Cap price as bond options

Question

I am currently struggling with model calibration of the Hull-White (or Vasicek) model to Caps and Floors. My main problem is that I am confused about the notation.

In Brigo & Mercurio (2006, p. 76) the Cap is viewed as a portfolio of zero-bond options:

$$ Cap(t, \tau, N, X) = N \sum_{i=1}^N (1 + X \tau_i) ZBP \left(t, t_{i-1}, t_i, \frac{1}{1 + X \tau_i} \right) $$

I need to get this into a more practical view. Concrete, I want to price a 1 year Cap from today. The payment of the first Caplet is known since the reset day is today and thus the payment is known.

How could I illustrate the next 3 Caplets (=Sum is 1 Year Cap) into the zero-bond notation?

My guess would be the following:

$$ Cap(0, \delta, N, K)=N\sum_{k=1}^n \left[ P(0, t_k)\Phi(-h_k + \sigma_P^k)-(1 + K \delta_k) P(0, t_{k+1})\Phi(-h_k)\right] $$

is $P(0, t_k)$ the value of the zero-bond on the reset day of the second caplet? and $P(0, t_{k+1})$ the value of the zero-bond at the payment of the second caplet?

Answer 1

Caplets as zero-bond puts

To simplify things, consider each caplet by itself, the value of the cap would be in that case the sum of the caplets' values.

So, let's take a single caplet on nominal $N$ and with strike $K$, Libor tenor $\delta$, expiry $T$ and payment date $T +\delta$.

If your pricing date is beyond the expiry but before the payment date: $T < t < T + \delta$ then the payoff is already known, and the value of the caplet is just the value of the flow multiplied by the zero-coupon bond:

$$ Caplet(t)= NP(t, T + \delta) \underbrace{(L(T, T+\delta) - K)^+}_{\text{already known if } t > T} $$

If the pricing date is before the expiry, then the caplet can be written as a put option on the zero-coupon bond with strike $X = \frac{1}{1+ \delta K}$ (as explained here for example Cap option on Libor), leading to:

$$ Caplet(t) = \frac{N}{X} P(t, T) \mathbb{E}^T \left[ \left(X - P(T, T+ \delta) \right)^+\right] $$

To price this option, a model is needed for the zero-coupon bond price.

Caplets pricing under Hull-White model

When the short rate follows Hull-White model dynamics with mean reversion $a$, and volatility $\sigma$, the zero-coupon bond distribution is lognormal: $$ \frac{dP(t, T)}{P(t,T)} = r(t)dt + \sigma(t) B(t, T) dW(t) $$

where: $$ B(u,T) = \frac{1 - e^{-a(T- u)}}{a} $$

As a result, under Hull-White, Black's formula gives a closed-form price to the option above:

$$ Caplet(t) = N(1 + \delta K) \left[ P(t, T + \delta) \Phi(d_+) - X P(t, T) \Phi(d_-) \right] $$

where:

• $d\pm=\frac{\log\left( \frac{P(t,T+\delta)}{X P(t,T)} \right)}{\Sigma} \pm \frac{\Sigma}{2}$
• $\Sigma^2 = B(T, T+\delta)^2 \int_t^T e^{-2a(T - u)} \sigma^2(u) du $
• $\Phi$ is the cumulative distribution function of the standard gaussian $\mathcal{N}(0, 1)$

Hull-White calibration on cap volatilities

The first step is to strip caps vol to get caplet vols. See for example: http://www.smileofthales.com/financial/cap-floor-pricing-stripping-the-basics/

Let's suppose you want to calibration on caplets with expiries $T_1 < T_2 < \dots < T_n$. Usually, the model's volatility term structure is assumed piecewise-constant, with the same pillars: $T_1, \dots, T_n$.

You start with the option with the nearest expiry $T_1$, then determine the volatility $\sigma(T_1)$ that enables you to match the $T_1$ caplets price.

Then, you move on to $T_2$, the caplet price is a function of $\sigma(T_1)$ that is already known and $\sigma(T_2)$, so you determine the value of $\sigma(T_2)$ enabling you to match the $T_2$ and so on, until you get to $T_n$, and you are done.