Optimal system, symmetry reductions and new closed form solutions for the geometric average Asian options

In this paper, the Lie group analysis method is applied to the geometric average Asian option pricing equation in financial problems. Firstly, the complete Lie symmetry group and infinitesimal generators of this equation are derived. Then the optimal system with one parameter for the Lie symmetry algebra are obtained, which gives the possibility to describe a complete set of invariant solutions to the pricing equation. Finally, based on the optimal system the symmetry reductions and corresponding closed form solutions for the pricing equation are proposed.

[1]  L. A. Bordag.  Study of the risk-adjusted pricing methodology model with methods of geometrical analysis , 2009 .

[2]  L. Rogers,et al.  The value of an Asian option , 1995, Journal of Applied Probability.

[3]  Ljudmila A. Bordag,et al.  Explicit solutions for a nonlinear model of financial derivatives , 2006, math/0604117.

[4]  P. Koehl,et al.  A P.D.E. Approach to Asian Options: Analytical and Numerical Evidence , 1997 .

[5]  Ljudmila A. Bordag,et al.  Models of Self-Financing Hedging Strategies in Illiquid Markets: Symmetry Reductions and Exact Solutions , 2011 .

[6]  Kamel Al-khaled,et al.  Computers and Mathematics with Applications Numerical Treatment of Stochastic Models Used in Statistical Systems and Financial Markets , 2022 .

[7]  Yang Xuan-liu,et al.  Symmetry Breaking for Black-Scholes Equations , 2007 .

[8]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[9]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[10]  Jeroen P. A. Devreese,et al.  Path integral approach to Asian options in the Black-Scholes model , 2009, 0906.4456.

[11]  M. Yor,et al.  BESSEL PROCESSES, ASIAN OPTIONS, AND PERPETUITIES , 1993 .

[12]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[13]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[14]  Changzheng Qu,et al.  Optimal Systems and Group Classification of (1+2)-Dimensional Heat Equation , 2004 .

[15]  San-Lin Chung,et al.  Efficient quadratic approximation of floating strike Asian option values , 2003 .

[16]  Lucas Jódar,et al.  A numerical method for European Option Pricing with transaction costs nonlinear equation , 2009, Math. Comput. Model..

[17]  A. Kemna,et al.  A pricing method for options based on average asset values , 1990 .

[18]  Changzheng Qu,et al.  A Note on Optimal Systems for the Heat Equation , 2001 .

[19]  M. Dai,et al.  CHARACTERIZATION OF OPTIMAL STOPPING REGIONS OF AMERICAN ASIAN AND LOOKBACK OPTIONS , 2006 .

[20]  S. Turnbull,et al.  A Quick Algorithm for Pricing European Average Options , 1991, Journal of Financial and Quantitative Analysis.

[21]  Chi-Fai Lo,et al.  Valuation of financial derivatives with time-dependent parameters: Lie-algebraic approach , 2001 .

[22]  A. Jamiołkowski.  Applications of Lie groups to differential equations , 1989 .

[23]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .

[24]  Dawn Hunter.  Pricing Asian and basket options via Taylor expansion , 2002 .

[25]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[26]  L. Bouaziz,et al.  The pricing of forward-starting asian options , 1994 .

[27]  Jaemin Ahn,et al.  A Laplace transform finite difference method for the Black-Scholes equation , 2010, Math. Comput. Model..