Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

We present a numerical scheme to calculate fluctuation identities for exponential Levy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Levy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener–Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-z domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

[1]  Steven G. Johnson,et al.  FFTW: an adaptive software architecture for the FFT , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[2]  Satya N. Majumdar,et al.  Persistence and first-passage properties in nonequilibrium systems , 2013, 1304.1195.

[3]  Liming Feng,et al.  PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH , 2008 .

[4]  Alexey Kuznetsov,et al.  Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes , 2010, 1011.1790.

[5]  L. Rogers.  Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains , 1994 .

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

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

[8]  Onno J. Boxma,et al.  Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks , 1996, Queueing Syst. Theory Appl..

[9]  Ward Whitt,et al.  Numerical inversion of probability generating functions , 1992, Oper. Res. Lett..

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

[11]  Guido Germano,et al.  Hilbert transform, spectral filters and option pricing , 2019, Ann. Oper. Res..

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

[13]  I. D. Abrahams,et al.  THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING , 2010 .

[14]  Ward Whitt,et al.  Numerical Inversion of Laplace Transforms of Probability Distributions , 1995, INFORMS J. Comput..

[15]  F. Spitzer.  A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[16]  I. David Abrahams,et al.  An exact analytical solution for discrete barrier options , 2006, Finance Stochastics.

[17]  W. Grassmann.  Chapter 5 Computational methods in probability theory , 1990 .

[18]  Dov Pekelman,et al.  LIFO Inventory Systems , 1978 .

[19]  Winfried K. Grassmann,et al.  Numerical Solutions of the Waiting Time Distribution and Idle Time Distribution of the Arithmetic GI/G/1 Queue , 1989, Oper. Res..

[20]  Cornelis W. Oosterlee,et al.  Pricing early-exercise and discrete barrier options by fourier-cosine series expansions , 2009, Numerische Mathematik.

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

[22]  J. W. Cohen.  The Wiener-Hopf Technique in Applied Probability , 1975 .

[23]  Vadim Linetsky,et al.  Computing exponential moments of the discrete maximum of a Lévy process and lookback options , 2008, Finance Stochastics.

[24]  Guido Germano,et al.  Pricing Credit Derivatives in a Wiener–Hopf Framework , 2012 .

[25]  C. Oosterlee,et al.  On the Application of Spectral Filters in a Fourier Option Pricing Technique , 2015 .

[26]  Guido Germano,et al.  Hilbert transform, spectral filters and option pricing , 2017, Ann. Oper. Res..

[27]  Zhenyu Cui,et al.  Equity-Linked Annuity Pricing with Cliquet-Style Guarantees in Regime-Switching and Stochastic Volatility Models with Jumps , 2017 .

[28]  Finbarr Murphy,et al.  Topics in numerical methods for finance , 2012 .

[29]  Yichun Chi,et al.  On the Threshold Dividend Strategy for a Generalized Jump-Diffusion Risk Model , 2009 .

[30]  Guido Germano,et al.  Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options , 2016, Eur. J. Oper. Res..

[31]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[32]  I. B. Russak,et al.  Manifolds , 2019, Spacetime and Geometry.

[33]  J. Kirkby.  Robust Barrier Option Pricing by Frame Projection Under Exponential Levy Dynamics , 2014 .

[34]  Herman Chernoff,et al.  Sequential Tests for the Mean of a Normal Distribution , 1965 .

[35]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

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

[37]  J. H. B. Kemperman.  A WIENER-HOPF TYPE METHOD FOR A GENERAL RANDOM WALK WITH A TWO-SIDED BOUNDARY' , 1963 .

[38]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[39]  M. Donsker,et al.  On the distribution of the supremum functional for processes with stationary independent increments , 1957 .

[40]  Cornelis W. Oosterlee,et al.  A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions , 2008, SIAM J. Sci. Comput..

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

[42]  Claudia Kluppelberg,et al.  Ruin probabilities and overshoots for general Lévy insurance risk processes , 2004 .

[43]  Alan L. Lewis.  A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes , 2001 .

[44]  B. S. Sathyaprakash,et al.  A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries , 2010, 1003.2939.

[45]  Sergei Levendorskii,et al.  Pricing Discrete Barrier Options and Credit Default Swaps Under Levy Processes , 2012 .

[46]  Hervé Vandeven,et al.  Family of spectral filters for discontinuous problems , 1991 .

[47]  N. U. Prabhu.  Wiener-Hopf Techniques in Queueing Theory , 1974 .