A Novel Reduction of the Simple Asian Option and Lie-Group Invariant Solutions

We develop the complete 6-dimensional classical symmetry group of the partial differential equation (PDE) that governs the fair price of a simple Asian option within a simple market model. The symmetries we expose include the 5-dimensional symmetry group partially noted by Rogers and Shi, and communicated implicitly by the change of numeraire arguments of Vecer (in which symmetries reduce the original 2 plus1 dimensional simple Asian option PDE to a 1 plus 1 dimensional PDE). Going beyond this previous work, we expose a new 1-dimensional space of symmetries of the Asian PDE that cannot reasonably be found by inspection. We demonstrate that the new symmetry could be used to formulate a new, "nonlinear" derivative security that has a 1 plus 1 dimensional PDE formulation. We indicate that this nonlinear security has a closed-form pricing formula similar to that of the Black-Scholes equation for a particular market dependent payoff, and show that hedging the short position in this particular exotic option is stable for all market parameters. We also demonstrate the patently Lie-algebraic method for obtaining the already well-known "Rogers Shi-Vecer" reduction