Using the same setup as Example 2

(a) Compute today's value of each of the four floorlets for a \$1M 1-year forward start 1 -year \(K=4 \%\) quarterly floor on 3 -month rates.

(b) Compute today's value to the payer of the fixed rate for a \(\$ 1 \mathrm{M} 1\)-year forward start 1 -year swap with quarterly payments at the fixed rate of \(4 \%\) p.a.

(c) Using the above results, show that put-call parity holds: Cap - Floor \(=\) Swap value to fixed rate payer.

Data from Example 2

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Let us consider the 10-year bond shown in Table 2.3. For a $1,000,000 face amount, its cash flows are $20,000 coupon payments every 6 months for 10 years (20 payments), plus $1,000,000 principal payment in 10 years. Since its coupon rate of 4% is below the market yield of 5%, it is trading at a discount: its price of 0.92205419 = 92.205419% is less than 1 = 100% (par), and one needs to pay $922,054.19 to buy this bond. TABLE 2.3 Sensitivity measures for three different bonds. I m N C y p dP/dy (dP/dy)/P dP/dC dP/dy 22 4 4% 3% 1.019272 -1.9507 5 2 10 4% 4% 11 -4.4913 -1.9139 1.9272 4.758 -4.4913 4.4913 23.499 -8.0542 7.7946 71.101 10 2 20 4% 5% 0.922054 -7.4264 If market yields change from 5% to 5.10%, then Ay= 0.0010, and yields have moved by 10 bps. Using the first two terms of Taylor Series, we expect the new price to be approximately P(y = 5.10%) ~ P(y = 5%) + (-7.4264) 0.0010+71.101 x (0.0010) = 0.91466330 If we compare this to the actual price at the new yield, P(y = 5.10%) = 0.91466318, we observe that the approximation error is quite small, amounting to $1,000,000 (0.91466330 -0.91466318) = $0.12 in market value change for the $1,000,000 face value.