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rates symmetry

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rates symmetry

Direct computation shows that the rates symmetry

$\begin{displaymath} \frac{\partial v}{\partial r_d}+\frac{\partial v}{\partial r_f}=-\tau v \end{displaymath}$

(33)

holds for vanilla options. This relationship, in fact, holds for a wider class of options, at least for bounded smooth path-independent payoffs

, because in this case we may write the value function

$\begin{displaymath} v=e^{-r_d\tau}I\!\!E[F(xe^{\sigma W_{\tau}+\sigma\theta_-\tau})], \end{displaymath}$

(34)

whence

$\displaystyle \frac{\partial v}{\partial r_d}$	$\textstyle =$	$\displaystyle -\tau v+\tau e^{-r_d\tau}I\!\!E[S_TF'(S_T)\vert S_t=x],$	(35)
$\displaystyle \frac{\partial v}{\partial r_f}$	$\textstyle =$	$\displaystyle -\tau e^{-r_d\tau}I\!\!E[S_TF'(S_T)\vert S_t=x].$	(36)

2000-06-11

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