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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 $

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  • $\begingroup$ What is your question actually? $\endgroup$ – siou0107 Apr 18 '20 at 21:52
  • $\begingroup$ Prove the above statement $\endgroup$ – Layan Apr 18 '20 at 21:59
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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.

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