fixed income - Cap option on Libor

We denote discount factor $D(t),$ zero coupon bond $B(t,T),$ $E_t[X] = E[X|\mathcal{F}(t)]$ and $T$-forward measure $E_t^{T}[\ ].$

First, let me fix the Libor and Forward Libor to avoid ambiguity

Libor $L(t,T):$

$$B(t, T)\cdot \Big(1 + (T-t) L(t, T)\Big) = 1.$$ Forward Libor $F(t,T-\delta,T):$ $$\Big(1 + (T-t)F(t,T-\delta,T)\Big)B(t,T) = B(t,T-\delta)$$

Now we see the cap

$$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$ We can change into forward measure $$C(t;T,L^*) = \delta B(t,T)E^T_t\left[\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$ and $F(t,T-\delta,T)$ is $T$-forward martingale, the above formula become the standard Black-Scholes.

But if we choose

$$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right]$$ then we can transform into $$C(t;T,L^*) = (1+\delta L^*)\cdot E^{T}_{t}\left[\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T)\right)^+\right]$$ it become a bond put option expiring at time $T - \delta$ maturing at time $T.$

But $B(t,T)$ is impossible log-normal under $T$-forward measure, then we can't use Black-Scholes. So how to deal with for this case?

Answer

Note that

$$\begin{align*} &\ \dfrac{1}{D(t)}E_t\left(D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right)\\ =&\ \dfrac{1}{D(t)}E\left(D(T-\delta) E\left(\frac{D(T)}{D(T-\delta)}\delta\Big(L(T-\delta,T) - L^*\Big)^+\mid\mathcal{F}_{T-\delta}\right) \mid \mathcal{F}_t\right)\\ =&\ \dfrac{1}{D(t)}E\left(D(T-\delta) B(T-\delta, T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\mid\mathcal{F}_t\right)\\ =&\ (1+\delta L^*)\dfrac{1}{D(t)}E\left(D(T-\delta)\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T) \right)^+\mid\mathcal{F}_t\right)\tag{1}\\ =&\ (1+\delta L^*)B(t, T)E^T_t\left(\frac{D(T-\delta)}{D(T)}\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T) \right)^+\right). \end{align*}$$ Your transformation from $$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left(D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right)$$ to $$C(t;T,L^*) = (1+\delta L^*)\cdot E^{T}_{t}\left(\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T)\right)^+\right)$$ does not appear correct.

We also note that $(1)$ is indeed the value of a put bond option with maturity $T-\delta$. Based on a certain short rate model such as the Hull-White model, this value can be computed analytically.