A second order L0 stable algorithm for evaluating European options

In this paper, we study the option pricing problem, one of the prominent and challenging problems in computational finance. Using the Pade approximation, we have developed a second order L0 stable discrete parallel algorithm for experimentation on advanced architectures. We have implemented the sequential version of this algorithm and evaluated the European Options. Numerical results are compared with those obtained using other commonly used numerical methods and shown that the new algorithm is robust and efficient than the traditional schemes. Also, using explicit Forward Time Centered Space (FTCS) on the reduced Black-Scholes partial differerential equation, we report pricing of European options. We have done our experiments on a shared memory multiprocessor machine using OpenMP and report a maximum speedup of 3.43 with 16 threads.

[1]  Stavros A. Zenios,et al.  High-performance computing in finance: The last 10 years and the next , 1999, Parallel Comput..

[2]  Robert G. McLeod,et al.  Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs. , 2006, Mathematical biosciences and engineering : MBE.

[3]  A. Mayo.  High-order accurate implicit finite difference method for evaluating American options , 2004 .

[4]  Seyed M. Moghadas,et al.  A qualitative study of a vaccination model with non-linear incidence , 2003, Appl. Math. Comput..

[5]  Abba B. Gumel,et al.  Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation , 2003 .

[6]  Edward H. Twizell,et al.  An unconditionally convergent finite-difference scheme for the SIR model , 2003, Appl. Math. Comput..

[7]  Seyed M. Moghadas,et al.  Dynamical and numerical analyses of a generalized food-chain model , 2003, Appl. Math. Comput..

[8]  HIV control in vivo: Dynamical analysis , 2004 .

[9]  Edward H. Twizell,et al.  Efficient parallel algorithm for the two-dimensional diffusion equation subject to specification of mass , 1997, Int. J. Comput. Math..

[10]  Guang R. Gao,et al.  Multithreaded algorithms for pricing a class of complex options , 2001, Proceedings 15th International Parallel and Distributed Processing Symposium. IPDPS 2001.

[11]  Abba B. Gumel,et al.  Removal of Contrived Chaos in Finite-difference Methods , 2002, Int. J. Comput. Math..

[12]  Abba B. Gumel,et al.  MATHEMATICAL STUDY OF THE IMPACT OF QUARANTINE, ISOLATION AND VACCINATION IN CURTAILING AN EPIDEMIC , 2007 .

[13]  A. B. Gumel,et al.  Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine , 2006, Bulletin of mathematical biology.

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

[15]  Kuldeep Shastri,et al.  Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques , 1985, Journal of Financial and Quantitative Analysis.

[16]  Carlos Castillo-Chavez,et al.  Mathematical studies on human disease dynamics : emerging paradigms and challenges : AMS-IMS-SIAM Joint Summer Research Conference on Modeling the Dynamics of Human Diseases, Emerging Paradigms and Challenges, July 17-21, 2005, Snowbird, Utah , 2006 .

[17]  Abba B. Gumel.  A competitive numerical method for a chemotherapy model of two HIV subtypes , 2002, Appl. Math. Comput..

[18]  Abba B. Gumel,et al.  Numerical and bifurcation analyses for a population model of HIV chemotherapy , 2000 .

[19]  Multiple interacting planar cracks in an anisotropic multilayered medium under an antiplane shear stress : a hypersingular integral approach , 1996 .

[20]  Abba B. Gumel,et al.  A New Mathematical Model for Assessing Therapeutic Strategies for HIV Infection , 2002 .

[21]  Abba B. Gumel,et al.  Mathematical Approaches for Emerging and Re-emerging Infectious Diseases: An Introduction , 2003 .

[22]  A. Gumel,et al.  Effect of a preventive vaccine on the dynamics of HIV transmission , 2004 .

[23]  Baojun Song,et al.  Existence of multiple-stable equilibria for a multi-drug-resistant model of Mycobacterium tuberculosis. , 2008, Mathematical biosciences and engineering : MBE.

[24]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1966 .

[25]  James Watmough,et al.  An sveir model for assessing potential impact of an imperfect anti-sars vaccine. , 2006, Mathematical biosciences and engineering : MBE.

[26]  Richard Baumgartner,et al.  Using multiple data features improved the validity of osteoporosis case ascertainment from administrative databases. , 2008, Journal of clinical epidemiology.

[27]  V. Zakian.  Author's reply to 'Comment on Rational approximants to the matrix exponential' , 1971 .

[28]  Abba B. Gumel,et al.  A boundary integral method for the three-dimensional heat equation subject to specification of energy , 2001 .

[29]  Numerical Methods for a Non-linear System Arising inChemical Kinetics , 1997 .

[30]  James Watmough,et al.  Role of incidence function in vaccine-induced backward bifurcation in some HIV models. , 2007, Mathematical biosciences.

