Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

References

• D.H. Bailey, R. Barrio and J.M. Borwein, High-precision computation: Mathematical physics and dynamics, Appl. Math. Comput. 218 (2012), pp. 10106–10121. doi:10.1016/j.amc.2012.03.087. MR 2921767
   Web of Science ®Google Scholar
• D.H. Bailey and J.M. Borwein, High-precision numerical integration: progress and challenges, J. Symbolic Comput. 46 (2011), pp. 741–754. doi:10.1016/j.jsc.2010.08.010. MR 2795208
   Web of Science ®Google Scholar
• D.H. Bailey and J.M. Borwein, Experimental mathematics: examples, methods and implications, Notices Amer. Math. Soc. 52 (2005), pp. 502–514. MR 2140093
   Google Scholar
• D.H. Bailey and J.M. Borwein, High-precision arithmetic in mathematical physics, Mathematics 3 (2015), pp. 337–367. doi:10.3390/math3020337
   Web of Science ®Google Scholar
• D.H. Bailey, K. Jeyabalan and X.S. Li, A comparison of three high-precision quadrature schemes, Experiment. Math. 14 (2005), pp. 317–329. doi:10.1080/10586458.2005.10128931. MR 2172710
   Web of Science ®Google Scholar
• D.H. Bailey and P.N. Swarztrauber, The fractional fourier transform and applications, SIAM Rev. 33 (1991), pp. 389–404. doi:10.1137/1033097. MR 1124359
   Web of Science ®Google Scholar
• D.H. Bailey and P.N. Swarztrauber, A fast method for the numerical evaluation of continuous fourier and laplace transforms, SIAM J. Sci. Comput. 15 (1994), pp. 1105–1110. doi:10.1137/0915067. MR 1289155
   Web of Science ®Google Scholar
• O.E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 (2001), pp. 167–241. doi:10.1111/1467-9868.00282. MR 1841412
   Web of Science ®Google Scholar
• D.S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, Rev. Financ. Stud. 9 (1996), pp. 69–107. doi:10.1093/rfs/9.1.69
   Web of Science ®Google Scholar
• F. Baustian, M. Mrázek, J. Pospíšil and T. Sobotka, Unifying pricing formula for several stochastic volatility models with jumps, Appl. Stoch. Models Bus. Ind. 33 (2017), pp. 422–442. doi:10.1002/asmb.2248. MR 3690484
   Web of Science ®Google Scholar
• S. Boyarchenko and S. Levendorskii, Efficient variations of the Fourier transform in applications to option pricing, J. Comput. Finance 18 (2014), pp. 57–90. doi:10.21314/jcf.2014.277
   Web of Science ®Google Scholar
• P. Carr and D.B. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance 2 (1999), pp. 61–73. doi:10.21314/JCF.1999.043
   Google Scholar
• G. Dahlquist and B. Åke, Numerical Methods in Scientific Computing, Vol. I, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi:10.1137/1.9780898717785. MR 2412832
   Google Scholar
• J. Daněk and J. Pospíšil, Numerical integration of inaccurately evaluated functions, in Technical Computing Prague 2015, University of Chemistry Technology, Prague, 2015, pp. 1–11. TCP 2015, November 4, 2015, Prague, Czech Republic.
   Google Scholar
• P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Dover Publications, Inc., Mineola, NY, 2007. Corrected reprint of the second edition (1984) MR 2401585.
   Google Scholar
• F. Fang and C.W. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2009), pp. 826–848. doi:10.1137/080718061. MR 2466138
   Web of Science ®Google Scholar
• W. Gander and W. Gautschi, Adaptive quadrature—revisited, Bit Numer. Math. 40 (2000), pp. 84–101. doi:10.1023/A:1022318402393. MR 1759036
   Web of Science ®Google Scholar
• W. Gautschi, Gauss-Kronrod quadrature—a survey, in Numerical Methods and Approximation Theory, III (Niš, 1987), Univ. Niš, Niš, 1988, pp. 39–66. MR 960329.
   Google Scholar
• W. Gautschi, Orthogonal polynomials: Computation and approximation, in Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. Oxford Science Publications. MR 2061539.
   Google Scholar
• W. Gautschi, Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range Gauss-Hermite Quadrature, ACM Trans. Math. Softw. 42 (2016), pp. 1–10. Available at http://doi.acm.org/10.1145/2735626. doi:10.1145/2735626
   Web of Science ®Google Scholar
• W. Gautschi, OPQ: A Matlab suite of programs for generating orthogonal polynomials and related quadrature rules (2017). Available at https://purr.purdue.edu/publications/1582/1. doi:10.4231/R7959FHP.
   Google Scholar
• P. Gonnet, Adaptive quadrature re-revisited, Ph.D. diss., ETH Zürich, 2009. doi:10.3929/ethz-a-005861903.
