# 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 $$T+\delta$$

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

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.