[31]  P. Wilmott,et al.  The Mathematics of Financial Derivatives: Preface , 1995 .

[32]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[33]  Seyed M. Moghadas,et al.  Could Condoms Stop the AIDS Epidemic , 2003 .

[34]  Eduardo S. Schwartz,et al.  The Valuation of American Put Options , 1977 .

[35]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[36]  E. H. Twizell,et al.  A second-order scheme for the “Brusselator” reaction–diffusion system , 1999 .

[37]  Bart W. Stuck,et al.  A Computer and Communication Network Performance Analysis Primer (Prentice Hall, Englewood Cliffs, NJ, 1985; revised, 1987) , 1987, Int. CMG Conference.

[38]  A. Gumel,et al.  The evolutionary consequences of vaccination. , 2008, Vaccine.

[39]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

[40]  Abba B. Gumel,et al.  Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation , 2002 .

[42]  Pei Yu,et al.  Numerical modelling of the perturbation of HIV-1 during combination anti-retroviral therapy , 2001, Comput. Biol. Medicine.

[43]  B. D. Corbett,et al.  SUBTHRESHOLD DOMAIN OF BISTABLE EQUILIBRIA FOR A MODEL OF HIV EPIDEMIOLOGY , 2003 .

[44]  O. Sharomi,et al.  Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs , 2008, Journal of biological dynamics.

[45]  Baojun Song,et al.  Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. , 2008, Mathematical biosciences and engineering : MBE.

[46]  E H Twizell,et al.  L 0 -stable splitting methods for the simple heat equation in two space dimensions with homogeneous boundary conditions , 1986 .

[47]  E. H. Twizell,et al.  Numerical Analysis of Defects Caused by Thermolysis in anInfinite Cylindrical Ceramic Moulding , 1999 .

[48]  Edward H. Twizell,et al.  Higher-order parallel methods for a model of percutaneous drug absorption , 1995, Int. J. Comput. Math..

[49]  Seyed M. Moghadas,et al.  Global stability of a two-stage epidemic model with generalized non-linear incidence , 2002, Math. Comput. Simul..

[50]  Neil Pomphrey,et al.  Diagonal Pade approximations for initial value problems , 1987 .

[51]  Huaiping Zhu,et al.  A mathematical model for assessing control strategies against West Nile virus , 2005, Bulletin of mathematical biology.

[52]  A. Gumel,et al.  To Cut or Not to Cut: A Modeling Approach for Assessing the Role of Male Circumcision in HIV Control , 2007, Bulletin of mathematical biology.

[53]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[54]  Edward H. Twizell,et al.  Parallel Algorithms for Second-order Hyperbolic Equations , 1995, Parallel Algorithms Appl..

[55]  Numerical solutions for a coupled non-linear oscillator , 2000 .

[56]  E. H. Twizell,et al.  A sequential algorithm for the non-linear dual-sorption model of percutaneous drug absorption. , 1998, Mathematical biosciences.

[57]  Salisu M. Garba,et al.  Backward bifurcations in dengue transmission dynamics. , 2008, Mathematical biosciences.

[58]  BIFURCATION AND STABILITY ANALYSES FOR A COUPLED BRUSSELATOR MODEL , 2001 .

[59]  Edward H. Twizell,et al.  Parallel Algorithms for Fourth-order Parabolic Equations , 1995, Parallel Algorithms Appl..

[60]  R. ZVANy,et al.  A GENERAL FINITE ELEMENT APPROACH FOR PDE OPTION PRICING MODELS , 2007 .

[61]  Curt Randall,et al.  Pricing Financial Instruments: The Finite Difference Method , 2000 .

[62]  H. Padé.  Sur la représentation approchée d'une fonction par des fractions rationnelles , 1892 .

[63]  M. E. Alexander,et al.  A Vaccination Model for Transmission Dynamics of Influenza , 2004, SIAM J. Appl. Dyn. Syst..

[64]  Abba B. Gumel,et al.  A Non-Standard Finite-Difference Scheme for a Model of HIV Transmission and Control , 2003 .

[65]  Abba B. Gumel,et al.  On the numerical solution of the diffusion equation subject to the specification of mass , 1999, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[66]  Sequential and parallel methods for solving first-order hyperbolic equations , 1996 .

[67]  Elamin H Elbasha,et al.  Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits , 2006, Bulletin of mathematical biology.

[68]  A. Gumel,et al.  Protecting residential care facilities from pandemic influenza , 2008, Proceedings of the National Academy of Sciences.

[69]  M. E. Alexander,et al.  Modelling strategies for controlling SARS outbreaks , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[70]  Georges Courtadon,et al.  A More Accurate Finite Difference Approximation for the Valuation of Options , 1982, Journal of Financial and Quantitative Analysis.

[71]  Seyed M. Moghadas,et al.  A mathematical study of a model for childhood diseases with non-permanent immunity , 2003 .