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

International Journal of Theoretical and Applied Finance, Vol. 12, No. 8, p. 1197, 2009

10 Pages Posted: 2 Apr 2011

See all articles by Stephen Michael Taylor

Stephen Michael Taylor

New Jersey Institute of Technology

Scott Alan Glasgow

Brigham Young University

Date Written: July 15, 2009

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 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.

Keywords: Asian Options, PDE Reductions, Vecer Reduction, Lie Symmetry Analysis

Suggested Citation

Taylor, Stephen Michael and Glasgow, Scott Alan, A Novel Reduction of the Simple Asian Option and Lie-Group Invariant Solutions (July 15, 2009). International Journal of Theoretical and Applied Finance, Vol. 12, No. 8, p. 1197, 2009. Available at SSRN: https://ssrn.com/abstract=1800442

Stephen Michael Taylor (Contact Author)

New Jersey Institute of Technology ( email )

University Heights
Newark, NJ 07102
United States

Scott Alan Glasgow

Brigham Young University ( email )

Provo, UT 84602
United States

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