Zero-Coupon Bond Volatility in the One-Factor Hull–White Model

# Zero-coupon bond price volatility with one factor Hull White interest rate model


# Zero-coupon bond price volatility with one factor Hull White interest rate model












I have been trying to understand the H&W model expression for zero coupon bond price volatilities:

$\nu_B(t_0,t_M)=-\frac{\nu_r}{m}(1-e^{-m\tau_{0,M}})$,

where $\nu_B(t_0,t_M)$ is zero coupon bond price volatility, $\nu_r$ is the short rate volatility, $m$ is the mean-reversion level (or speed?) and $\tau_{0,M}$ is the time to maturity.

I have looked in all the associated papers but found no exact match for this expression. What is the intuition and how exactly do you get this expression?

Edit: made notation clearer

## Answer by Gordon (score 5)

https://quant.stackexchange.com/a/34819

Based on this question, for the Hull-White model of the form \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is a standard Brownian motion, the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is given by \begin{align*} P(t, T) &= A(t, T) e^{-B(t, T) r_t}, \end{align*} where \begin{align*} B(t, T) = \frac{1}{a}\Big(1-e^{-a(T-t)} \Big), \end{align*} and \begin{align*} A(t, T) &= \exp\left(- \int_t^T \theta(u) B(u, T) du -\frac{\sigma^2}{2a^2}\big(B(t, T) -T+t\big)-\frac{\sigma^2}{4a}B(t, T)^2\right). \end{align*} Note that \begin{align*} \ln P(t, T) = \ln A(t, T) -B(t, T) r_t. \end{align*} Therefore, \begin{align*} d\ln P(t, T) &=\frac{\partial \ln A(t, T)}{\partial t}dt - r_t \frac{\partial B(t, T)}{\partial t} dt- B(t, T) dr_t\\ &=\frac{\partial \ln A(t, T)}{\partial t}dt - r_t \frac{\partial B(t, T)}{\partial t} dt- B(t, T) (\theta(t)-a r_t) dt - \sigma B(t, T) dW_t. \end{align*} That is, the zero-coupon bond price volatility is of the form \begin{align*} \sigma B(t, T) = \frac{\sigma}{a}\Big(1-e^{-a(T-t)} \Big). \end{align*}