Repeated convolution and truncation of a truncated fat-tailed distribution, instead of Monte Carlo simulation, for pricing a discrete, simple barrier option is presented. The parameters for the truncated fat-tailed distribution are obtained by fitting the sum of a Student’s t distribution plus a normal distribution to the one-day returns obtained from the adjusted closing values for the S&P 500. It is argued that truncation of the fat-tailed, one-day returns distribution is a physically reasonable action and evidence to support this truncation is provided. The steps to price a discrete up-and-out barrier on an European call option through repeated convolution and truncation are given.
References
[1]
Kabby, W. (2022) The Valuations of Barrier Options: A Method Review. Revue Maghrébine Management des Organisations, 7, 1-14.
[2]
Wang, H., Zhang, J. and Zhou, K. (2022) On Pricing of Vulnerable Barrier Options and Vulnerable Double Barrier Options. Finance Research Letters, 44, Article 102100. https://doi.org/10.1016/j.frl.2021.102100
[3]
Blanda, V. (2023) FX Barrier Option Pricing. MA Thesis, Imperial College.
[4]
Guarch Termens, M.E. (2022) Pricing Methods for Barrier Options. MA Thesis, Universitat de Barcelona.
[5]
Nouri, K. and Abbasi, B. (2017) Implementation of the Modified Monte Carlo Simulation for Evaluate the Barrier Option Prices. Journal of Taibah University for Science, 11, 233-240. https://doi.org/10.1016/j.jtusci.2015.02.010
[6]
Shevchenko, P.V. and Del Moral, P. (2015) Valuation of Barrier Options Using Sequential Monte Carlo. Journal of Computational Finance. https://doi.org/10.2139/ssrn.2529539
[7]
Li, Y. and Yan, K. (2023) Prediction of Barrier Option Price Based on Antithetic Monte Carlo and Machine Learning Methods. Cloud Computing and Data Science, 4, 77-86. https://doi.org/10.37256/ccds.4120232110
[8]
Lee, H., Lee, G. and Song, S. (2022) Multi‐Step Reflection Principle and Barrier Options. Journal of Futures Markets, 42, 692-721. https://doi.org/10.1002/fut.22306
[9]
Guardasoni, C., Rodrigo, M.R. and Sanfelici, S. (2018) A Mellin Transform Approach to Barrier Option Pricing. IMA Journal of Management Mathematics, 31, 49-67. https://doi.org/10.1093/imaman/dpy016
[10]
Mandelbrot, B. (1963) The Variation of Certain Speculative Prices. The Journal of Business, 36, 394-419. https://doi.org/10.1086/294632
[11]
Fama, E.F. (1965) The Behavior of Stock-Market Prices. The Journal of Business, 38, 34-105. https://doi.org/10.1086/294743
[12]
Cont, R. (2001) Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1, 223-236. https://doi.org/10.1080/713665670
[13]
Basnarkov, L., Stojkoski, V., Utkovski, Z. and Kocarev, L. (2019) Option Pricing with Heavy-Tailed Distributions of Logarithmic Returns. International Journal of Theoretical and Applied Finance, 22, Article 1950041. https://doi.org/10.1142/s0219024919500419
[14]
Dreier, I. and Kotz, S. (2002) A Note on the Characteristic Function of the t-Distribution. Statistics & Probability Letters, 57, 221-224. https://doi.org/10.1016/s0167-7152(02)00032-9
[15]
Oppenheim, V. and Schafer, R.V. (1975) Digital Signal Processing. Prentice Hall, Inc.
[16]
Bracewell, R.N. (2000) The Fourier Transform and Its Applications. 3rd Edition, McGraw-Hill Companies.
[17]
Papoulis (1965) Probability, Random Variables, and Stochastic Processes. McGraw Hill, Inc.
