A Method of Reducing Dimension of Space Variablesin Multi-dimensional Black-Scholes Equations

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