TY - JOUR
T1 - A novel reduction of the simple asian option and lie-group invariant solutionsa
AU - Taylor, Stephen
AU - Glasgow, Scott
PY - 2009/12
Y1 - 2009/12
N2 - 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.
AB - 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.
KW - RogersShiVee reduction
KW - Simple Asian option
KW - Symmetry analysis
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U2 - 10.1142/S0219024909005634
DO - 10.1142/S0219024909005634
M3 - Article
AN - SCOPUS:75649119302
SN - 0219-0249
VL - 12
SP - 1197
EP - 1212
JO - International Journal of Theoretical and Applied Finance
JF - International Journal of Theoretical and Applied Finance
IS - 8
ER -