Pricing Call Option on Coupon Bond under Vasicek

Question

Consider a the Vascicek model, and let A and B denote the functions such that $P(t,T)=\exp(A(t,T)-B(t,T)r(t))$. We now look at a coupon bond that makes deterministic payments $\alpha_1,...,\alpha_N$ at dates $T_1,...,T_N$. Clearly the price of this coupon bond is $$\pi^c(t)=\sum_{i|T_i<t}\alpha_i P(t,T_i)$$.

Assume $K$ as the strike of an expiry-T European call option on the coupon bond.

Show that there exists $r*\in\mathbb{R}$ such that $\pi^c(T)\geqslant K$ if and only if $r(T)\leqslant r*$. Define the adjusted strikes via $K_i=\exp(A(T,T_i)-B(T,T_i)r*)$ Show that the pay-off of the call can be writte as:

$$(\pi^c(T)-K)^+=\sum_{i|T_i<t}\alpha_i (P(T,T_i)-K_i)^+$$

My attempt: $\sum_{i|T_i<t}\alpha_i P(t,T_i)\geqslant K \implies\sum_{i|T_i<t}\alpha_i\exp(A(T,T_i)-B(T,T_i)r(T)) \geqslant K $

I tried to solve this equation for r(T) by taking the logratihtms but it does not work because I get $log(\sum_{i|T_i<t}\alpha_i\exp(A(T,T_i)-B(T,T_i)r(T)))$.

Question:

How should I solve this problem?

Thanks in advance!

Answer 1

It seems to me what you want to prove is the Jamshidian's trick.

We know that the function $\Bbb R \ni r \to \exp(A(t,T)-B(t,T)r)$ is monotone and if $B(t,T) \neq 0$ (If my memory is good, normally, $B(t,T)>0$) then this function gets value in $(0,+\infty)$.

Then the function $\Bbb R \ni r \to \pi^c(t,r)=\sum_{i|T_i<t}\alpha_i P(t,T_i,r)$ is also decreasing (because $B(t,T)>0$). Then for all $K \in \Bbb R^*$, there exists one and only one value $r^*$ such that $\pi^c(t,r^*) = K$. And because the function $\pi^c(t,r)$ is decresing then $\pi^c(t,r^*) \ge K$ for all $r <r^*$ (1).

From (1), it's evident that \begin{align} (\pi^c(T)-K)^+ &= (\sum_{i|T_i<t}\alpha_i P(t,T_i,r)-\sum_{i|T_i<t}\alpha_i P(t,T_i,r^*))^+ \\ &=\left( \sum_{i|T_i<t}\alpha_i \left( P(t,T_i,r)- P(t,T_i,r^*)\right) \right)^+ \\ &= \sum_{i|T_i<t}\alpha_i \left( P(t,T_i,r)- P(t,T_i,r^*)\right)^+ \tag{2} \\ \end{align}

If we denote $K_i=\exp(A(T,T_i)-B(T,T_i)r*)$, then (2) is equivalent to $$(\pi^c(T)-K)^+ = \sum_{i|T_i<t}\alpha_i \left( P(t,T_i,r)- K_i\right)^+$$