Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns

This paper proposes a stylized model that reconciles several seemingly conflicting findings on financial security returns and option prices. The model is based on a pure jump Levy process, wherein the jump arrival rate obeys a power law dampened by an exponential function. The model allows for different degrees of dampening for positive and negative jumps, and also different pricing for upside and downside market risks. Calibration of the model to the S&P 500 index shows that the market charges only a moderate premium on upward index movements, but the maximally allowable premium on downward index movements.

[1]  Svetlana Boyarchenko,et al.  OPTION PRICING FOR TRUNCATED LÉVY PROCESSES , 2000 .

[2]  R. Miller,et al.  On the Stable Paretian Behavior of Stock-Market Prices , 1974 .

[3]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[4]  R. Engle,et al.  Empirical Pricing Kernels , 1999 .

[5]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study , 2001 .

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

[7]  Laurent E. Calvet,et al.  Multifractality in Asset Returns: Theory and Evidence , 2002, Review of Economics and Statistics.

[8]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[9]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[10]  M. Yor,et al.  Stochastic Volatility for Levy Processes , 2001 .

[11]  A. Coughlan,et al.  An Empirical Investigation * , 2002 .

[12]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[13]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[14]  A. Barnea,et al.  A Reexamination of the Empirical Distribution of Stock Price Changes , 1973 .

[15]  Jan Kallsen,et al.  The cumulant process and Esscher's change of measure , 2002, Finance Stochastics.

[16]  A. Weron,et al.  Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes , 1993 .

[17]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[18]  A. Lo,et al.  Nonparametric Estimation of State‐Price Densities Implicit in Financial Asset Prices , 1998 .

[19]  Liuren Wu,et al.  Crash-O-Phobia: A Domestic Fear or a Worldwide Concern? , 2005 .

[20]  David S. Bates.  Post-'87 crash fears in the S&P 500 futures option market , 2000 .

[21]  R. Hagerman.  More Evidence on the Distribution of Security Returns , 1978 .

[22]  J. Jackwerth.  Recovering Risk Aversion from Option Prices and Realized Returns , 1998 .

[23]  Jun Pan.  The Jump-Risk Premia Implicit in Options : Evidence from an Integrated Time-Series Study , 2001 .

[24]  M. Brenner.  On the Stability of the Distribution of the Market Component in Stock Price Changes , 1974, Journal of Financial and Quantitative Analysis.

[25]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[26]  Y. Miyahara.  Minimal Entropy Martingale Measures of Jump Type Price Processes in Incomplete Assets Markets , 1999 .

[27]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

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

[29]  S. Irwin,et al.  The Distribution of Futures Prices: A Test of the Stable Paretian and Mixture of Normals Hypotheses , 1989, Journal of Financial and Quantitative Analysis.

[30]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[31]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[32]  D. Madan,et al.  1option Pricing with V. G. Martingale Components , 1991 .

[33]  P. Carr,et al.  The Finite Moment Log Stable Process and Option Pricing , 2003 .

[34]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

[35]  Ultra-slow convergence to a Gaussian: The truncated Lévy flight , 1995 .

[36]  John Teichmoeller.  A Note on the Distribution of Stock Price Changes , 1971 .

[37]  Bruce D. Fielitz,et al.  Stable Distributions and the Mixtures of Distributions Hypotheses for Common Stock Returns , 1983 .

[38]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[39]  Feng Li.  Option Pricing , 2000 .

[40]  Liuren Wu,et al.  Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes , 2003 .

[41]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study $ , 2002 .

[42]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[43]  Hans U. Gerber,et al.  Option pricing by Esscher transforms. , 1995 .

[44]  R. Officer.  The Distribution of Stock Returns , 1972 .

[45]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[46]  Liuren Wu,et al.  Accounting for Biases in Black-Scholes , 2004 .

[47]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .