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Pricing discrete double barrier options with a numerical method

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Abstract

Most of traded double barrier options are monitored in discrete time, their pricing being more challenging than in continuous time. A few solutions are analytical with a correction for continuity; the remaining solutions are numerical with lattices, grids or Monte Carlo (MC) simulation. We apply an original variance reduction technique to the pricing of European double barrier options monitored in discrete time. This technique speeds up significantly the MC simulation. The computational algorithm repeats each experiment with an increasing number of trials at a logarithmic rate and calculates a weighted average of the options values. We test the accuracy of the Login (Logarithmic Increment) model with an analytical solution adjusted for discretization and a naïve numerical model using Monte Carlo simulation. We show that the Login model accurately prices double barrier options. Market participants in need of selecting a reliable numerical method for pricing double barrier options monitored in discrete time will find our article appealing. Moreover, the idea behind the method is simple and can be applied to the pricing of plain-vanilla or more complex derivatives, easing and speeding the valuation step significantly.

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References

  • Ahn, D.H., Figlewski, S. and Gao, B. (1999) Pricing discrete barrier options with an adaptive mesh model. Journal of Derivatives 6 (4): 33–43.

    Article  Google Scholar 

  • Antonov, I.A. and Saleev, V.M. (1979) An economic method of computing LP tau-sequences. USSR Computational Mathematics and Mathematical Physics 19 (1): 252–256.

    Article  Google Scholar 

  • Asmussen, S. and Glynn, P.W. (2007) Stochastic Simulation, New York: Springer-Verlag.

    Google Scholar 

  • Berry, A.C. (1941) The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society 49 (1): 122–136.

    Article  Google Scholar 

  • Boyle, P.P. (1977) Options: A Monte Carlo approach. Journal of Financial Economics 4 (3): 323–338.

    Article  Google Scholar 

  • Boyle, P.P. and Lau, S.H. (1994) Bumping up against the barrier with the binomial method. Journal of Derivatives 1 (4): 6–14.

    Article  Google Scholar 

  • Bratley, P. and Fox, B. (1988) Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software 14 (1): 88–100.

    Article  Google Scholar 

  • Bratley, P., Fox, B. and Niederreiter, H. (1994) Algorithm 738: Programs to generate niederreiter’s low-discrepancy sequences. ACM Transactions on Mathematical Software 20 (4): 494–495.

    Article  Google Scholar 

  • Broadie, M., Glasserman, P. and Kou, S. (1997) A continuity correction for discrete barrier options. Mathematical Finance 7 (4): 325–348.

    Article  Google Scholar 

  • Burkardt, J. (2012) The sobol quasirandom sequence. http://people.sc.fsu.edu/~jburkardt/m_src/sobol/sobol.html, accessed 22 February 2015.

  • Cheuk, T. and Vorst, T. (1996) Complex barrier options. Journal of Derivatives 4 (1): 8–22.

    Article  Google Scholar 

  • Clewlow, L. and Carverhill, A. (1992) Efficient Monte Carlo Valuation and Hedging of Contingent Claims. Technical Report, Financial Options Research Center, University of Warwick, Coventry, U.K., 92/30.

  • Clewlow, L. and Carverhill, A. (1994) Quicker on the curves. Risk 7 (5): 63–64.

    Google Scholar 

  • Clewlow, L. and Strickland, C. (2000) Implementing Derivative Models, Chichester, UK: Wiley Publications.

    Google Scholar 

  • Costabile, M. (2001) A discrete-time algorithm for pricing double barrier options. Decisions in Economics and Finance 24 (1): 49–59.

    Article  Google Scholar 

  • Cox, J.C., Ross, S.A. and Rubinstein, M. (1979) Option pricing: A simplified approach. Journal of Financial Economics 7 (3): 229–263.

    Article  Google Scholar 

  • Cox, J.C. and Rubinstein, M. (1985) Options Markets, Englewood Cliffs, NJ: Prentice-Hall Publications.

    Google Scholar 

  • Curran, M. (1994) Rata gems. Risk 7 (3): 70–71.

    Google Scholar 

  • Dupuis, P.G., Leder, K. and Wang, H. (2007) Importance sampling for sums of random variables with heavy tails. ACM Transactions on Modeling and Computer Simulation 17 (3): 1–21.

