Bond prices under EFM(1)

In this section we toggle the superscript stars $\star$ towards a slimmer notation, i.e. in this section the risk neutral parameters carry no star whereas the objective measure parameters obtain the $\star$. I.e. now

$$A = \left( \begin{array}{ccc} -\kappa_r & \kappa^\star_r & \kappa... ...ppa_r\,\alpha_r\\ \kappa_x\,\alpha_x \\ \kappa_y\,\alpha_y \end{array}\right)$$ (106)

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

$$\frac{\partial u}{\partial t} - A^\prime u = (1,0,0)^\prime$$ (107)

$$\frac{\partial v}{\partial t} = -u^\prime a + {1\over 2} u^\prime V u \qquad .$$ (108)

The function $u(t)$ has a natural interpretation as duration with respect to the rates $\xi$, since $\partial\ln(P)/\partial\xi = u$. The first ODE can be solved right away,

$$u=\left( \begin{array}{c} \frac{1}{\kappa_r}(1-e^{-\kappa... ...{-\kappa_r t}-e^{-\kappa_y t}) \right) \end{array} \right),$$ (109)

solving the second ODE amounts to integration over $u$ and $u^2$ terms. To do this simplify notation: $\kappa_i=\kappa_r, \kappa_x, \kappa_y$, i.e. $i=r,x,y$. Introduce the two functions

$$f_i = \frac{1}{\kappa_i}\left(1-e^{-\kappa_i t}\right)$$ (110)

$$g_{ij} = \frac{1}{\kappa_i-\kappa_j}\left(e^{-\kappa_i t}-e^{-\kappa_j t}\right)$$ (111)

which gives $u$ as

$$u=\left( \begin{array}{c} f_r\\ \frac{\kappa_r^\star}{\... ...{\kappa_r^\star}{\kappa_r}(f_y+g_{ry}) \end{array} \right).$$ (112)

The limiting cases of $f_i$ and $g_{ij}$ must be taken into account:

$$\lim_{\kappa_i\to 0}f_i = t$$ (113)

$$g_{ii}:=\lim_{\kappa_j\to\kappa_i}g_{ij} = -t e^{-\kappa_i t}$$ (114)

Five integrals are required (the $\mathrm{d}t$'s in $\int\!\dots$ are implicit throughout):

$$\int\!\! f_i = \frac{1}{\kappa_i} \left(t-f_i\right)$$ (115)

$$\int\!\! g_{ij} = \frac{1}{\kappa_i-\kappa_j} \left(f_i-f_j\right)$$ (116)

$$\int\!\! f_i\, f_j = \frac{1}{\kappa_i \kappa_j} \left(t-f_i-f_j+f_{i+j}\right)$$ (117)

$$\int\!\! f_i\, g_{jk} = \frac{1}{\kappa_i(\kappa_j-\kappa_k)} \left(f_j-f_k-f_{i+j}+f_{i+k}\right)$$ (118)

$$\int\!\! g_{ij}\, g_{kl} = \frac{1}{(\kappa_i-\kappa_j)(\kappa_k-\kappa_l)} \left(f_{i+k}-f_{i+l}-f_{j+k}+f_{j+l}\right)$$ (119)

where the subscript $i+j$ stand for $\kappa_i+\kappa_j$, etc. Four limiting cases need to be considered:

$$\int\!\! g_{ii} = -\int\! t e^{-\kappa_i t} = \frac{1}{\kappa_i^2}\left(e^{-\kappa_i t}(\kappa_i\, t+1)-1\right)$$ (120)

$$\int\!\! f_i\, g_{jj} = \frac{1}{\kappa_i}\left[\int\!\! g_{ii}-\int\!\! g_{i+j,i+j}\right]$$ (121)

$$\int\!\! g_{ij}\, g_{kk} = \frac{1}{\kappa_i-\kappa_j}\left[\int\!\! g_{i+k,i+k}-\int\!\! g_{j+k,j+k}\right]$$ (122)

$$\int\!\! g_{ii}\, g_{kk} = \int\! t^2 e^{-(\kappa_i+\kappa_k)} = \left.\frac{1}{\kappa^3}\le... ...ppa t}(-t^2\kappa^2-\kappa t-1)+1\right)\right\vert _{\kappa=\kappa_i+\kappa_k}$$ (123)

In terms of the five integrals $v$ can be integrated to (set $\rho=\frac{\kappa_r^\star}{\kappa_r}$)

$$\begin{eqnarray*} v &=& -\, \kappa_r\,\alpha_r\,\int\!\! f_r -\, \kappa_x\,\... ...g_{ry} +\, \sigma_y^2\,\rho^2\int\!\! g_{ry}\,g_{ry} \qquad . \end{eqnarray*}$$

Note that $\kappa_x\to\kappa_y$ is not a singularity of $v$. Table summarizes the correct combinations. From the viewpoint of computational effectiveness calculate the quantities in the following order:

1. $e_i=e^{-\kappa_i t}$
2. $f_i$
3. $\int\! f_i$
4. $\int\! f_i f_j$
5. $\int\! f_i g_{jk}\quad$ : $\kappa_j\ne\kappa_k\quad$ : $\int\! f_i g_{jk}$ $\kappa_j\to\kappa_k\quad$ : $\int\! f_i g_{jj}$
6. $\int\! g_{ij}\, g_{kl}\quad$ : $\kappa_i\ne\kappa_j\quad$ : $\int\! g_{ij}\, g_{kl}\quad$ $\kappa_i\to\kappa_j\quad$ : $\int\! g_{ii}\, g_{kl}\quad$ $\kappa_k\to\kappa_l\quad$ : $\int\! g_{ij}\, g_{kk}\quad$
7. $v=\dots$

Table: Integrating the $u^\prime a$ and $u^\prime V u$-term

$\int\!\dots$	$f_r$	$f_x+g_{rx}$	$f_y+g_{ry}$
$f_r$	$\sigma_r^2$	$2\, v_{rx}\,\rho$	$2\, v_{ry}\,\rho$
$f_x+g_{rx}$		$\sigma_r^2\,\rho^2$	$2\, v_{xy}\,\rho^2$
$f_y+g_{ry}$			$\sigma_r^2\,\rho^2$

        

$\int\!\dots$	1
$f_r$	$\kappa_r\,\alpha_r\,\rho$
$f_x+g_{rx}$	$\kappa_x\,\alpha_x\,\rho$
$f_y+g_{ry}$	$\kappa_y\,\alpha_y\,\rho$