   Google Scholar
• P. Gonnet, A review of error estimation in adaptive quadrature, ACM Comput. Surv. 44 (2012), pp. 1–36. Available at http://doi.acm.org/10.1145/2333112.2333117. doi:10.1145/2333112.2333117
   Web of Science ®Google Scholar
• S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), pp. 327–343. doi:10.1093/rfs/6.2.327
   Web of Science ®Google Scholar
• C. Kahl and P. Jäckel, Not-so-complex logarithms in the Heston model, Wilmott Magazine 2005 (2005), pp. 94–103.
   Google Scholar
• V.I. Krylov, Approximate calculation of integrals, Translated by Arthur H. Stroud, The Macmillan Co., New York, London, 1962. MR 0144464.
   Google Scholar
• D.P. Laurie, Computation of Gauss-type quadrature formulas, J. Comput. Appl. Math. 127 (2001), pp. 201–217. doi:10.1016/S0377-0427(00)00506-9
   Web of Science ®Google Scholar
• S. Levendorskii, Efficient pricing and reliable calibration in the Heston model., Int. J. Theor. Appl. Finance 15 (2012), pp. 1–44. doi:10.1142/S0219024912500501
   Google Scholar
• A.L. Lewis, Option Valuation Under Stochastic Volatility: With Mathematica code, Finance Press, Newport Beach, CA, 2000. MR 1742310 (2000m:91068)
   Google Scholar
• A.L. Lewis, Option Valuation Under Stochastic Volatility II: With Mathematica code, Finance Press, Newport Beach, CA, 2016. MR 3526206
   Google Scholar
• E. Lindström, J. Ströjby, M. Brodén, M. Wiktorsson and J. Holst, Sequential calibration of options, Comput. Statist. Data Anal. 52 (2008), pp. 2877–2891. doi:10.1016/j.csda.2007.08.009
   Web of Science ®Google Scholar
• M. Mrázek and J. Pospíšil, Calibration and simulation of Heston model, Open Math. 15 (2017), pp. 679–704. doi:10.1515/math-2017-0058. MR 3657941
   Web of Science ®Google Scholar
• M. Mrázek, J. Pospíšil and T. Sobotka, On calibration of stochastic and fractional stochastic volatility models, European J. Oper. Res. 254 (2016), pp. 1036–1046. doi:10.1016/j.ejor.2016.04.033. MR 3508893
   Web of Science ®Google Scholar
• L. Ortiz-Gracia and C.W. Oosterlee, A highly efficient Shannon wavelet inverse Fourier technique for pricing European options, SIAM J. Sci. Comput. 38 (2016), pp. B118–B143. doi:10.1137/15M1014164. MR 3452255
   Web of Science ®Google Scholar
• S.P. Pal, R.K. Koul, F. Musadeekh, P.H.D. Ramakrishna and H. Basu, Computations that require higher than double precision for robust and exact decision making, Int. J. Comput. Math. 81 (2004), pp. 595–605. doi:10.1080/00207160410001684235
   Web of Science ®Google Scholar
• J. Pospíšil and T. Sobotka, Market calibration under a long memory stochastic volatility model, Appl. Math. Finance 23 (2016), pp. 323–343. doi:10.1080/1350486X.2017.1279977. MR 3615548
   Google Scholar
• J. Pospíšil and T. Sobotka, Test data sets for calibration of stochastic and fractional stochastic volatility models, Data Brief 8 (2016), pp. 628–630. doi:10.1016/j.dib.2016.06.016
   PubMed Web of Science ®Google Scholar
• W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes: The art of scientific computing, 3rd ed., Cambridge University Press, Cambridge, 2007. MR 2371990
   Google Scholar
• H.T. Rathod, R.D. Sathish, M.S. Islam and A.K. Gali, Application of MATLAB symbolic maths with variable precision arithmetic (vpa) to compute some high order Gauss Legendre quadrature rules, Ganit 29 (2011), pp. 117–125.
   Google Scholar
• M. Schmelzle, Option pricing formulae using Fourier transform: Theory and application (2010). Available at http://pfadintegral.com/articles/.
   Google Scholar
• J. Stoer and R. Bulirsch, Introduction to numerical analysis, 3rd ed., Texts in Applied Mathematics Vol. 12, Springer-Verlag, New York, NY, 2002. doi:10.1007/978-0-387-21738-3. MR 1923481.
   Google Scholar
• G. Yan and F.B. Hanson, Option pricing for a stochastic-volatility jump-diffusion model with log-uniform jump-amplitude, in Proceedings of American Control Conference, Piscataway, NJ. IEEE, 2006, pp. 2989–2994. doi:10.1109/acc.2006.1657175.
   Google Scholar