    Article  Google Scholar 

  • Esseen, C.-G. (1942) On the Liapunoff limit of error in the theory of probability. Arkiv för matematik, astronomi och fysik A28: 1–19. ISSN 0365-4133.

  • Esseen, C.-G. (1956) A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr 39 (1): 160–170.

    Google Scholar 

  • Faure, H. (2006) Selection Criteria for (Random) Generation of Digital (0, s)-Sequences. Working Paper, Institut de Mathématiques de Luminy, UMR 6206 CNRS, Marseille, France.

  • Feng, L. and Linetsky, V. (2008) Pricing discretely monitored barrier options and defaultable bonds in Levy process models: A fast Hilbert transform approach. Mathematical Finance 18 (3): 337–384.

    Article  Google Scholar 

  • Fox, B. (1986) Algorithm 647: Implementation and relative efficiency of quasirandom sequence generators. ACM Transactions on Mathematical Software 12 (4): 362–376.

    Article  Google Scholar 

  • Fusai, G. and Roncoroni, A. (2008) Implementing Models in Quantitative Finance: Methods and Cases, Berlin, Germany: Springer Finance.

    Google Scholar 

  • Geman, H. and Yor, M. (1996) Pricing and hedging double barrier options: A probabilistic approach. Mathematical Finance 6 (4): 365–378.

    Article  Google Scholar 

  • Giles, M.B. (2008) Multilevel Monte Carlo path simulation. Operations Research 56 (3): 607–617.

    Article  Google Scholar 

  • Glasserman, P., Heidelberger, P., Shahabuddin, P. and Zajic, T. (1999) Multilevel splitting for estimating rare event probabilities. Operations Research 47 (4): 585–600.

    Article  Google Scholar 

  • Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000) Efficient Monte Carlo methods for Value at Risk, https://www0.gsb.columbia.edu/faculty/pglasserman/Other/masteringrisk.pdf, accessed 22 February 2015.

  • Glasserman, P. (2003) Monte Carlo Methods in Financial Engineering, New York: Springer-Verlag.

    Book  Google Scholar 

  • Golbabai, A., Ballestra, L.V. and Ahmadian, D. (2014) A highly accurate finite element method to price discrete double barrier options. Computational Economics 44 (2): 153–173.

    Article  Google Scholar 

  • Halton, J. (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2 (1): 84–90.

    Article  Google Scholar 

  • Halton, J. and Smith, G.B. (1964) Algorithm 247: Radical-inverse quasi-random point sequence. Communications of the ACM 7 (12): 701–702.

    Article  Google Scholar 

  • Hammersley, J.M. and Mortonal, K.W. (1956) A new Monte Carlo technique: Antithetic variates. Mathematical Proceedings of the Cambridge Philosophical Society 52 (3): 449–475.

    Article  Google Scholar 

  • Härdle, W., Kleinow, T. and Stahl, G. (2002) Applied Quantitative Finance – Theory and Computational Tools, Berlin, Germany: Springer Verlag.

    Google Scholar 

  • Haug, E. (2006) The Complete Guide to Option Pricing Formulas, New York: McGraw-Hill Publications.

    Google Scholar 

  • Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (2): 327–343.

    Article  Google Scholar 

  • Howison, S. and Steinberg, M. (2007) A matched asymptotic expansions approach to continuity corrections for discretely sampled options. 1: Barrier options. Applied Mathematical Finance 14 (1): 63–89.

    Article  Google Scholar 

  • Hull, J. and White, A. (1988) The use of control variate techniques in option pricing. Journal of Financial and Quantitative Analysis. 23 (3): 237–251.

    Article  Google Scholar 

  • Huynh, H.T., Lai, V.S. and Soumar, I. (2008) Stochastic Simulation and Applications in Finance with Matlab Programs, Chichester, UK: John Wiley & Sons.

    Book  Google Scholar 

  • Ikeda, M. and Kunimoto, N. (1992) Pricing options with curve boundaries. Mathematical Finance 2 (4): 275–298.

    Article  Google Scholar 

  • Joy, C., Boyle, P.P. and Tan, K.S. (1996) Quasi-Monte Carlo methods in numerical finance. Management Science 42 (6): 926–938.

    Article  Google Scholar 

  • Kleijnen, J., Ridder, A. and Rubinstein, R. (2010) Variance Reduction Techniques in Monte Carlo Methods, SSRN Working Papers, Discussion Paper Series No. 2010-117, Tilburg University, Center for Economic Research, https://ideas.repec.org/p/tiu/tiucen/87680d1a-53c1-4107-ada4-71b0a2bf46a0.html, accessed 22 February 2015.

  • Kloeden, P.E. and Platen, E. (1992) Numerical solution of stochastic differential equations, Berlin Heidelberg, Germany: Springer-Verlag.

    Book  Google Scholar 

  • Kocis, L. and Whiten, W. (1997) Computational investigations of low-discrepancy sequences. ACM Transactions on Mathematical Software 23 (2): 266–294.

    Article  Google Scholar 

  • Korn, R., Korn, E. and Kroisandt, G. (2010) Monte Carlo Methods and Models in Finance and Insurance, Boca Raton, FL: Chapman and Hall.

    Book  Google Scholar 

  • Lagoze, C. and Van de Sompel, H. (2007) Compound information objects: The OAI-ORE perspective. White Paper for the Open Archives Initiative, http://www.openarchives.org/ore/documents/CompoundObjects-200705.html, accessed 19 April 2014.

  • Lepage, P. (1978) A new algorithm for adaptive multidimensional integration. Journal of Computational Physics 27 (2): 192–203.

    Article  Google Scholar 

  • Liu, J.S. and Chen, R. (1998) Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association 93 (443): 1032–1044.

    Article  Google Scholar 

  • Longstaff, F.A. and Schwartz, E.S. (2001) Valuing American options by simulations: A simple least-squares approach. Review of Financial Studies 14 (1): 113–147.

    Article  Google Scholar 

  • Matsumoto, M. and Nishimura, T. (1998) Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM transactions on Modeling and Computer simulation 8 (1): 3–30.

    Article  Google Scholar 

  • McKaya, M.D., Beckmana, R.J. and Conoverb, W.J. (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 (2): 239–245.

    Google Scholar 

  • McLeish, D.L. and Rollans, S. (1992) Conditioning for variance reduction in estimating the sensitivity of simulations. Annals of Operations Research 39 (1): 157–172.

    Article  Google Scholar 

  • Merton, R.C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 (1): 125–144.

    Article  Google Scholar 

  • Milstein, G.N. (1988) Numerical Integration of Stochastic Differential Equations. in Russian Sverdlovsk, Russia: Urals University Press.

    Google Scholar 

  • Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods, Philadelphia, PA: SIAM.

    Book  Google Scholar 

  • Park, S.K. and Miller, K.W. (1988) Random number generators: Good ones are hard to find. Communications of the ACM 31 (10): 1192–1201.

    Article  Google Scholar 

  • Perninge, M., Amelin, M. and Knazkins, V. (2008) Comparing variance reduction techniques for Monte Carlo simulation of trading and Security in a three-area power system. Transmission and Distribution Conference and Exposition: Bogota, Latin America, 2008 IEEE/PES, DOI: 10.119/TDC-LA.2008.464173.

  • Platen, E. and Bruti-Liberati, N. (2010) Numerical solution of stochastic differential equations with jumps in finance. Stochastic Modelling and Applied Probability 64 (1): 637–695.

    Article  Google Scholar 

  • Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran, The Art of Scientific Computing, New York: Cambridge University Press.

    Google Scholar 

  • Rostan, P. and Rostan, A. (2013) Testing quasi-random versus pseudorandom numbers on bond option pricing. Aestimatio, the IEB International Journal of Finance 6 (1): 96–115.

    Google Scholar 

  • Rostan, P., Rostan, A. and Racicot, F.E. (2014) A probabilistic Monte Carlo model for pricing discrete barrier options and compound real options. The Journal of Derivatives and Hedge Funds 20 (2): 113–126.

    Article  Google Scholar 

  • Sabuncuoglu, I., Fadiloglu, M. and Çelik, S. (2008) Variance reduction techniques: Experimental comparison and analysis for single systems. IIE Transactions 40 (5): 538–551.

