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Euro related symmetries of value, delta and leverage next up previous
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Euro related symmetries of value, delta and leverage

Let us now consider the example of $S_t$ modelling the exchange rate GBP/DEM. After the currency Euro has been introduced, we need to know how to relate options written on GBP/DEM to options on EUR/GBP. We denote by $E=1.95583$ the fixed exchange rate EUR/DEM. Then $E/S_t$ serves as model for EUR/GBP. Combining the foreign-domestic symmetry (37) with the space-homogeneity (28) we obtain

\begin{displaymath} v(x,K,T,t,\sigma,r_d,r_f,\phi)=\frac{Kx}{E}v(\frac{E}{x},\frac{E}{K},T,t,\sigma,r_f,r_d,-\phi). \end{displaymath} (38)

Taking the derivative with respect to $x$ on both sides results in
$\displaystyle v_x(x,K,T,t,\sigma,r_d,r_f,\phi)$ $\textstyle =$ $\displaystyle \frac{K}{E}v(\frac{E}{x},\frac{E}{K},T,t,\sigma,r_f,r_d,-\phi)$  
  $\textstyle -$ $\displaystyle \frac{K}{x}v_x(\frac{E}{x},\frac{E}{K},T,t,\sigma,r_f,r_d,-\phi).$ (39)

In particular the deltas of identical options are not exactly negatives of each other. This is only approximately correct. The right quantities to compare are not the deltas, but the dimensionless leverages, because (39) implies
\begin{displaymath} \frac{xv_x(x,K,T,t,\sigma,r_d,r_f,\phi)}{v(x,K,T,t,\sigma,r_... ...,-\phi)}{v(\frac{E}{x},\frac{E}{K},T,t,\sigma,r_f,r_d,-\phi)}. \end{displaymath} (40)

This means that the leverages of a GBP call and an identical EUR put add up to one. Note the the factor $E$ could be cancelled on the right hand side to produce a plain foreign-domestic leverage symmetry.

next up previous
Next: quotation Up: identities Previous: foreign-domestic symmetry

2000-06-11
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