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Invariant solutions of the Black-Scholes equation. (English) Zbl 1200.91309
Summary: As the Black-Scholes equation can be transformed into the one-dimensional linear heat equation via two sets of transformations, an optimal system of one-dimensional subalgebras for the one-dimensional heat equation is exploited to obtain two classes of optimal systems of one-dimensional subalgebras for the well-known Black-Scholes equation of the mathematics of finance. Two methods for the derivation of the two classes of optimal systems of group-invariant solutions for this model are available. We present the simpler approach.
MSC:
91G80Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
35A30Geometric theory for PDE, characteristics, transformations
35Q91PDEs in connection with game theory, economics, social and behavioral sciences
91G10Portfolio theory