Deriving the Caplet and Bond Put Payoff Equivalence
Summary
The document explains an equivalence between a caplet paying a floating rate above a fixed strike and a put-style payoff on a zero-coupon bond. Its economic interpretation treats the caplet as the right to sell a one-period coupon bond with a predetermined coupon at par. The proceeds can be invested at the period’s known Libor rate, leaving the caplet’s excess-rate payoff after the bond obligation is repaid.
The mathematical argument rewrites the rate difference using the bond price and discounts the terminal payoff back to the reset date at the Libor rate known then. This produces the corresponding bond-option payoff. The result relies on the standard relation between the forward Libor rate and the discount bond price for the period, as well as the stated payment and valuation dates. The short answer gives an intuition and algebraic outline, but it does not discuss alternative rate conventions, day-count details, or market model assumptions.
Key ideas
- A caplet can be interpreted as an option to sell a coupon bond at par.
- The forward Libor rate is linked to the period bond price by a reciprocal relation.
- Discounting the caplet payoff at the known reset-date Libor rate yields a bond put payoff.
- The payoff equivalence depends on consistent period, strike, and payment-date conventions.
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Full text
# The NA price of a caplet with payoff
# The NA price of a caplet with payoff
Prove the following statement: The NA price of a caplet with payoff $$\delta \cdot (L(T;T,T+\delta)-k)^{+} $$ at time $T+\delta$ equals the NA price of a put option with the payoff $$(1+\delta \cdot k)\cdot( \frac{1}{1+\delta \cdot k}-p(T,T+\delta) )^{+}$$ at time T.
My idea is: I try to start using the definition of NA which is $$\sum_{I=1}^n c_I p(t,T_i)+ K p(t,T_n).$$ Also, a caplet is derivative with payoff $$Cpl(T,T+\delta ):= \delta \cdot (L(T;T,T+\delta )-k)^{+} $$ at time $T+ \delta .$ A floorlet is an interest rate derivative with payoff $$\delta \cdot(k- L(T;T,T+\delta ))^{+}$$ at time $T+\delta $
## Answer by siou0107 (score 0, accepted)
https://quant.stackexchange.com/a/53394
Intuitively, you can think of a caplet as the option to sell at time $T$ a one-period ($\delta$) coupon bond with a predetermined coupon rate $k$ for par (say nominal is 1). With the proceeds of the bond sale, you invest in the money market and at time $T + \delta$ you get $1 + \delta L \left(T, T + \delta\right)$. After repaying the principal of the bond with interest $\delta k$, you have locked in the payoff $\delta \left[L \left(T, T + \delta\right) - k \right]^+$. This is an economic justification.
For the maths justification: with the caplet, you receive at time $T + \delta$ the payoff \begin{align} \delta \left[L \left(T, T + \delta\right) - k \right]^+ &= \left[1 + \delta L \left(T, T + \delta\right) - \left(1 + \delta k\right) \right]^+\\ & = \left[\frac{1}{P \left(T, T + \delta\right)} - \left(1 + \delta k\right) \right]^+ \end{align} To get the payoff at time $T$, you discount at the relevant Libor rate $L\left(T, T + \delta\right)$, which is known at time $T$: that yields $ \left[1 - P\left(T, T + \delta\right) \left(1 + \delta k\right)\right] $ and your formula comes immediately.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.