Model specification

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Model specification

The Economic Factor Model in its second specifation, EFM(2), is defined under objective measure

-dynamics as follows:

$\displaystyle \mathrm{d}r$	$\textstyle =$	$\displaystyle \kappa_r(x-r) \mathrm{d}t + (\Sigma \mathrm{d}W)_r$	(42)
$\displaystyle \mathrm{d}x$	$\textstyle =$	$\displaystyle \kappa_x(y-x) \mathrm{d}t + (\Sigma \mathrm{d}W)_x$	(43)
$\displaystyle \mathrm{d}y$	$\textstyle =$	$\displaystyle \kappa_y(\alpha_y-y) \mathrm{d}t + (\Sigma \mathrm{d}W)_y$	(44)

where

is a 3-d BM entering through full covariance

$\begin{displaymath} V:=\Sigma \Sigma^\prime= \left( \begin{array}{ccc} \sigm... ... \sigma_x \sigma_y c_{xy} & \sigma_y^2 \end{array} \right). \end{displaymath}$

(45)

Let $\xi=(r,x,y)^\prime$ and introduce

$\begin{displaymath} A=\left( \begin{array}{ccc} -\kappa_r & \kappa_r & 0 \\ ... ...}{c} 0 \\ 0 \\ \kappa_y\,\alpha_y \end{array} \right) \end{displaymath}$

which gives the

-SDE

$\begin{displaymath} \mathrm{d}\xi = (A\xi+a) \mathrm{d}t + \Sigma \mathrm{d}W \end{displaymath}$

(46)

Specify the risk premium $\lambda = (\lambda_r,\lambda_x,\lambda_y)^\prime$

$\begin{displaymath} \lambda = \Sigma^{-1}(\Lambda\xi + \lambda^1) \end{displaymath}$

(47)

with matrix $\Lambda$ given as

$\begin{displaymath} \Lambda = \left( \begin{array}{ccc} \lambda_r^0 & -\lam... ... -\lambda_x^0 \\ 0 & 0 & \lambda_y^0 \end{array} \right). \end{displaymath}$

(48)

Under the risk-neutral measure $P^\star$ the dynamics reads

$\displaystyle P^\star:\,\mathrm{d}\xi$	$\textstyle =$	$\displaystyle (A\xi+a+\Lambda\xi+\lambda^1) \mathrm{d}t + \Sigma \mathrm{d}W^\star$	(49)
	$\textstyle =$	$\displaystyle (A+\Lambda)\xi \mathrm{d}t +(a+\lambda^1) \mathrm{d}t + \Sigma \mathrm{d}W^\star$	(50)

where $\mathrm{d}W^\star = -\lambda \mathrm{d}t + \mathrm{d}W$ . By introduciong new parameters the SDE can now be brought into the following $P^\star$ -form

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

(51)

where $A^\star=A+\Lambda$ and $a^\star=a+\lambda^1$ , i.e.

$\begin{displaymath} A^\star=\left( \begin{array}{ccc} -\kappa_r+\lambda^0_r &... ...appa_x^\star \\ 0 & 0 & -\kappa_y^\star \end{array}\right) \end{displaymath}$

(52)

and

$\begin{displaymath} a^\star = \left( \begin{array}{c} \lambda_r^1 \\ \lambda_... ...x^\star \\ \kappa_y^\star\,\alpha_y^\star \end{array}\right) \end{displaymath}$

(53)

i.e. with this particular affine choice of $\lambda$ the structure of the dynamics is maintained under risk neutral measure $P^\star$ .

Next: Diagonalisation of and Up: EFM(2) Previous: Introduction

Markus Mayer 2009-06-22