    Article  Google Scholar 

  • Salta, E.R. (2008) Variance reduction techniques in pricing financial derivatives. The Florida State University, Tallahassee, FL, U.S.A., UMI Dissertations Publishing, Document 3348535.

  • Shiraya, K., Takahashi, A. and Yamada, T. (2012) Pricing discrete barrier options under stochastic volatility. Asia-Pacific Financial Markets 19 (3): 205–232.

    Article  Google Scholar 

  • Sobol, I.M. and Levitan, Y.L. (1976) The Production of Points Uniformly Distributed in a Multidimensional Cube. (in Russian) Moscow, Russia: Preprint IPM Akademii Nauk SSSR, 40.

    Google Scholar 

  • Sobol, I.M. (1977) Uniformly distributed sequences with an additional uniform property. USSR Computational Mathematics and Mathematical Physics 16 (5): 236–242.

    Article  Google Scholar 

  • Tchebichef, P. (1846) Démonstration élémentaire d’une proposition générale de la théorie des probabilités. Journal für die reine und angewandte Mathematik (Crelles Journal) 33 (1): 259–267.

    Article  Google Scholar 

  • Tezuka, S. (1993) Polynomial arithmetic analogue of Halton sequences. ACM Transactions on Modeling and Computer Simulation 3 (2): 99–107.

    Article  Google Scholar 

  • Tian, X. and Benkrid, K. (2010) High-performance Quasi-Monte Carlo financial simulation: FPGA vs. GPP vs. GPU. Journal of Transactions on Reconfigurable Technology and Systems 3 (4): 26.

    Google Scholar 

  • Traub, J.F. and Paskov, S. (1995) Faster evaluation of financial derivatives. Journal of Portfolio Management 22 (1): 113–120.

    Article  Google Scholar 

  • Wade, B.A., Khaliq, A.Q.M., Yousuf, M., Vigo-Aguiar, J. and Deininger, R. (2007) On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options. Journal of Computational and Applied Mathematics 204 (1): 144–158.

    Article  Google Scholar 

  • Wright, R.D. and Ramsay Jr. T.E. (1979) On the effectiveness of common random numbers. Management Science 25 (7): 649–656.

    Article  Google Scholar 

  • Yousuf, M. (2008) Efficient smoothing of Crank-Nicolson method for pricing barrier options under stochastic volatility, Published Online: 10 July doi: 10.1002/pamm.200700249, 1081101–1081102.

  • Yousuf, M. (2009) A fourth-order smoothing scheme for pricing barrier options under stochastic volatility. International Journal of Computer Mathematics 86 (6): 1054–1067.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Management, American University in Cairo, AUC Avenue, PO Box 74, New Cairo, 11835, Egypt

    Pierre Rostan

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  1. Pierre Rostan
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  2. Alexandra Rostan
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  3. François-Éric Racicot
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Correspondence to Pierre Rostan.

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1worked for more than 12 years in industry, having developed expertise in derivatives and risk management. He has been New Products Manager in the R&D Department of the Montreal Exchange, Canada; Risk Consultant at APT Consulting Paris and Analyst at Clearstream Luxembourg. He holds a PhD in Administration from the University of Quebec, Canada. His major areas of research are numerical methods applied to derivatives and yield curve forecasting. As Associate Professor at the American University in Cairo, Egypt, he teaches courses in the area of financial markets.

2has international experience in financial analysis and consulting (Derivatives Analyst at the Montreal Exchange, Consultant in Business Creation in Paris, France) as well as experience in teaching on four continents: at the University of Quebec, Canada; SRMS International Business School, Lucknow, India; Royal Melbourne Institute of Technology, Vietnam; Montpellier Business School, France and American University, Cairo, Egypt. She holds a Master 2 in Management from the University of Nantes, France. Her major topics of research focus on derivative products.

3PhD, is Associate Professor of Finance at the Telfer School of Management, University of Ottawa. His research interests focus on the problems of measurement errors, specification errors and endogeneity in financial models of returns. He is also interested in developing new methods for forecasting financial time series, especially with regard to hedge fund risk. He has published several books and many articles on quantitative finance and financial econometrics.

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Rostan, P., Rostan, A. & Racicot, FÉ. Pricing discrete double barrier options with a numerical method. J Asset Manag 16, 243–271 (2015). https://doi.org/10.1057/jam.2015.6

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