Jamshidian’s Trick for Calls on Coupon Bonds in the Vasicek Model
Summary
The document presents a coupon bond composed of deterministic cash flows, with each payment discounted using zero-coupon bond prices from the Vasicek short-rate model. For a European call on the coupon bond, the answer identifies the argument as Jamshidian’s trick: because each component bond price decreases with the short rate, the coupon bond price is also decreasing. A critical short-rate level therefore defines when the option finishes in the money.
At that threshold, each cash flow can be assigned an adjusted strike equal to its zero-coupon bond price. The coupon bond option payoff can then be represented as a weighted sum of options on the individual zero-coupon bonds. The explanation relies on positive cash flows and positive rate sensitivities so that monotonicity and a unique threshold hold. The response sketches the payoff decomposition but provides no numerical example or option valuation formula.
Key ideas
- In the Vasicek model, each zero-coupon bond price decreases as the short rate rises when its bond sensitivity is positive.
- A coupon bond with positive deterministic payments inherits this monotonicity.
- A short-rate threshold determines whether the coupon bond call finishes in the money.
- Setting each component bond's adjusted strike at the threshold decomposes the coupon bond call payoff into bond option payoffs.
- The argument depends on the monotonicity conditions and positive cash flows.
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# Pricing Call Option on Coupon Bond under Vasicek
# 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 $$\pi^c(t)=\sum_{i|T_i<t}\alpha_i P(t,T_i)$$.
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 by NN2 (score 4, accepted)
https://quant.stackexchange.com/a/61557
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 $$(\pi^c(T)-K)^+ = \sum_{i|T_i<t}\alpha_i \left( P(t,T_i,r)- K_i\right)^+$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.