Model specification - EFM(1)

EFM(1) is specified under objective measure $P$-dynamics as follows:

$$\mathrm{d}r = \kappa_r(x+y-r) \mathrm{d}t + (\Sigma \mathrm{d}W)_r$$ (91)

$$\mathrm{d}x = \kappa_x(\alpha_x-x) \mathrm{d}t + (\Sigma \mathrm{d}W)_x$$ (92)

$$\mathrm{d}y = \kappa_y(\alpha_y-y) \mathrm{d}t + (\Sigma \mathrm{d}W)_y$$ (93)

where $W$ is a 3-d BM entering through full covariance $V:=\Sigma \Sigma^\prime$ Let $\xi=(r,x,y)^\prime$ and introduce

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

which gives the $P$-SDE

$$\mathrm{d}\xi = (A\xi+a) \mathrm{d}t + \Sigma \mathrm{d}W$$ (94)

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

$$\lambda = \Sigma^{-1}(\Lambda\xi + \lambda^1)$$ (95)

with diagonal $\Lambda = \textrm{diag}(\lambda_r^1, \lambda_x^1,\lambda_y^1)$. Under the risk-neutral measure $P^\star$ the dynamics reads

$$\mathrm{d}\xi = (A\xi+a+\Lambda\xi+\lambda^1) \mathrm{d}t + \Sigma \mathrm{d}W^\star$$ (96)

$$= (A+\Lambda)\xi \mathrm{d}t +(a+\lambda^1) \mathrm{d}t + \Sigma \mathrm{d}W^\star$$ (97)

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

$$\mathrm{d}\xi = (A^\star\xi+a^\star)\,\mathrm{d}t + \Sigma\,\mathrm{d}W^\star$$ (98)

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

$$A^\star=\left( \begin{array}{ccc} -\kappa_r+\lambda^0_r &... ...a_x^\star& 0 \\ 0 & 0 & -\kappa_y^\star \end{array}\right)$$ (99)

and

$$a^\star = \left( \begin{array}{c} \lambda_r^1 \\ \kappa_x... ...x^\star \\ \kappa_y^\star\,\alpha_y^\star \end{array}\right)$$ (100)

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