Approximations for Asian Options in Local Volatility Models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.

[1]  J. Vecer.  A new PDE approach for pricing arithmetic average Asian options , 2001 .

[2]  H. O. Fattorini,et al.  The time-optimal control problem in Banach spaces , 1974 .

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

[4]  M. Yor,et al.  Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques , 1992 .

[5]  D. Dufresne.  Laguerre Series for Asian and Other Options , 2000 .

[6]  William T. Shaw,et al.  Modelling financial derivatives with Mathematica : mathematical models and benchmark algorithms , 1998 .

[7]  K. Govinder,et al.  Solving The Asian Option Pde Using Lie Symmetry Methods , 2010 .

[8]  A. Pascucci,et al.  Path dependent volatility , 2008 .

[9]  P. Glasserman,et al.  Monte Carlo methods for security pricing , 1997 .

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  William T. Shaw,et al.  Differential equations and asymptotic solutions for arithmetic Asian options: ‘Black–Scholes formulae’ for Asian rate calls , 2008, European Journal of Applied Mathematics.

[12]  Emilio Barucci,et al.  Some Results on Partial Differential Equations and Asian Options , 2001 .

[13]  Jean-Pierre Fouque,et al.  Pricing Asian options with stochastic volatility , 2003 .

[14]  Andrea Pascucci,et al.  Free boundary and optimal stopping problems for American Asian options , 2007, Finance Stochastics.

[15]  Steven Kou,et al.  Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model , 2012, Oper. Res..

[16]  Bara Kim,et al.  Pricing of geometric Asian options under Heston's stochastic volatility model , 2014 .

[17]  V. Henderson,et al.  On the equivalence of floating- and fixed-strike Asian options , 2002, Journal of Applied Probability.

[18]  Friedrich Hubalek,et al.  On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps , 2011, J. Comput. Appl. Math..

[19]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[20]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[21]  Michael Schröder,et al.  ON CONSTRUCTIVE COMPLEX ANALYSIS IN FINANCE : EXPLICIT FORMULAS FOR ASIAN OPTIONS , 2008 .

[22]  J. V. Leeuwen.  The domino effect , 2004, physics/0401018.

[23]  Vadim Linetsky,et al.  Spectral Expansions for Asian (Average Price) Options , 2004, Oper. Res..

[24]  M. Yor.  On some exponential functionals of Brownian motion , 1992, Advances in Applied Probability.

[25]  Akihiko Takahashi,et al.  Pricing Barrier and Average Options Under Stochastic Volatility Environment , 2009 .

[26]  L. Rogers,et al.  Complete Models with Stochastic Volatility , 1998 .

[27]  Andrea Pascucci,et al.  Expansion formulae for local Levy models , 2011 .

[28]  S. Pagliarani,et al.  Analytical approximation of the transition density in a local volatility model , 2012 .

[29]  Claudio Albanese,et al.  Coherent Global Market Simulations and Securitization Measures for Counterparty Credit Risk , 2010 .

[30]  Erhan Bayraktar,et al.  Pricing Asian Options for Jump Diffusions , 2007, ArXiv.

[31]  Emmanuel Gobet,et al.  Weak approximation of averaged diffusion processes , 2014 .

[32]  M. Yor.  Sur certaines fonctionnelles exponentielles du mouvement brownien réel , 1992, Journal of Applied Probability.

[33]  P. Wilmott,et al.  A Note on Average Rate Options with Discrete Sampling , 1995, SIAM J. Appl. Math..

[34]  A note on pricing Asian derivatives with continuous geometric averaging , 1999 .

[35]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[36]  D. Dufresne.  The integrated square-root process , 2001 .

[37]  Mohamed Karim Sbai,et al.  Exact retrospective Monte Carlo computation of arithmetic average Asian options , 2007, Monte Carlo Methods Appl..

[38]  Andrea Pascucci,et al.  Parametrix Approximation of Diffusion Transition Densities , 2010, SIAM J. Financial Math..

[39]  J. Coron.  Control and Nonlinearity , 2007 .

[40]  A. Pascucci,et al.  The obstacle problem for a class of hypoelliptic ultraparabolic equations , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[41]  Jin E. Zhang.  A Semi-Analytical Method for Pricing and Hedging Continuously Sampled Arithmetic Average Rate Options , 2001 .

[42]  Lower bounds for densities of Asian type stochastic differential equations , 2010 .

[43]  M. Fu,et al.  Pricing Continuous Asian Options: A Comparison of Monte Carlo and Laplace Transform Inversion Methods , 1998 .

[44]  N. Yoshida,et al.  Monte Carlo Simulation with Asymptotic Method , 2005 .

[45]  A Novel Reduction of the Simple Asian Option and Lie-Group Invariant Solutions , 2009 .

[46]  R. E. Kalman,et al.  Controllability of linear dynamical systems , 1963 .

[47]  A. Dassios,et al.  The square-root process and Asian options , 2006 .

[48]  Akihiko Takahashi,et al.  Pricing Average Options on Commodities , 2011 .

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

[50]  Paolo Guasoni,et al.  Optimal importance sampling with explicit formulas in continuous time , 2007, Finance Stochastics.