In: Finance
Consider a 3-period interest-rate tree with the following values. i0 = 5% i1L = 4.3% i1H = 6.3% i2LL = 4.2% i2HL = 5.7% i2HH = 8.2% In this tree, find the price of
a) A 3-year, annual-pay bond with a coupon of 7.25% using the tree
b) A 3-year, annual-pay bond with a coupon of 7.25% using pathwise valuation
c) A 3-year, annual-pay bond with a coupon of 7.25% that is callable in years 1 and 2 at 101
(A) and (C) are answered in image
(A) The price of a 3-year, annual-pay bond with a coupon of 7.25% using the tree is 105.05
(C) The price of a 3-year, annual-pay bond with a coupon of 7.25% that is callable in years 1 and 2 at 101 is 103.07
(B) There are 4 possible paths for a 3-year bond and each cash flow will be discounted at each respective path-specified rate. The price of A 3-year, annual-pay bond with a coupon of 7.25% using pathwise valuation is 105.7629
Path | Year 1 | Year 2 | Year 3 | Value |
1 | 5% | 4.30% | 4.20% | 108.0909 |
2 | 5% | 4.30% | 5.70% | 106.7571 |
3 | 5% | 6.30% | 5.70% | 105.152 |
4 | 5% | 6.30% | 8.20% | 103.0516 |
Average | 105.7629 |
Note: Value of Path 1 = ((7.25/1.05)+(7.25/1.05*1.043)+(107.25/(1.05*1.043*1.042))) = 108.09
Value at path 2 = ((7.25/1.05)+(7.25/1.05*1.043)+(107.25/(1.05*1.043*1.057))) = 106.7571
Value at path 3 = ((7.25/1.05)+(7.25/1.05*1.063)+(107.25/(1.05*1.063*1.057))) = 105.152
Value at path 4 = ((7.25/1.05)+(7.25/1.05*1.063)+(107.25/(1.05*1.063*1.082))) = 103.0516