# Pricing Call Option on Coupon Bond under Vasicek

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.

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

My attempt: $$\sum_{i|T_i

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.

Question:

How should I solve this problem?

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 $$\Bbb R \ni r \to \pi^c(t,r)=\sum_{i|T_i is also decreasing (because $$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 (1).
From (1), it's evident that \begin{align} (\pi^c(T)-K)^+ &= (\sum_{i|T_i
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