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Next: put-call symmetry Up: identities Previous: space-homogeneity

time-homogeneity

We can perform a similar computation for the time-affected parameters and obtain the obvious equation
\begin{displaymath}
v(x,K,T,t,\sigma,r_d,r_f,\phi)=v(x,K,\frac{T}{a},\frac{t}{a},\sqrt{a}\sigma,ar_d,ar_f,\phi) \mbox{ for all } a>0.
\end{displaymath} (30)

Differentiating both sides with respect to $a$ and then setting $a=1$ yields
\begin{displaymath}
0=\tau v_t+\frac{1}{2}\sigma v_{\sigma}+r_dv_{r_d}+r_fv_{r_f}.
\end{displaymath} (31)

Of course, this can also be verified by direct computation. The overall use of such equations is to generate double checking benchmarks when computing Greeks. These homogeneity methods can easily be extended to other more complex options.




2000-06-11
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