Chapter 16

We take as given an interest rate model with the following \(P\)-dynamics for the short rate. $$ d r(t)=\mu(t, r(t)) d t+\sigma(t, r(t)) d \bar{W}(t) $$ Now consider a T-claim of the form \(\chi=\Phi(r(T))\) with corresponding price process \(\Pi(t)\). (a) Show that, under any martingale measure \(Q\), the price process \(\Pi(t)\) has a local rate of return equal to the short rate of interest. In other words, show that the stochastic differential of \(\Pi(t)\) is of the form $$ d \Pi(t)=r(t) \Pi(t) d t+\sigma_{\Pi} \Pi(t) d W(t) . $$ (b) Show that the normalized price process $$ Z(t)=\frac{\Pi(t)}{B(t)} $$ is a Q-martingale.

The object of this exercise is to connect the forward rates defined in Chapter 15 to the framework above. (a) Assuming that we are allowed to differentiate under the expectation sign, show that $$ f(t, T)=\frac{E_{t, r(t)}^{Q}\left[r(T) \exp \left\\{-\int_{t}^{T} r(s) d s\right\\}\right]}{{E_{l, r}^{Q}\left(0\left[\operatorname{cxp}\left\\{-\int_{t}^{T} r(s) d s\right)\right]\right.}} $$ (b) Check that indeed \(r(t)=f(t, t)\).

Consider the following version of an interest rate swap. The contract is made between two parties, \(\mathrm{A}\) and \(\mathrm{B}\), 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 \(\mathrm{R}\) (to be determined below) over the time interval \([0, T]\). \- At time \(T\) the principal will have grown to \(K_{A}\) SEK. A will then subtract the principal amount and pay the surplus \(K-K_{A}\) to \(\mathrm{B}\) (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 \(K_{B}\) SEK. B will then subtract the principal amount and pay the surplus \(K-K_{\mathrm{B}}\) to \(\mathrm{A}\) (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.