  We study a method of reducing space dimension in multi-dimensional
Black-Scholes partial differential equations as well as in multi-dimensional
parabolic equations. We prove that a multiplicative transformation of space
variables in the Black-Scholes partial differential equation reserves the form
of Black-Scholes partial differential equation and reduces the space dimension.
We show that this transformation can reduce the number of sources of risks by
two or more in some cases by giving remarks and several examples of financial
pricing problems. We also present that the invariance of the form of
Black-Scholes equations is based on the invariance of the form of parabolic
equation under a change of variables with the linear combination of variables.
