A novel reduction of the simple asian option and lie-group invariant solutionsa

Stephen Taylor, Scott Glasgow

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 languageEnglish (US)
Pages (from-to)1197-1212
Number of pages16
JournalInternational Journal of Theoretical and Applied Finance
Volume12
Issue number8
DOIs
StatePublished - Dec 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Finance
  • Economics, Econometrics and Finance(all)

Keywords

  • RogersShiVee reduction
  • Simple Asian option
  • Symmetry analysis

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