Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

[1]  Hans-Peter Scheffler,et al.  Stochastic solution of space-time fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[3]  Karina Weron,et al.  Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Jean-Philippe Bouchaud,et al.  A non-Gaussian option pricing model with skew , 2004, cond-mat/0403022.

[5]  Andrey G. Cherstvy,et al.  Random diffusivity from stochastic equations : comparison of two models for Brownian yet non-Gaussian diffusion , 2018 .

[6]  K. L. Sebastian,et al.  Diffusion in a Crowded, Rearranging Environment. , 2016, The journal of physical chemistry. B.

[7]  F. Ren,et al.  Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime , 2012 .

[8]  Roberto Garrappa,et al.  The Prabhakar or three parameter Mittag-Leffler function: Theory and application , 2017, Commun. Nonlinear Sci. Numer. Simul..

[9]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[10]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[11]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[12]  Jim Gatheral,et al.  Pricing under rough volatility , 2015 .

[13]  A. Jurlewicz,et al.  Accelerating subdiffusions governed by multiple-order time-fractional diffusion equations: Stochastic representation by a subordinated Brownian motion and computer simulations , 2013 .

[14]  Fritz Oberhettinger,et al.  Tables of Mellin Transforms , 1974 .

[15]  B. I. Henry,et al.  Continuous Time Random Walks with Reactions Forcing and Trapping , 2013 .

[16]  Marcin Magdziarz,et al.  A weighted finite difference method for subdiffusive Black Scholes Model , 2019, Comput. Math. Appl..

[17]  Sidney Redner,et al.  Random multiplicative processes: An elementary tutorial , 1990 .

[18]  Exact moment scaling from multiplicative noise. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  M. Magdziarz.  Black-Scholes Formula in Subdiffusive Regime , 2009 .

[20]  Holger Kantz,et al.  Distributed-order diffusion equations and multifractality: Models and solutions. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  R. Metzler,et al.  Aging renewal theory and application to random walks , 2013, 1310.1058.

[22]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[23]  William Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 2 , 1967 .

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

[25]  Andrew Matacz,et al.  Financial Modeling and Option Theory with the Truncated Levy Process , 1997, cond-mat/9710197.

[26]  Christopher Angstmann,et al.  Generalized Continuous Time Random Walks, Master Equations, and Fractional Fokker-Planck Equations , 2015, SIAM J. Appl. Math..

[27]  Daniel T. Cassidy,et al.  Pricing European options with a log Students t-distribution: A Gosset formula , 2009, 0906.4092.

[28]  I. Sokolov,et al.  Beyond monofractional kinetics , 2017 .

[29]  L. Kocarev,et al.  Hitting times in turbulent diffusion due to multiplicative noise. , 2020, Physical review. E.

[30]  H. Iemoto.  Modelling the persistence of conditional variances , 1986 .

[31]  Marcin Magdziarz,et al.  Pricing of basket options in subdiffusive fractional Black–Scholes model , 2017 .

[32]  Fogedby.  Langevin equations for continuous time Lévy flights. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  Murad S. Taqqu,et al.  Non-Markovian diffusion equations and processes: Analysis and simulations , 2007, 0712.0240.

[34]  O. Peters.  Optimal leverage from non-ergodicity , 2009, 0902.2965.

[35]  Andrey G. Cherstvy,et al.  Time averaging, ageing and delay analysis of financial time series , 2017 .

[36]  G. Schehr,et al.  Two stock options at the races: Black–Scholes forecasts , 2010, 1005.1760.

[37]  E. Barkai,et al.  Fractional Fokker-Planck equation, solution, and application. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  C. Angstmann,et al.  Time-fractional geometric Brownian motion from continuous time random walks , 2019, Physica A: Statistical Mechanics and its Applications.

[39]  A. Chechkin,et al.  From continuous time random walks to the generalized diffusion equation , 2018 .

[40]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[41]  L. Kocarev,et al.  OPTION PRICING WITH HEAVY-TAILED DISTRIBUTIONS OF LOGARITHMIC RETURNS , 2018, International Journal of Theoretical and Applied Finance.

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

[43]  Ljupco Kocarev,et al.  Cooperation dynamics in networked geometric Brownian motion. , 2019, Physical review. E.

[44]  H. Kantz,et al.  Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel , 2015 .

[45]  Bruno Dupire.  Pricing with a Smile , 1994 .

[46]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[47]  J. Klafter,et al.  Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: the case of a time-dependent force. , 2008, Physical review letters.

[48]  E. Barkai,et al.  Infinite ergodic theory for heterogeneous diffusion processes. , 2018, Physical review. E.

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

[50]  D. Sornette,et al.  Stock Market Crashes, Precursors and Replicas , 1995, cond-mat/9510036.

[51]  I. M. Sokolov,et al.  Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities , 2016, 1611.06202.

[52]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[53]  J. Hull.  Options, Futures, and Other Derivatives , 1989 .

[54]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[55]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[56]  J. Hull.  Fundamentals of Futures and Options Markets , 2001 .

[57]  Kevin E. Bassler,et al.  Hurst exponents, Markov processes, and fractional Brownian motion , 2006, cond-mat/0609671.

[58]  A. Iomin,et al.  Superdiffusion on a comb structure. , 2004, Physical review letters.

[59]  H. G. E. Hentschel,et al.  Fractal nature of turbulence as manifested in turbulent diffusion , 1983 .

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

[61]  Gianni Pagnini,et al.  Characterizations and simulations of a class of stochastic processes to model anomalous diffusion , 2008, 0801.4879.

[62]  J. Aitchison,et al.  The lognormal distribution : with special reference to its uses in economics , 1957 .

[63]  Nassim Nicholas Taleb,et al.  The Black Swan: The Impact of the Highly Improbable , 2007 .

[64]  L. Moriconi.  Delta hedged option valuation with underlying non-Gaussian returns , 2007 .

[65]  F. Seno,et al.  Unexpected crossovers in correlated random-diffusivity processes , 2020, New Journal of Physics.

[66]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[67]  O. Peters,et al.  Ergodicity breaking in geometric Brownian motion. , 2012, Physical review letters.

[68]  Chao-Wen Li.  Option pricing with generalized continuous time random walk models , 2016 .

[69]  M. Magdziarz,et al.  Anomalous dynamics of Black–Scholes model time-changed by inverse subordinators , 2012 .

[70]  A. Wyłomańska,et al.  Geometric Brownian Motion with Tempered Stable Waiting Times , 2012 .

[71]  Omar El Euch.  Quantitative Finance under rough volatility , 2018 .

[72]  On option pricing models in the presence of heavy tails , 2007 .

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

[74]  Lasko Basnarkov,et al.  Evolution of cooperation in populations with heterogeneous multiplicative resource dynamics , 2019, 1912.09205.

[75]  Andrey G. Cherstvy,et al.  Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise. , 2020, Physical review. E.

[76]  M. Heidernätsch.  On the diffusion in inhomogeneous systems , 2014 .

[77]  Giacomo Bormetti,et al.  Minimal model of financial stylized facts. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[78]  L. Borland.  A theory of non-Gaussian option pricing , 2002, cond-mat/0205078.