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space-homogeneity

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space-homogeneity

We may wish to measure the value of the underlying in a different unit. This will obviously effect the option pricing formula as follows.

$\begin{displaymath} av(x,K,T,t,\sigma,r_d,r_f,\phi)=v(ax,aK,T,t,\sigma,r_d,r_f,\phi) \mbox{ for all } a>0. \end{displaymath}$

(28)

Differentiating both sides with respect to

and then setting

yields

$\begin{displaymath} v=xv_x+Kv_K. \end{displaymath}$

(29)

Comparing the coefficients of

and

in equations (3) and (29) leads to suggestive results for the delta

and dual delta

. This homogeneity is the reason behind the simplicity of the delta formulas, whose tedious computation can be saved this way.

2000-06-11

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