Deriving the ATM Caplet Black Formula Under the Forward Measure

# Caplet volatility formula


# Caplet volatility formula












Consider an ATM caplet with maturity $T$ and delivery $T+\tau$. In the book Interest Rate Models (Brigo and Mercurio), page 81, the authors define the model caplet volatility as the unique value of $\sigma$ such that the formula $$P(0,T+\tau)\cdot \tau f_\tau(T;0)\left(2\Phi \left( \frac{\sqrt{T}\sigma}2\right)-1\right) = (P(0,T) - P(0,T+\tau))\left(2\Phi \left( \frac{\sqrt{T}\sigma}2\right)-1\right) $$ is equal to the caplet price produced by the model in question. Here $f_\tau(T;t)$ is defined (page 12, (1.20)) by

$$ f_\tau(T;t) = \frac{1}{\tau} \left( \frac{P(t,T)}{P(t,T+\tau)} -1 \right).$$

Then, they say

> The left-hand side is the market Black's formula for a $T$-expiry $(T+\tau)$-maturity at-the-money caplet.

So my question is, where is this expression coming from in the author's book? Why is the forward rate $f_\tau(T;0)$ showing up?

Note: the error was using the incorrect model, but I leave the computation here (for log-normal bond prices, not forward rates). If we consider the martingale $F_t = \frac{P(t,T+\tau)}{P(t,T)}$ in the $T$-forward measure, and notice that $1+\tau f_\tau(T;t) = F_t^{-1}$ then it is clear that an ATM caplet gives an ATM bond put, and the formula for the bond put is $$ P(t) = P(t,T) (K \Phi(-z_-) - F_t \Phi(-z_+)) $$ where $z_{\pm} = \dfrac z \sigma - \dfrac \sigma 2$ and $z = \log \frac{F_t}K$ is the log-moneyness. When the option is at the money, $z=0$, $K = F_t$ and $F_t P(t,T) = P(t, T+\tau)$, so $$ P_{\mathsf{ATM}}(0) = P(0,T+\tau) \left(2\Phi \left(\frac\sigma 2\right) - 1\right) $$ and multiplying by $F_0^{-1}$ gives the pre-factor $P(0,T)$ instead for the caplet.

## Answer by Pedro (score 0, accepted)

https://quant.stackexchange.com/a/81293

To clear any doubt, I will write down the calculation here.

To be concrete, we let $f(t) = f_\tau(T;t)$ for some interval $[T,T+\tau]$, we assume that under the $T+\tau$ forward measure we have that $df(t) = \sigma(t) f(t) dW(t)$ where $\sigma(t)$ is some deterministic function of $t$. Thus, our numeraire is the process $P(t,T+\tau)$.

> Under the Black model, the price of a $T$ maturity $T+\tau$ delivery caplet with strike $K$ is given by the Black formula $$ \mathsf{BlackCaplet}_\tau(T,K;t) = P(t,T+\tau) \tau \left( f(t) \Phi\left( z_+ \right) - K \Phi\left( z_{-}\right) \right). $$ where $z_\pm = \frac{z}{\sigma} \pm \frac{\sigma}{2}$, $z = \log \frac{f(t)}K$ is the log-moneyness and $\sigma^2 = \int_t^T \sigma(s)^2 ds$ is the term volatility.

Proof. Using Ito's formula we see that $$d \log f(t) = \sigma dW(t) - \frac 12 \sigma^2 dt$$ so that $$ \log f(T) = \log f(t) + \int_t^T \sigma(s)dW(s) - \frac 12 \int_t^T \sigma(s)^2 ds.$$ The payoff of a $T$-maturity caplet with strike $K$ at time $T$ is $\tau (f(T) - K)^+$, so we know that the fair price of a caplet at time $t$ is $$ \mathsf{BlackCaplet}_\tau(T,K;t) = \tau P(t,T+\tau) \mathbb{E}((f(T)-K)^+ \mid \mathcal F_t) $$ where the expected value is taken under the $T+\tau$ forward measure. Since, conditional on $\mathcal F_t$, $\log f(T)$ is normal, we see that we want to compute $$ \mathbb{E}((X-K)^+) \text{ where }\log X \sim N(\mu, \sigma^2) $$ where $\mu = \log f(t) - \frac 12 \int_t^T \sigma(s)^2 ds$ and $\sigma^2 = \int_0^T \sigma(t)^2 dt$. A routine calculation shows that in this case $$ \mathbb E((X-K)^+) = f(t) \Phi\left( z_+ \right) - K \Phi\left( z_{-}\right) . $$ where $z_\pm = \frac{z}{\sigma} \pm \frac{\sigma}{2}$ and $z = \log \frac{f(t)}K$ is the log-moneyness. This gives the formula.

When the caplet is at the money, we get

$$ \mathsf{BlackCapletATM}_\tau(T;t) = P(t,T+\tau) f_\tau(T;t) \tau \left( 2\Phi\left(\frac\sigma 2\right) -1 \right) $$

which is the formula in the book.