Question on current fair value. Excel is ok.

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\fCollared Floater and Participating Cap Valuation of Collared floater: A collared floater is like a variable rate bond, but with an upper limit and a lower limit on the rate of coupon payment. Consider, for example, annual rates coupon payment and one-year spot interest rates. In the following table, the current one-year spot interest rate is denoted by r in column (1). Depending on the current spot interest rate, the collared floater with a face value of $100 will pay the holder of this floater a sum a year later, as given in column (2) of the table. The binomial tree below gives the evolution of one-year spot interest rates over a four year period. What is the current fair value of the collared floater if the risk-neutral probabilities in the binomial tree are 0.5 for every move? The time to maturity of the collared floater is 4 years. Current Rate, r (1) r r > 4% r > 7% Collared Floater with face value $100 pays at the end of the year (2) $4 100 x r% $7 9.64% 7.69% 5.52% 3.5% 7.14% 5.70% 4.09% 5.29% 4.22% 3.92% 1 97.59 = 107 1.0964 107 99.87 = 107 1.0714 107 98.18 99.11 100.05 99.94 100 = 99.99 105.29 1.0529 105.29 100.04 100.08 = 2 104 1.0392 104 98.18 = 97.59 .5 + 99.87 .5 + 7 1.0769 99.87 .5 + 100 .5 + 5.7 99.94 = 1.057 100.04 = 100 .5 + 100.08 .5 + 4.22 1.0422 99.11 = 98.18 .5 + 99.94 .5 + 5.52 1.0552 99.99 = 99.94 .5 + 100.04 .5 + 4.09 1.0409 100.05 = 99.11 .5 + 99.99 .5 + 4 1.035 Solution: Working step-by-step as above the current fair value of the collared floater is $100.05. 3 Participating Cap Valuation of Participating Cap: The participating cap pays an interest based on a nominal principal, say $100, at the end of the year if the current annual spot interest rate, r, is higher than 6%. If the current annual spot rate, r, is less than 6%, the participating cap obligates its holder to lose 40% of (6%-r%) on the nominal principal at the end of the year. The payments are given in column (2) of the following table, given the rates in column (1). The binomial tree below gives the evolution of one-year spot interest rates over a four year period. What is the current fair value of the participating cap if the risk-neutral probabilities in the binomial tree are 0.5 for every move? The time to maturity of the cap is 4 years. Current rate, r (1) r > 6% r 6% Buyer gets at end of year (2) (r% - 6%)100 -.40(6% - r%)100 9.62% 7.79% 6.27% 5.0% 7.12% 5.77% 4.64% 5.28% 4.28% 3.91% 4 3.3023 = 3.62 1.0962 3.62 1.0456 = 1.12 1.0712 1.12 .2736 = .288 1.05298 -.288 .8045 = .836 1.0391 3.6775 2.1151 .1742 .2780 -.9493 -1.1767 5 .836 3.6775 = 3.3023 .5 + 1.0456 .5 + 1.79 1.0779 1.0456 .5 .2736 .5 .092 .2780 = 1.0577 1.1767 = .2736 .5 .8045 .5 .688 1.0428 2.1151 = 3.6775 .5 + .2780 .5 + .27 1.0627 .9493 = .2780 .5 1.1767 .5 .544 1.0464 .1742 = 2.1151 .5 .9493 .5 .4 1.05 Solution: Working backwards step-by-step as shown above, we can find the current fair value of the participating cap as $0.1742. 6