Bond prices

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Bond prices

Consider the general Gaussian real-world n-factor model under objective measure

$\begin{displaymath} P:\quad \mathrm{d}\xi = (A\xi+a)\,\mathrm{d}t + \Sigma\,\ma... ...{d}W\, , A\in\mathbb{R}^{n\times n}, a\in\mathbb{R}^n \quad . \end{displaymath}$

(1)

Via Girsanov transform to risk-neutral measure $P^\star$ with $\mathrm{d}W = \lambda(\xi) \mathrm{d}t+ \mathrm{d}W^\star$ and $\lambda(\xi)$ affine:

$\begin{displaymath} \lambda(\xi)=\Sigma^-(\Lambda\xi+\lambda^1)\, , \Lambda\in\mathbb{R}^{n\times n}\quad . \end{displaymath}$

(2)

Introducing $A^\star=A+\Lambda$ and $a^\star=a+\lambda^1$ leads to risk-neutral dynamics

$\begin{displaymath} P^\star:\quad \mathrm{d}\xi = (A^\star\xi+a^\star)\,\mathrm{d}t + \Sigma\,\mathrm{d}W^\star \quad , \end{displaymath}$

(3)

The price $p(\tau,T)$ of a T-Bond (i.e. a zero-bond of maturity

) at time $\tau$ then satisfies the PDE

$\begin{displaymath} -r p + \frac{\partial p}{\partial\tau} + \left(\frac{\parti... ...ac{\partial^2 p}{\partial \xi\partial \xi}\right) \Sigma = 0. \end{displaymath}$

(4)

Setting $t=T-\tau$ the ansatz $p(\tau,T) = p(t) = \exp\left(-u(t)^\prime\xi + v(t)\right)$ leads to the following two ODEs

$\displaystyle \frac{\partial u}{\partial t} - {A^\star}^\prime u$	$\textstyle =$	$\displaystyle (1,0,0)^\prime$	(5)
$\displaystyle \frac{\partial v}{\partial t}$	$\textstyle =$	$\displaystyle -u^\prime a^\star + {1\over 2} u^\prime V u \qquad ,$	(6)

with initial conditions

and

. The function

has a natural interpretation as duration with respect to the rates $\xi$ , since $\partial\ln(P)/\partial\xi = u$ . The solutions to the two ODEs are ¹

$\displaystyle u$	$\textstyle =$	$\displaystyle \left(e^{t {A^\star}^\prime} -1\right) {{A^\star}^\prime}^{-1} e_1$	(7)
$\displaystyle v$	$\textstyle =$	$\displaystyle \int_0^t ds\left[-{a^\star}^\prime u + {1\over 2} u^\prime V u\right]\quad .$	(8)

The asymptotics of

for $t\to 0$ will be useful:

$\displaystyle u = \left(t\,\mathrm{\bf 1}+{t^2\over 2}\,{A^\star}^\prime\right)\,e_1 + o(t^3) \quad .$

(9)

We turn to the calculation of and via diagonalization of ${A^\star}^\prime$ . To avoid clutter the star $^\star$ will be dropped for the remainder of this subsection, i.e. we write for $A^\star$ , etc.

Assuming $A^\prime$ can be diagonalised $A^\prime = Q D Q^{-1}$ with diagonal the solution may be rewritten

$\begin{eqnarray*} u &=& Q\left(e^{t D} -1\right) b\\ v &=& \int_0^t ds\left[a... ...\over 2} b^\prime e^{s D} Q^\prime V Q e^{s D} b\right] \quad , \end{eqnarray*}$

where the following abbrevation was introduced:

$\begin{displaymath} b := D^- Q^- e_1\quad . \end{displaymath}$

(10)

Denote the eigenvalues of $A^\prime$ by $\nu=(\nu_1,\nu_2,\dots,\nu_n)$ . Then $D=\rm {diag}(\nu)$ and $e^{s D}=\rm {diag}(e^{s \nu_i})$ and

$\begin{displaymath} \left(e^{s D} Q^\prime V Q e^{s D}\right)_{ij} = \left(Q^\prime V Q\right)_{ij} e^{s(\nu_i+\nu_j)} \end{displaymath}$

(11)

which makes integration towards

straightforward:

$\begin{eqnarray*} v = && t\left[a^\prime Q b + {1\over 2} b^\prime Q^\prime V Q... ..._{ij} \frac{e^{t(\nu_i+\nu_j)}-1}{\nu_i+\nu_j}\right) b \quad . \end{eqnarray*}$

This can be written in another form,

$\begin{eqnarray*} v = && a^\prime Q\left(t-\left(e^{tD}-1\right)D^-\right)b \\ ... ..._j)}-1}{\nu_i+\nu_j}-\frac{e^{t\nu_j}-1}{\nu_j}\right]\right) b \end{eqnarray*}$

or, using the result for $y(\infty)$ from section

below,

$\begin{eqnarray*} v = && -t\,y(\infty) + a^\prime Q\left(-\left(e^{tD}-1\right)... ...\nu_i+\nu_j}-\frac{e^{t\nu_j}-1}{\nu_j}\right]\right) b \quad . \end{eqnarray*}$

Next: Properties of the term Up: The general -factor Gaussian Previous: Notation

Markus Mayer 2009-06-22