Exercise 23.3 (Swap a fixed rate vs. a short rate)

Consider the following version of an interest rate swap. The contract is made between two parties, A and B, and the payments are made as follows:

• A (hypothetically) invests the principal amount K at time 0 and lets it grow at a fixed rate of interest R (to be determined below) over the time interval [0,T].

• At time T the principal will have grown to KA SEK. A will then subtract the principal amount and pay the surplus K - KA to B (at time T).

• B (hypothetically) invests the principal at the stochastic short rate of interest over the interval [0,T].

• At time T the principal will have grown to KB SEK. B will then subtract the principal amount and pay the surplus K - KB to A (at time T).

The swap rate for this contract is now defined as the value, R, of the fixed rate which gives this contract the value zero at t=0. Your task is to compute the swap rate.