[18]
Blattberg, R.C. and Gonedes, N.J. (1974) A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices. The Journal of Business, 47, 244-280. https://doi.org/10.1086/295634
[19]
Amaral, L.A.N., Plerou, V., Gopikrishnan, P., Meyer, M. and Stanley, H.E. (2000) The Distribution of Returns of Stock Prices. International Journal of Theoretical and Applied Finance, 3, 365-369. https://doi.org/10.1142/s0219024900000218
[20]
de Jong, C. and Huisman, R. (2000) From Skews to a Skewed-T: Modelling Option-Implied Returns by a Skewed Student-T. Proceedings of the IEEE/IAFE/INFORMS 2000 Conference on Computational Intelligence for Financial Engineering (CIFEr), New York, 28-28 March 2000, 132-142. https://doi.org/10.1109/cifer.2000.844611
[21]
Platen, E. and Rendek, R. (2008) Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices. Journal of Statistical Theory and Practice, 2, 233-251. https://doi.org/10.1080/15598608.2008.10411873
[22]
Galbraith, J.W. and Zhu, D. (2009) A Generalized Asymmetric Student-t Distribution with Application to Financial Econometrics. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.1499914
[23]
Cassidy, D.T., Hamp, M.J. and Ouyed, R. (2010) Pricing European Options with a Log Student’s t-Distribution: A Gosset Formula. Physica A: Statistical Mechanics and its Applications, 389, 5736-5748. https://doi.org/10.1016/j.physa.2010.08.037
[24]
Cassidy, D.T. (2011) Describing n-Day Returns with Student’s t-Distributions. Physica A: Statistical Mechanics and Its Applications, 390, 2794-2802. https://doi.org/10.1016/j.physa.2011.03.019
[25]
Bevington, P.R. and Robinson, D.K. (1992) Data Reduction and Error Analysis for the Physical Sciences. 2nd Edition, McGraw Hill, Inc.
[26]
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran: The Art of Scientific Computing. 2nd Edition, Cambridge University Press.
[27]
Bouchaud, J.-P. and Sornette, D. (1994) The Black-Scholes Option Pricing Problem in Mathematical Finance: Generalization and Extensions for a Large Class of Stochastic Processes. Journal de Physique I, 4, 863-881. https://doi.org/10.1051/jp1:1994233
[28]
McCauley, J.L., Gunaratne, G.H. and Bassler, K.E. (2007) Martingale Option Pricing. Physica A: Statistical Mechanics and its Applications, 380, 351-356. https://doi.org/10.1016/j.physa.2007.02.038
[29]
Bouchaud, J. and Potters, M. (2003) Theory of Financial Risk and Derivative Pricing. 2nd Edition, Cambridge University Press. https://doi.org/10.1017/cbo9780511753893
[30]
Evans, M., Hastings, N. and Peacock, B. (1993) Statistical Distributions. 2nd Edition, John Wiley and Sons.
[31]
Lax, M., Cai, W. and Xu, M. (2006) Random Processes in Physics and Finance. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198567769.001.0001
[32]
Coffey, W.T., Kalmykov, Y.P. and Waldron, J.T. (2004) The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering. 2nd Edition, World Scientific Publishing Co. Pte. Ltd. https://doi.org/10.1142/9789812795090
[33]
Cassidy, D.T. (2016) Student’s t Increments. Open Journal of Statistics, 6, 156-171. https://doi.org/10.4236/ojs.2016.61014
[34]
Cassidy, D.T., Hamp, M.J. and Ouyed, R. (2013) Log Student’s t-Distribution-Based Option Sensitivities: Greeks for the Gosset Formulae. Quantitative Finance, 13, 1289-1302. https://doi.org/10.1080/14697688.2012.744087
[35]
Moriconi, L. (2007) Delta Hedged Option Valuation with Underlying Non-Gaussian Returns. Physica A: Statistical Mechanics and its Applications, 380, 343-350. https://doi.org/10.1016/j.physa.2007.01.018
[36]
Lim, G.C., Martin, G.M. and Martin, V.L. (2006) Pricing Currency Options in the Presence of Time-Varying Volatility and Non-Normalities. Journal of Multinational Financial Management, 16, 291-314. https://doi.org/10.1016/j.mulfin.2005.08.004
[37]
Cassidy, D.T. (2012) Effective Truncation of a Student’s t-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture. Open Journal of Statistics, 2, 519-525. https://doi.org/10.4236/ojs.2012.25067
[38]
Ross, S.M. (2007) Introduction to Probability Models. 9th Edition, Elsevier.
[39]
Cassidy, D.T. (2018) Risk-Neutral Pricing of European Call Options: A Specious Concept. Journal of Mathematical Finance, 8, 335-348. https://doi.org/10.4236/jmf.2018.82022
[40]
Greenwood, P.E. and Nikulin, M.S. (1996) A Guide to Chi-Squared Testing. John Wiley and Sons.
[41]
Ross, S. (2006) A First Course in Probability. 7th Edition, Pearson Prentice Hall.
