Jamshidian's Decomposition for Swaptions under One-Factor Hull-White

Scenario

A European payer swaption expiring at time $T_0$ gives the holder the right to enter an interest rate swap with payment dates $T_1, T_2, \dots, T_n$ and fixed rate $R$. Under the one-factor Hull-White model, the short rate $r(t)$ is normally distributed.

Your Task

1. Express the terminal swaption payoff as $\left( 1 - \sum_{i=1}^n c_i P(T_0, T_i) \right)^+$ where $c_i = R \tau_i$ for $i < n$ and $c_n = 1 + R \tau_n$.
2. Prove Jamshidian's theorem: Because $P(T_0, T_i; r)$ is strictly monotonically decreasing in short rate $r(T_0)$, there exists a unique critical rate $r^*$ such that $\sum_{i=1}^n c_i P(T_0, T_i; r^*) = 1$.
3. Conclude that the swaption decomposes into a strip of European put options on zero-coupon bonds with strikes $K_i = P(T_0, T_i; r^*)$.
4. For a 1-year swaption on a 2-year swap with annual payments ($T_0 = 1, T_1 = 2, T_2 = 3$), fixed rate $R = 4\%$, $a = 0.10, \sigma = 0.015$, and flat zero curve $r_0 = 3\%$, solve for $r^*$ and the individual bond put strikes $K_i$.

Constraints

• One-factor Gaussian term structure model.
• No-arbitrage replication.

Examples The swaption is in-the-money if and only if each individual bond put option is simultaneously in-the-money.