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A Novel Reduction Of The Simple Asian Option And Lie-Group Invariant Solutions

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    (Department of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, N.Y. 11794-3651, USA)



    (Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA)

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    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 Večeř (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 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–Večeř" reduction.

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    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

    Volume (Year): 12 (2009)
    Issue (Month): 08 ()
    Pages: 1197-1212

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    Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:08:p:1197-1212
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