Pricing and hedging bonds and options in the Vasicek model

Damien LAMBERTON and Bernard LAPEYRE

July 20, 2007

pdf version of the document

Sampling the short rate

The Vasicek model is presented in Chapter [*]. The short rate $r_t$ follows the stochastic differential equation

\begin{displaymath} dr(t) = a \left(b - r(t)\right) dt + \sigma dW_t, \end{displaymath} (1)

where $a$, $b$, $\sigma$ are positive constants and $W$ is a standard Brownian motion under $\P$. In computer experiments, one can choose $a= 10 / \mbox{year}$, $r_0=b=0.05 / \mbox{year}$, $\sigma= 0.1/\sqrt{\mbox{year}}$.
  1. Show that $r_{h}$ follows a Gaussian distribution with mean $b+e^{-ah}(r_0-b)$ and variance $\sigma^2\frac{1-e^{-2at}}{2a}$
  2. What is the conditional distribution of $r_{t+h}$ given $r_t=r$?
  3. Explain how to sample exactly the vector $(r_{kh},0 \leq k \leq N)$.
  4. Implement the suggested algorithm and plot the trajectory $(r_{kh},0 \leq k \leq N)$ for $h=1$ hour, $h=1$ day, $h=1$ week and $N=100$.

Sampling the zero-coupon bond dynamics

We denote by $P(t,T)$ the price at time $t$ of a zero-coupon with maturity date $T$. We assume that $\P$ is a probability under which all the discounted bond price

\begin{displaymath} \tilde{P}(t,T) = e^{-\int_0^t r_s ds} P(t,T), \end{displaymath}

are martingales.
  1. We know (see chapter [*]) that the zero coupon bonds can be written as

    \begin{displaymath} P(t,T)=\exp\left[-(T-t)R(T-t,r(t))\right] \end{displaymath}

    with :

    \begin{displaymath} R(\theta,r) = R_{\infty} - \frac{1}{a\theta} \left( \lef... ...frac{\sigma^2}{4a^2}\left(1 - e^{-a\theta}\right)^2 \right). \end{displaymath}

    and $R_{\infty} = \lim_{\theta \to \infty} R(\theta,r) = b - \frac{\sigma^2}{2 a^2}$.

    Sample the discretised trajectory price of a bond with maturity $T=1$ $(P(kh,T),0\leq k \leq N$ where $h=1$ day and is such that $Nh=1$ year.

Pricing a zero-coupon bond option

Here, we consider a call option in the Vasicek model, with maturity $\theta$ on a zero-coupon bond with maturity $T$, $T>\theta$. We want to implement a hedging strategy for this option.

  1. Show, using the results of Chapter [*], that

    \begin{displaymath} \frac{d\tilde{P}(t,T)}{\tilde{P}(t,T)} = \sigma^T_s dW_t \end{displaymath}

    where

    \begin{displaymath} \sigma^T_s= - \sigma\frac{1-e^{-a(T-s)}}{a}. \end{displaymath}

  2. Show using Proposition [*] that the price $C_t$ of the call option at time $t$, is given by

    \begin{displaymath} C_t= P(t,\theta) B\left(t,\frac{P(t,T)}{P(t,\theta)}\right), \end{displaymath}

    with

    \begin{displaymath} B(t,x) = xN(d_1(t,x))-KN(d_2(t,x)). \end{displaymath}

    where $N$ is the cumulative normal distribution function,

    \begin{displaymath} d_1(t,x)=\frac{\log (x/K)+( \Sigma^{2}(t,\theta)/2) }{\Sigm... ...ad {\rm and} \quad d_{2}(t,x)=d_{1}(t,x) - \Sigma(t,\theta). \end{displaymath}

    and

    \begin{displaymath} \Sigma^2(t,\theta) =\int_t^\theta\left(\sigma^{T}_{s}-\sigma^{\theta}_{s}\right)^2ds. \end{displaymath}

    Implement this formula and plot the option price at time $0$ as a function of the strike $K$.
  3. Using Exercise [*] of this chapter, show that

    \begin{displaymath} C_t = P(t,T)H^T_t+P(t,\theta)H^\theta_t, \end{displaymath}

    $H^T_t=N\left(d_1\left(t,\frac{P(t,T)}{P(t,\theta)}\right)\right)$ and $H^\theta_t=-KN\left(d_2\left(t,\frac{P(t,T)}{P(t,\theta)}\right)\right)$.

    Implement these formulae and plot the values of $H^T_0$ and $H^\theta_0$ as a function of the strike $K$.

    Give a perfect hedging portfolio for the call option using only zero coupon bonds with maturity $T$ and zero coupon bonds with maturity $\theta$.

  4. We are interested in studying discrete approximation of this perfect hedging portfolio in which the quantity ${\bar H}^T_{s}$ of zero coupon bonds with maturity $T$ remains constant on the interval $[kh,(k+1)h]$ and equal to $H^T_{kh}$. ${\bar H}^\theta_{s}$, the quantity of zero coupon bonds with maturity $\theta$, is determined using the discrete self-financing condition at times $kh$.

    For a given $h$ (successively chosen to be $h=1$ day, $h=1$ week, $h=1$ month) sample the residual risk of this approximated hedging portfolio. Plot a histogram of the residual risk and study the values of its mean and its variance when $h$ decreases to $0$.