Model specification

The Economic Factor Model in its second specifation, EFM(2), is defined under objective measure $P$-dynamics as follows:

$$\mathrm{d}r = \kappa_r(x-r) \mathrm{d}t + (\Sigma \mathrm{d}W)_r$$ (42)

$$\mathrm{d}x = \kappa_x(y-x) \mathrm{d}t + (\Sigma \mathrm{d}W)_x$$ (43)

$$\mathrm{d}y = \kappa_y(\alpha_y-y) \mathrm{d}t + (\Sigma \mathrm{d}W)_y$$ (44)

where $W$ is a 3-d BM entering through full covariance

$$V:=\Sigma \Sigma^\prime= \left( \begin{array}{ccc} \sigm... ... \sigma_x \sigma_y c_{xy} & \sigma_y^2 \end{array} \right).$$ (45)

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

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

which gives the $P$-SDE

$$\mathrm{d}\xi = (A\xi+a) \mathrm{d}t + \Sigma \mathrm{d}W$$ (46)

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

$$\lambda = \Sigma^{-1}(\Lambda\xi + \lambda^1)$$ (47)

with matrix $\Lambda$ given as

$$\Lambda = \left( \begin{array}{ccc} \lambda_r^0 & -\lam... ... -\lambda_x^0 \\ 0 & 0 & \lambda_y^0 \end{array} \right).$$ (48)

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

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

$$= (A+\Lambda)\xi \mathrm{d}t +(a+\lambda^1) \mathrm{d}t + \Sigma \mathrm{d}W^\star$$ (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

$$P^\star:\,\mathrm{d}\xi = (A^\star\xi+a^\star)\,\mathrm{d}t + \Sigma\,\mathrm{d}W^\star$$ (51)

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

$$A^\star=\left( \begin{array}{ccc} -\kappa_r+\lambda^0_r &... ...appa_x^\star \\ 0 & 0 & -\kappa_y^\star \end{array}\right)$$ (52)

and

$$a^\star = \left( \begin{array}{c} \lambda_r^1 \\ \lambda_... ...x^\star \\ \kappa_y^\star\,\alpha_y^\star \end{array}\right)$$ (53)

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