¨ ¨ 6.83% 9.38% ¨ 8.44% ¨ 7.59% ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨¨.75% ¨¨¨.08% ¨¨¨.47% ¨¨.92% 4 7.5% ¨ 6 6 5 ¨ ¨ ¨¨ ¨¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ 6% ¨ 5.4% ¨ 4.86% ¨ 4.37% ¨ 3.94% ¨ 3.54% ¨ ¨ ¨ ¨ ¨ t=0 t=1 t=2 t=3 t=4 t=5 The short-rate, r , grows by a factor of u = 1.25 or d = .9 in each period – not very realistic but more than suﬃcient for our purposes. 6 Pricing a ZCB that Matures at Time t=4 100 ¨ ¨¨ 89.51 ¨ 100 ¨ ¨¨ ¨ ¨ ¨¨92.22¨¨ 100 83.08 ¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨¨ 79.27 ¨ 87.35 ¨ 94.27 ¨ 100 ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨ 84.43¨¨ 90.64¨¨ 95.81¨¨ 100 77.22 ¨ t=0 t=1 e.g. 83.08 = t=2 1 1 + .0938 t=3 t=4 1 1 × 89.51 + × 92.22 . 2 2 Can compute the term-structure by pricing ZCB’s of every maturity and then backing out the spot-rates for those maturities - so s4 = 6.68% assuming per-period compounding, i.e., 77.22(1 + s4 )4 = 100. 7 Pricing a ZCB that Matures at Time t=4 100 ¨ ¨¨ 89.51 ¨ 100 ¨ ¨¨ ¨ ¨ ¨¨92.22¨¨ 100 83.08 ¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨¨ 79.27 ¨ 87.35 ¨ 94.27 ¨ 100 ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨ 84.43¨¨ 90.64¨¨ 95.81¨¨ 100 77.22 ¨ t=0 t=1 t=2 t=3 t=4 1 2 3 4 Therefore can compute compute Z0 , Z0 , Z0 and Z0 – and then compute s1 , s2 , s3 and s4 to obtain the term-structure of interest rates at time t = 0. At t = 1 we will compute new ZCB prices and obtain a new term-structure – model for the short-rate, rt , therefore deﬁnes a model for the term-structure! 8 Financial Engineering & Risk Management Fixed Income Derivatives: Options on Bonds 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 ZCB that Matures at Time t=4 100 ¨ ¨¨ 89.51 ¨ 100 ¨ ¨¨ ¨ ¨¨92.22¨¨¨ 100 83.08 ¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨¨ 79.27 ¨ 87.35 ¨ 94.27 ¨ 100 ¨ ¨¨ ¨ ¨¨ ¨¨ ¨¨84.43¨¨¨90.64¨¨¨95.81¨¨¨ 100 77.22 ¨ t=0 t=1 e.g. 83.08 = t=2 1 1 + .0938 t=3 t=4 1 1 × 89.51 + × 92.22 . 2 2 3 Pricing a European Call Option on the ZCB 0 1.56 2.97 ¨ t=0 ¨ ¨¨ ¨ ¨ ¨¨ ¨ ¨¨ 4.74 ¨ ¨ 3.35 ¨ ¨¨ ¨ ¨ t=1 e.g. 1.56 = ¨¨ 1 1 + .075 Strike = \$84 Option Expiration at t = 2 4 Option Payoﬀ = max 0, Z2,. − 84 Underlying ZCB Matures at t = 4 6.64 t=2 1 1 ×0 + × 3.35 . 2 2 4 Pricing an American Put Option on a ZCB Strike = \$88 Expirati...
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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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