10.1007/s40096-018-0267-z

Computational technique for simulating variable-order fractional Heston model with application in US stock market

HTML PDF
  • Zeinab Salamat Mostaghim1,
  • Behrouz Parsa Moghaddam*1,
  • Hossein Samimi Haghgozar2,
  1. Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, IR
  2. Department of Statistics, Faculty of Mathematical Sciences, University of Guilan, Rasht, IR
Cover Image

Published in Issue 2018-10-22

How to Cite

Mostaghim, Z. S., Moghaddam, B. P., & Haghgozar, H. S. (2018). Computational technique for simulating variable-order fractional Heston model with application in US stock market. Mathematical Sciences, 12(4 (December 2018). https://doi.org/10.1007/s40096-018-0267-z
Crossref
Scopus
Google Scholar
Europe PMC

HTML views: 21

PDF views: 92

Abstract

Abstract In this paper, a numerical technique is developed to discretize variable-order fractional Heston differential equation. The proposed strategy is followed by an optimization technology, genetic algorithm, for tuning the unknown parameters in the proposed model. The performance of the model is analyzed to profit and loss 500 close index from the US stock markets. Simulations illustrate the application of the proposed technique.

Keywords

  • Fractional calculus,
  • Stochastic calculus,
  • Computational techniques,
  • Optimization,
  • Variable-order fractional Heston model,
  • Stock price
HTML PDF

