Abstract
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 numéraire arguments of Vee (in which symmetries reduce the original 2 + 1 dimensional simple 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 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 + 1 dimensional PDE formulation. We indicate that this nonlinear security has a closed-form pricing formula similar to that of the BlackScholes 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-Večě" reduction.
Original language | English (US) |
---|---|
Pages (from-to) | 1197-1212 |
Number of pages | 16 |
Journal | International Journal of Theoretical and Applied Finance |
Volume | 12 |
Issue number | 8 |
DOIs | |
State | Published - Dec 2009 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Finance
- Economics, Econometrics and Finance(all)
Keywords
- RogersShiVee reduction
- Simple Asian option
- Symmetry analysis