A NOVEL REDUCTION OF THE SIMPLE ASIAN OPTIONAND LIE-GROUP INVARIANT SOLUTIONS

摘要

We develop the complete 6-dimensional classical symmetry group of the partial differ-ential equation (PDE) that governs the fair price of a simple Asian option within asimple market model. The symmetries we expose include the 5-dimensional symmetrygroup partially noted by Rogers and Shi, and communicated implicitly by the change ofnumeraire arguments of Vecer (in which symmetries reduce the original 2+1 dimensionalsimple Asian option PDE to a 1+1 dimensional PDE). Going beyond this previous work,we expose a new 1-dimensional space of symmetries of the Asian PDE that cannot rea-sonably be found by inspection. We demonstrate that the new symmetry could be usedto formulate a new, "nonlinear" derivative security that has a 1 + 1 dimensional PDEformulation. We indicate that this nonlinear security has a closed-form pricing formulasimilar to that of the Black–Scholes equation for a particular market dependent pay-off, and show that hedging the short position in this particular exotic option is stablefor all market parameters. We also demonstrate the patently Lie-algebraic method forobtaining the already well-known "Rogers–Shi–Vecer" reduction.