General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Levy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases.

[1]  Eric Benhamou,et al.  Fast Fourier Transform for Discrete Asian Options , 2000 .

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

[3]  Alexander Novikov,et al.  Lower and upper bounds for prices of Asian-type options , 2014 .

[4]  Jan Kallsen,et al.  A Didactic Note on Affine Stochastic Volatility Models , 2006 .

[5]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[6]  Stewart Hodges,et al.  A general closed-form spread option pricing formula , 2013 .

[7]  Kendall Kim Electronic and Algorithmic Trading Technology: The Complete Guide , 2007 .

[8]  N. Shephard,et al.  Some recent developments in stochastic volatility modelling , 2002 .

[9]  A. Cerný,et al.  An Improved Convolution Algorithm for Discretely Sampled Asian Options , 2011 .

[10]  J. Andreasen The pricing of discretely sampled Asian and lookback options: a change of numeraire approach , 1998 .

[11]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

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

[13]  M. Yor,et al.  Stochastic Volatility for Lévy Processes , 2003 .

[14]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[15]  Serge Darolles,et al.  Improving VWAP. Strategies: A Dynamical Volume Approach , 2006 .

[16]  E. Nicolato,et al.  Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type , 2003 .

[17]  Paul Glasserman,et al.  Sensitivity estimates from characteristic functions , 2007, 2007 Winter Simulation Conference.

[18]  Jean-Charles Rochet,et al.  Changes of numéraire, changes of probability measure and option pricing , 1995, Journal of Applied Probability.

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

[20]  William T. Shaw Modelling Financial Derivatives with MATHEMATICA , 1999 .

[21]  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.

[22]  Ioannis Kyriakou,et al.  Monte Carlo Simulation of the CGMY Process and Option Pricing , 2013 .

[23]  Micah S. Officer The Market Pricing of Implicit Options in Merger Collars , 2006 .

[24]  J. Wan,et al.  Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation , 2010 .

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

[26]  Noel A Cressie,et al.  The moment generating function has its moments , 1986 .

[27]  R. Lord Partially Exact and Bounded Approximations for Arithmetic Asian Options , 2005 .

[28]  Sai Hung Marten Ting,et al.  Asian and Australian options: A common perspective , 2013 .

[29]  A. Yamazaki Pricing average options under time-changed Lévy processes , 2014 .

[30]  David S. Bates Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[31]  E. Eberlein,et al.  Equivalence of floating and fixed strike Asian and lookback options , 2005 .

[32]  Jan Kallsen,et al.  Variance-Optimal Hedging in General Affine Stochastic Volatility Models , 2010, Advances in Applied Probability.

[33]  Cornelis W. Oosterlee,et al.  Efficient Pricing of European-Style Asian Options under Exponential Lévy Processes Based on Fourier Cosine Expansions , 2013, SIAM J. Financial Math..

[34]  Chao Shi,et al.  Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models , 2014, Math. Oper. Res..

[35]  A. Meucci,et al.  Pricing discretely monitored Asian options under Levy processes , 2008 .

[36]  Dawn Hunter Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods , 2000 .

[37]  Leif Andersen Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[38]  W. Schoutens Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

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

[40]  Gianluca Fusai Pricing Asian options via Fourier and Laplace transforms , 2004 .

[41]  K. Sandmann,et al.  Pricing Bounds on Asian Options , 2003, Journal of Financial and Quantitative Analysis.

[42]  Michael Curran Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price , 1994 .

[43]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[44]  Gianluca Fusai,et al.  Pricing exotic derivatives exploiting structure , 2014, Eur. J. Oper. Res..

[45]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

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

[47]  Mark Broadie,et al.  Exact simulation of option Greeks under stochastic volatility and jump diffusion models , 2004, Proceedings of the 2004 Winter Simulation Conference, 2004..

[48]  Chihoon Lee,et al.  Single-Transform Formulas for Pricing Asian Options in a General Approximation Framework under Markov Processes , 2015 .

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

[50]  Friedrich Hubalek,et al.  Geometric Asian option pricing in general affine stochastic volatility models with jumps , 2014, 1407.2514.

[51]  S. Heston A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[52]  Lorenzo Mercuri PRICING ASIAN OPTIONS IN AFFINE GARCH MODELS , 2011 .

[53]  Gianluca Fusai,et al.  Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets , 2008 .

[54]  G. Thompson Fast narrow bounds on the value of Asian options , 2002 .

[55]  S. Zahra,et al.  Asian real option: New approach to project economic valuation , 2012, 2012 International Conference on Information Management, Innovation Management and Industrial Engineering.

[56]  A. D. Schepper,et al.  Pricing bounds for discrete arithmetic Asian options under Lévy models , 2010 .

[57]  W. Whitt,et al.  Multidimensional Transform Inversion with Applications to the Transient M/G/1 Queue , 1994 .

[58]  David C. Emanuel,et al.  Further Results on the Constant Elasticity of Variance Call Option Pricing Model , 1982, Journal of Financial and Quantitative Analysis.

[59]  Steven Kou,et al.  A General Framework for Pricing Asian Options Under Markov Processes , 2015, Oper. Res..

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

[61]  Jan Vecer,et al.  Pricing Asian options in a semimartingale model , 2004 .

[62]  T Driouchi,et al.  Capacity planning under uncertainty: an Asian option approach , 2006 .

[63]  Guofu Zhou,et al.  Technical analysis: An asset allocation perspective on the use of moving averages , 2009 .

[64]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

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

[66]  Luc Devroye,et al.  On the computer generation of random variables with a given characteristic function , 1981 .

[67]  Gianluca Fusai,et al.  Asian Options with Jumps , 2013 .

[68]  M. Bartlett The Characteristic Function of a Conditional Statistic , 1938 .

[69]  Bernard Lapeyre,et al.  Introduction to Stochastic Calculus Applied to Finance , 2007 .