References

  1. Black and Scholes (1973) The pricing of options and corporate liabilities 81(3) (pp. 637-654) https://doi.org/10.1086/260062
  2. Papi et al. (2017) Weighted average price in the Heston stochastic volatility model 40(1–2) (pp. 351-373) https://doi.org/10.1007/s10203-017-0197-5
  3. Vajargah and Shoghi (2015) Simulation of stochastic differential equation of geometric Brownian motion by quasi- Monte Carlo method and its application in prediction of total index of stock market and value at risk 9(3) (pp. 115-125) https://doi.org/10.1007/s40096-015-0158-5
  4. Cox et al. (1985) A theory of the term structure of interest rates (pp. 385-407) https://doi.org/10.2307/1911242
  5. Hull and White (1987) The pricing of options on assets with stochastic volatilities 42(2) (pp. 281-300) https://doi.org/10.2307/2328253
  6. Heston (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options 6(2) (pp. 327-343) https://doi.org/10.1093/rfs/6.2.327
  7. Ballestra et al. (2007) A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model 31(11) (pp. 3420-3437) https://doi.org/10.1016/j.jbankfin.2007.04.013
  8. Atiya and Wall (2009) An analytic approximation of the likelihood function for the Heston model volatility estimation problem 9(3) (pp. 289-296) https://doi.org/10.1080/14697680802595601
  9. Hout and Foulon (2010) ADI finite difference schemes for option pricing in the Heston model with correlation 7(2) (pp. 303-320)
  10. Forde et al. (2011) A note on essential smoothness in the Heston model 15(4) (pp. 781-784) https://doi.org/10.1007/s00780-011-0162-z
  11. Hout, K.I.: Finite difference approximation of hedging quantities in the Heston model. AIP (2012).
  12. https://doi.org/10.1063/1.4756108
  13. Lenkšas and Mackevičius (2015) A second-order weak approximation of Heston model by discrete random variables 55(4) (pp. 555-572) https://doi.org/10.1007/s10986-015-9298-4
  14. Boguslavskaya and Muravey (2016) An explicit solution for optimal investment in Heston model 60(4) (pp. 679-688) https://doi.org/10.1137/s0040585x97t987946
  15. Cui et al. (2016) Variable annuities with VIX-linked fee structure under a Heston-type stochastic volatility model 21(3) (pp. 458-483) https://doi.org/10.2139/ssrn.2862657
  16. Cui et al. (2017) Full and fast calibration of the Heston stochastic volatility model 263(2) (pp. 625-638) https://doi.org/10.1016/j.ejor.2017.05.018
  17. Altmayer and Neuenkirch (2017) Discretising the Heston model: an analysis of the weak convergence rate 37(4) (pp. 1930-1960) https://doi.org/10.1093/imanum/drw063
  18. Canale et al. (2017) Analytic approach to solve a degenerate parabolic PDE for the Heston model 40(13) (pp. 4982-4992) https://doi.org/10.1002/mma.4363
  19. Shreve (2004) Springer
  20. Broadie and Kaya (2006) Exact simulation of stochastic volatility and other affine jump diffusion processes 54(2) (pp. 217-231) https://doi.org/10.1287/opre.1050.0247
  21. Andersen (2008) Simple and efficient simulation of the Heston stochastic volatility model 11(3) (pp. 1-42) https://doi.org/10.21314/jcf.2008.189
  22. Alfonsi (2010) High order discretization schemes for the CIR process: application to affine term structure and Heston models 79(269) (pp. 209-209) https://doi.org/10.1090/s0025-5718-09-02252-2
  23. Kahl et al. (2008) Structure preserving stochastic integration schemes in interest rate derivative modeling 58(3) (pp. 284-295) https://doi.org/10.1016/j.apnum.2006.11.013
  24. Dabiri and Butcher (2018) Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods (pp. 424-448) https://doi.org/10.1016/j.apm.2017.12.012
  25. Al-Khaled and Alquran (2014) An approximate solution for a fractional model of generalized Harry Dym equation 8(4) (pp. 125-130) https://doi.org/10.1007/s40096-015-0137-x
  26. Arshed (2016) Quintic B-spline method for time-fractional superdiffusion fourth-order differential equation 11(1) (pp. 17-26) https://doi.org/10.1007/s40096-016-0200-2
  27. Bhrawy and Zaky (2016) Numerical algorithm for the variable-order Caputo fractional functional differential equation 85(3) (pp. 1815-1823) https://doi.org/10.1007/s11071-016-2797-y
  28. Li and Yang (2017) Error estimates of finite element methods for stochastic fractional differential equations 35(3) (pp. 346-362) https://doi.org/10.4208/jcm.1607-m2015-0329
  29. Ahmadi et al. (2017) An efficient approach based on radial basis functions for solving stochastic fractional differential equations 11(2) (pp. 113-118) https://doi.org/10.1007/s40096-017-0211-7
  30. Zaky (2017) A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations (pp. 1-14) https://doi.org/10.1007/s40314-017-0530-1
  31. Dabiri et al. (2018) Optimal variable-order fractional PID controllers for dynamical systems (pp. 40-48) https://doi.org/10.1016/j.cam.2018.02.029
  32. Keshi et al. (2018) A numerical approach for solving a class of variable-order fractional functional integral equations 37(4) (pp. 4821-4834) https://doi.org/10.1007/s40314-018-0604-8
  33. Zaky et al. (2018) New recursive approximations for variable-order fractional operators with applications 23(2) (pp. 227-239) https://doi.org/10.3846/mma.2018.015
  34. Machado and Moghaddam (2018) A robust algorithm for nonlinear variable-order fractional control systems with delay 19(3–4) (pp. 231-238) https://doi.org/10.1515/ijnsns-2016-0094
  35. Dabiri et al. (2017) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation (pp. 230-244) https://doi.org/10.1016/j.jsv.2016.10.013
  36. Feng, X., Quan, S.: Pricing of option with power payoff driven by mixed fractional Brownian motion. In: 2010 3rd International Conference on Business Intelligence and Financial Engineering, IEEE, 2010, pp. 170–173.
  37. https://doi.org/10.1109/bife.2010.48
  38. Ballestra et al. (2016) A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion (pp. 240-248) https://doi.org/10.1016/j.chaos.2016.04.008
  39. Bondarenko et al. (2017) Forecasting of time data with using fractional Brownian motion (pp. 44-50) https://doi.org/10.1016/j.chaos.2017.01.013
  40. Panov (2017) Springer https://doi.org/10.1007/978-3-319-65313-6
  41. Mostaghim et al. (2018) Numerical simulation of fractional-order dynamical systems in noisy environments (pp. 1-15) https://doi.org/10.1007/s40314-018-0698-z
  42. Lorenzo and Hartley (2002) Variable order and distributed order fractional operators 29(1–4) (pp. 57-98) https://doi.org/10.1023/A:1016586905654
  43. Zaky et al. (2018) Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection-diffusion equation 2018(1) https://doi.org/10.1186/s13662-018-1561-7
  44. Moghaddam and Machado (2017) SM-algorithms for approximating the variable-order fractional derivative of high order 151(1–4) (pp. 293-311) https://doi.org/10.3233/fi-2017-1493
  45. Moghaddam and Machado (2017) A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels 20(4) (pp. 1023-1042) https://doi.org/10.1515/fca-2017-0053
  46. Zaky (2018) A research note on the nonstandard finite difference method for solving variable-order fractional optimal control problems 24(11) (pp. 2109-2111) https://doi.org/10.1177/1077546318761443
  47. Moghaddam et al. (2017) A computationally efficient method for tempered fractional differential equations with application 37(3) (pp. 3657-3671) https://doi.org/10.1007/s40314-017-0522-1
  48. Moghaddam and Machado (2016) Extended algorithms for approximating variable order fractional derivatives with applications 71(3) (pp. 1351-1374) https://doi.org/10.1007/s10915-016-0343-1
  49. Hossein-Zadeh (2016) Application of growth models to describe the lactation curves for test-day milk production in Holstein cows 45(1) (pp. 145-151) https://doi.org/10.1080/09712119.2015.1124336
  50. Guthery et al. (2003) Model selection and multimodel inference: a practical information-theoretic approach 67(3) https://doi.org/10.2307/3802723
  51. Wang et al. (2017) Parameter estimations of Heston model based on consistent extended Kalman filter 50(1) (pp. 14100-14105) https://doi.org/10.1016/j.ifacol.2017.08.1850
  52. Zhang, J.E., Shu, J.: Pricing S&P 500 index options with Heston’s model. In: 2003 IEEE International Conference on Computational Intelligence for Financial Engineering, 2003. Proceedings., IEEE, pp. 85–92 (2003)