Caplet Pricing and Delta Under Black's Model

Worked Solution

How to Think About It: A caplet is just a call option on a forward interest rate, so the pricing structure mirrors Black-Scholes almost exactly. The key difference from equity options is the discounting: instead of $e^{-rT}$, you use the zero-coupon bond price $P(0,T)$, which is the natural numeraire under the $T$-forward measure. The forward rate $F$ plays the role of the stock price, and because $L$ is a martingale under the $T$-forward measure, the drift drops out and you get the same $N(d_1)$/$N(d_2)$ structure as Black-Scholes. The subtlety -- and what makes this interesting -- is the delta behavior near zero rates, where lognormality breaks down economically.

Quick Sanity Checks: Before writing formulas, check limiting cases. If $F \gg K$ (deep in-the-money), the caplet should be worth roughly $N \cdot \tau \cdot P(0,T) \cdot (F - K)$ -- just the discounted intrinsic value. If $F \ll K$ (deep out-of-the-money), it should be worth nearly zero. Delta should approach 1 deep ITM and 0 deep OTM. These are identical to equity call limits, and they should be -- the math is the same until we get to the near-zero-rate regime.

Derivation:

Part (a): Closed-Form Caplet Price

Under the $T$-forward measure $\mathbb{Q}^T$, the forward rate $L$ is a martingale starting at $F = F(0, T-\tau, T)$, the current forward rate for the period $[T-\tau, T]$. Black's model assumes:

$dL = \sigma L \, dW^T$

so that $L$ is lognormal at time $T - \tau$ with:

$L(T - \tau) = F \exp\!\left(-\frac{1}{2}\sigma^2 (T - \tau) + \sigma W^T_{T-\tau}\right)$

The caplet price is:

$\text{Price} = P(0, T) \cdot \mathbb{E}^{\mathbb{Q}^T}\!\left[N \cdot \tau \cdot \max(L - K, 0)\right]$

Note: the $\tau$ factor converts the rate difference into a dollar amount (since LIBOR is a simple rate over $\tau$ years). Applying the standard lognormal call formula:

$\boxed{\text{Price} = N \cdot \tau \cdot P(0,T) \left[ F \cdot N(d_1) - K \cdot N(d_2) \right]}$

where:

$d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2(T-\tau)}{\sigma\sqrt{T-\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{T-\tau}$

Here $N(\cdot)$ is the standard normal CDF. This is Black's formula for a caplet.

Part (b): Delta with Respect to F

Differentiate the price with respect to $F$, holding $P(0,T)$, $K$, $\sigma$, $\tau$ fixed. Using the same calculation as for Black-Scholes equity delta (the $d_1$ and $d_2$ terms contribute via the chain rule, but the $N'(d_1)$ and $N'(d_2)$ terms cancel):

$\boxed{\Delta = \frac{\partial \text{Price}}{\partial F} = N \cdot \tau \cdot P(0,T) \cdot N(d_1)}$

This is bounded between 0 and $N \cdot \tau \cdot P(0,T)$, approaching $N \cdot \tau \cdot P(0,T)$ deep ITM and 0 deep OTM.

Part (c): Near-Zero Rates and Displaced Diffusion

The issue with lognormality near zero: Black's model assumes $dL = \sigma L \, dW^T$. When $F \to 0$, the absolute volatility $\sigma L$ also goes to zero, so the model effectively says the forward rate cannot go negative and barely moves when rates are low. This is the right behavior if rates must stay positive, but in practice:

• Rates can and do go negative (as seen in Europe and Japan post-2008).
• Near zero, a 1% absolute move represents an enormous percentage change, so lognormal vols become unstable and enormous.
• Hedging ratios ($N(d_1)$ terms) become unreliable because $d_1$ blows up when $F$ is small.

For equity, these issues do not arise -- stock prices are bounded below by zero and rarely get close to it in liquid markets. The equity lognormal model is well-behaved for typical parameter ranges.

The displaced-diffusion (DD) fix: replace the lognormal assumption with:

$d(L + \alpha) = \sigma_{\text{DD}} (L + \alpha) \, dW^T$

where $\alpha > 0$ is the displacement. This shifts the forward rate up by $\alpha$, so the effective "stock" in the Black formula is $F + \alpha$ and the effective strike is $K + \alpha$:

$\text{Price}_{\text{DD}} = N \cdot \tau \cdot P(0,T) \left[ (F + \alpha) N(d_1^{\text{DD}}) - (K + \alpha) N(d_2^{\text{DD}}) \right]$

with $d_1^{\text{DD}}$ and $d_2^{\text{DD}}$ using $(F+\alpha)/(K+\alpha)$ and $\sigma_{\text{DD}}$ in place of $F/K$ and $\sigma$. The displacement $\alpha$ allows rates to go as low as $-\alpha$ while preserving analytical tractability. The delta becomes:

$\Delta_{\text{DD}} = N \cdot \tau \cdot P(0,T) \cdot N(d_1^{\text{DD}})$

which is better behaved near zero and allows for non-zero sensitivity even when $F$ is very small.

Answer: The caplet price under Black's model is $N \cdot \tau \cdot P(0,T)[F \cdot N(d_1) - K \cdot N(d_2)]$ with standard log-moneyness $d_1$, $d_2$. Delta is $N \cdot \tau \cdot P(0,T) \cdot N(d_1)$. Near zero rates, lognormality is problematic because absolute vol collapses and rates cannot go negative; displaced diffusion shifts the process by $\alpha$ to restore tractability and allow negative rates.