A swaption is an option to enter a swap arrangement in the future. Suppose that company B has a debt of \(\$ 10\) million financed over 6 years at a fixed rate of interest of \(8.64 \%\). Company A offers to sell company B a swaption to swap the fixed rate obligation for a floating rate obligation, with payments equal to the short rate, with the same principal and the same termination date. The swaption can be exercised at the beginning of year 2 (just after the payment for the previous year and when the short rate for the coming year is known). Assuming that the short rate process is that of Example 16.1, how much is this swaption worth?

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Example 16.1 (A simple short rate lattice) Figure 16.2 shows a short rate lattice giving the rates for 6 years. (The period length is 1 year.) The figure was constructed by using an up factor of u = 1.3 and a flat (or down) factor of d = .9. Risk-neutral probabilities for the lattice were assigned as q = .5 for up and 1-q= .5 for flat. The entire term structure of interest rates can be determined from this lattice by computing the prices of the zero-coupon bonds of various maturities. An example of such a calculation is shown in the lower part of the figure for a bond maturing at time 4. The value is computed by moving backward through the lattice in the familiar way, at each period weighting the next period's values by the risk-neutral probabilities and discounting by the one-period rate. For example, the top entry in the third column is P22(4)=(.8667+.9038)/1.1183= .7916. The value of the bond at time zero is found to be .7334 times its face value. This corresponds to a spot rate from time zero to time 4 of $4 = (1/.7334) 25-1=.0806. The other spot rates can be calculated in a similar way by constructing a lattice of the corresponding length with I's in the final column. If this is done, the resulting term structure is found to be (.0700, .0734, .0769, .0806, .0844, .0882). Note how the term structure rises smoothly in a manner that is fairly characteristic of actual term structures.