Parametrix Approximation of Diffusion Transition Densities

A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

[1]  Mark Schroder.  Computing the Constant Elasticity of Variance Option Pricing Formula , 1989 .

[2]  C. F. Lee,et al.  Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation , 2008, Math. Comput. Simul..

[3]  T. McMillen.  Simulation and Inference for Stochastic Differential Equations: With R Examples , 2008 .

[4]  Yacine Aït-Sahalia.  Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed‐form Approximation Approach , 2002 .

[5]  Jin E. Zhang.  Pricing continuously sampled Asian options with perturbation method , 2003 .

[6]  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.

[7]  Rachel Kuske,et al.  Optimal exercise boundary for an American put option , 1998 .

[8]  V. Isakov,et al.  TOPICAL REVIEW: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets , 1999 .

[9]  Sam Howison,et al.  Matched Asymptotic Expansions in Financial Engineering , 2005 .

[10]  Christian Kahl,et al.  Fast strong approximation Monte Carlo schemes for stochastic volatility models , 2006 .

[11]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .

[12]  P. Duck,et al.  SINGULAR PERTURBATION TECHNIQUES APPLIED TO MULTIASSET OPTION PRICING , 2009 .

[13]  Ronnie Sircar,et al.  Singular Perturbations in Option Pricing , 2003, SIAM J. Appl. Math..

[14]  Eugenio Elia Levi,et al.  Sulle equazioni lineari totalmente ellittiche alle derivate parziali , 1907 .

[15]  S. Menozzi,et al.  Explicit parametrix and local limit theorems for some degenerate diffusion processes , 2008, 0802.2229.

[16]  F. Vega-Redondo.  Complex Social Networks: Econometric Society Monographs , 2007 .

[17]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

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

[19]  Jordan Stoyanov,et al.  Simulation and Inference for Stochastic Differential Equations: with R Examples , 2011 .

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

[21]  Pricing Derivative Securities , 2000 .

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

[23]  Andrea Pascucci,et al.  Calibration of a path-dependent volatility model: Empirical tests , 2009, Comput. Stat. Data Anal..

[24]  P. Glasserman,et al.  A Continuity Correction for Discrete Barrier Options , 1997 .

[25]  R. Elliott,et al.  Approximations for the values of american options , 1991 .

[26]  JAN OB LÓJ.  FINE-TUNE YOUR SMILE CORRECTION TO HAGAN ET AL , 2008 .

[27]  S. Basov.  Simulation and Inference for Stochastic Differential Equations: With R Examples , 2010 .

[28]  M. Broadie,et al.  The Valuation of American Options on Multiple Assets , 1997 .

[29]  P. Wilmott,et al.  An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs , 1997 .

[30]  P. Foschi,et al.  Parametrix approximations for non constant coefficient parabolic PDEs , 2008 .

[31]  Andrea Pascucci,et al.  On a class of degenerate parabolic equations of Kolmogorov type , 2005 .

[32]  P. Hagan,et al.  Equivalent Black volatilities , 1999 .

[33]  C. J. Harwood.  Modelling Financial Derivatives with Mathematica , 2000 .

[34]  Martin Widdicks,et al.  THE BLACK‐SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS , 2005 .

[35]  G. Papanicolaou,et al.  Derivatives in Financial Markets with Stochastic Volatility , 2000 .

[36]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[37]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .