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

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## 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## Mathematics Subject Classification

26A33 34A08 60H35 93B40 49J15## Introduction

The Black–Scholes model has been introduced based on the geometric Brownian motion to providing a closed-form pricing formula for the European options and describing the behavior of underlying asset prices [1, 2, 3]. The assumptions of the Black–Scholes model are unrealistic due to its inability to generate volatility satisfying the market observations, being nonnegative and mean-reverting [4, 5, 6]. To overcome this limitation, in 1993, Heston suggested Heston’s stochastic volatility (HSV) model [6]. The HSV model has been extended in finance for modeling the dynamics of implied volatilities and providing their user with simple breakeven accounting conditions for the profit and loss (S&P) of a hedged position [2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

*S*(

*t*) denotes the stock price, and

*V*(

*t*) denotes it’s return variance at time

*t*that referred to in the literature as volatility [18]. The parameters

*r*and \(\theta\) show the risk-free rate and the volatility long-run mean, respectively. Moreover, the parameter \(\kappa\) reveals the mean reversion rate toward \(\theta\) that controls the speed volatility going back to its average value. The volatility of volatility, \(\sigma\), is a significant risk factor that depicts the kurtosis in the stock return rate distribution. The Brownian motions, \(\omega _1(t)\) and \(\omega _2(t)\) with correlation coefficient, \(\rho \in [-1,1]\), are in the risk-neutral measure.

Despite its popularity, research on efficient discretizations of the continuous time dynamics of the Heston model has generated less attention by researchers. In 2006, an exact simulation technique was suggested by Broadie and Kaya for the HSV model [19]. Subsequently, in [20], a simulation scheme including the quadratic-exponential technique has been developed, and in [21], a second-order discretization scheme has been discussed. Along the same line of thought, the drift-implicit Milstein algorithm for the volatility [22], a Euler discretization for the log-Heston price [16] and an analytic approach for degenerate parabolic problem associated with the HSV model [17] were presented.

In the last decades, the theory of fractional calculus was motivated as a useful mathematical tool to handle application of associated concepts in the areas of physics, chemistry, economics, finance and engineering sciences [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. To the author’s best knowledge, while numerous stochastic volatility models driven by fractional Brownian motion have been considered [35, 36, 37, 38], stochastic volatility models with fixed-order and variable-order fractional derivative operators have not been carried out and this topic is far from being fully explored. More recently, just a numerical discretization technique has been proposed for approximation of a class of fractional stochastic differential equations driven by Brownian motion [39].

*u*(

*t*) is \((m-1)\), \(m\in {\mathbb {N}}\), times continuously differentiable and \(u^{(m)}(t)\) is once integrable, \(\zeta\) is an auxiliary variable that belongs to the interval [0,

*t*], and \({\varGamma }(\cdot )\) denotes the Gamma function. The concept and properties of variable-order fractional derivative operators and their application have been discussed in [41, 42, 43, 44, 45].

The outline of the paper is organized as follows. In “Optimal VOF-HSV model” section, the tuning technique is described for designing optimal VOF-HSV model for US stock market. For this propose, a discretization algorithm for the VOF-HSV model is proposed and followed by an optimization technique, genetic algorithm (GA), for tuning the unknown parameters in the VOF-HSV model. The advantages of using the VOF-HSV model are shown and verified by considering different performance criteria in “Numerical results” section. Finally, “Conclusion” section draws the conclusions.

## Optimal VOF-HSV model

Throughout this paper, we let \({\varLambda }=[0,T]\) with a uniform grid \(t_{j}=jh\), \(j=0,1,\ldots ,n\), \(n\in {\mathbb {Z}}^+\), such that \(n+1=\frac{T}{h}\), and \(\gamma _n=\gamma (t_n)\), \(\beta _n=\beta (t_n)\), \(S_n=S(t_n)\) and \(V_n=V(t_n)\). In this section, an optimization strategy is formulated based on discretizing the equations in (2) and minimizing performance index under the following propositions.

### Proposition 2.1

*The discretized expression of variable-order fractional derivative*\(_{0}^{\nu o}{\mathscr {D}}^{\gamma (\cdot )}_{t} [\cdot ]\)

*in*(3)

*is obtained by the forward finite-difference approximation*(

*M*-

*algorithm*)

*as*

*where*

*h*

*denotes the uniform step size,*\(\psi _{m,n,j}=[{(j+1)}^{m-\gamma _n}-j^{m-\gamma _n}]\),

*where*\({\mathcal {O}}(\cdot )\)

*denotes convergence order of approximation.*

### Proposition 2.2

*Let*\(0<\gamma (t)\le 1\).

*Assume that*

*u*(

*t*)

*be a function in*\({\mathbb {L}}^2({\varOmega }, {\mathcal {F}}_t,{\mathbb {P}})\)

*and for every subinterval*\(t\in [t_j,t_{j+1}]\subseteq {\varLambda }\), \(u(t)\in C^2[t_j,t_{j+1}]\)

*and*\(\Vert u''(t)\Vert \le \varrho\), \(j=0,1,\ldots ,n\).

*Then, there exists an*\(\gamma _{n+1}\)

*dependent constant, i.e.,*\(C_{\gamma _{n+1}}>0\),

*such that the truncated error of the variable-order fractional derivative operator obtained by the*

*M*-

*algorithm satisfies*

*where*\(\displaystyle {C_{\gamma _{n+1}}=\frac{{(n+1)}^{1-\gamma _{n+1}}\varrho }{2{\varGamma }(2-\gamma _{n+1})}}\).

### *Proof*

Using Propositions 2.1, the discretized equations of VOF-HSV model (2) are derived as follows

*N*and

*p*are the number of observations and parameters in the equation, respectively [47].

## Numerical results

In this section, fixed-order and VOF-HSV models are used to estimate the behavior of the price of S&P 500 index of American options. The S&P 500 is a US stock market index that covers all the large market public companies recorded on the New York Stock Exchange (NYSE) or National Association of Securities Dealers Automated Quotations (NASDAQ) [49]. For practical applicability, in this work, the parameters of the financial market model are chosen based on [50]. In [50], authors obtained estimate values of parameters for the typical Heston model by considering S&P 500 index returns. Throughout the numerical analysis, the main parameters satisfy that the mean reversion rate converges to \(\kappa =2.75\), the long-run mean of the volatility converges to \(\theta =0.035\), the volatility of volatility converges to \(\sigma =0.425\), and the correlation coefficient converges to \(\rho =-0.4644\).

For constructing VOF-HSV model, \(\gamma (t)=c_1+c_2t\) and \(\beta (t)=c_3+c_4t\) functions where \(0\le {c_i}\le 1\), \(i=1,\ldots ,4\), with four unknown parameters being considered. Next, an optimization method, GA, is used to find the optimal values of the unknowns \(c_i\), \(i=1,\ldots ,4\), by minimizing the mean absolute error (MAE) at the discretized points goodness, that is, by minimizing \(\text {MAE}=\frac{1}{N}\sum _{n=1}^{N}|S(t_n)-\hat{S}_n|\), where \(S(t_n)\) and \(\hat{S}_n\) are the discretized value of the model and experimental value, respectively. The optimization algorithm is based on discretizing (6) by using the *M*-algorithm formulated in Propositions 2.1 with \(n =1227\) equal mesh points, corresponding to observed data, so that approximation error can be determined by using Proposition 2.2. All the computations are performed under Maple v18 on an Intel (R) Core (TM) i7-7500U CPU @ 2.70 GHz machine.

*S*(

*t*), and it’s return variance,

*V*(

*t*), with integer-order (i.e., \(\gamma (t)=\beta (t)=1\)), the fixed-order fractional (i.e., \(\gamma (t)=0.8960\) and \(\beta (t)=0.9292\)), and the variable-order fractional (i.e., \(\gamma (t)=0.9010+0.0890t\) and \(\beta (t)=0.9088+0.0869t\)) HSV models with \(n =1227\) equal mesh points in the interval [0, 1].

The MAEs and optimal parameters of the integer-order, the fixed-order fractional and VOF-HSV models for \(\gamma (t)=c_1+c_2t\) and \(\beta (t)=c_3+c_4t\) in the interval [0, 1]

HSV models | MAE | \(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) |
---|---|---|---|---|---|

Integer-order | 69.43743 | 1 | 0 | 1 | 0 |

Fixed-oder fractional | 44.95400 | 0.8960 | 0 | 0.9292 | 0 |

Variable-order fractional | 42.10175 | 0.9010 | 0.0890 | 0.9088 | 0.0869 |

Comparison of RMSE, AIC and BIC of integer-order and optimal fixed- and VOF-HSV models with \(n =1227\) equal mesh points in the interval [0, 1]

HSV models | RMSE | AIC | BIC |
---|---|---|---|

Integer-order | \(2.9030\times 10^5\) | 19787.83 | 11094.67 |

Fixed-oder fractional | \(1.3501\times 10^5\) | 18849.18 | 10156.02 |

Variable-order fractional | \(1.1700\times 10^5\) | 18673.73 | 9980.580 |

Approximation values of the mean, median, first and third quartiles, kurtosis, skewness, STD and 95% CI of the 50 simulated trajectories for stock prices for various values of \(\gamma (t)\) and \(\beta (t)\), with \(h =\frac{1}{1227}\) at \(T=1\)

Statistical indicators | HSV model | ||
---|---|---|---|

Integer-order | Fixed-order fractional | Variable-order fractional | |

Mean | 1279.50 | 1411.67 | 1367.36 |

Median | 1306.74 | 1434.41 | 1386.61 |

First quartile | 1136.37 | 1235.72 | 1171.84 |

Third quartile | 1424.69 | 1596.35 | 1525.60 |

Kurtosis | 2.630 | 2.640 | 2.811 |

Skewness | \(-\,0.113\) | \(-\,0.151\) | \(-\,0.066\) |

STD | 203.02 | 263.55 | 237.48 |

95% CI | [881.57, 1677.42] | [895.12, 1928.23] | [901.90, 1832.82] |

## Conclusion

It has been shown that using fixed-order and variable-order fractional derivative operators instead of an integer-order derivative operator in models can apparently lead to better results since one has an extra freedom degree. In this paper, the objective of introducing modified Heston model was to capture the complexities in simulating the scenarios of equity movement in finance. The proposed Heston’s stochastic volatility models, due to random volatility and memory or hereditary properties, have more flexibility with market data than the constant volatility models. In this study, an optimization technique, genetic algorithm, was utilized to find an optimal set of unknown parameters of models. For this propose, an explicit numerical technique has been developed for solving Heston’s stochastic volatility models. Furthermore, experimental stock price tests of a US stock market were used to estimate fixed-order and variable-order fractional Heston’s stochastic volatility models. It was also shown that proposed fractional Heston’s stochastic volatility models are significantly better in estimating the stock price in comparison to integer-order Heston’s stochastic volatility models. Moreover, variable-order fractional Heston’s stochastic volatility model is superior to fixed-order fractional Heston’s stochastic volatility model.

## Notes

## References

- 1.Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ.
**81**(3), 637–654 (1973). https://doi.org/10.1086/260062 MathSciNetCrossRefzbMATHGoogle Scholar - 2.Papi, M., Pontecorvi, L., Donatucci, C.: Weighted average price in the Heston stochastic volatility model. Decis. Econ. Finance
**40**(1–2), 351–373 (2017). https://doi.org/10.1007/s10203-017-0197-5 MathSciNetCrossRefzbMATHGoogle Scholar - 3.Vajargah, K.F., Shoghi, M.: 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. Math. Sci.
**9**(3), 115–125 (2015). https://doi.org/10.1007/s40096-015-0158-5 CrossRefzbMATHGoogle Scholar - 4.Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica
**53**, 385–407 (1985)MathSciNetCrossRefGoogle Scholar - 5.Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance
**42**(2), 281–300 (1987). https://doi.org/10.2307/2328253 CrossRefzbMATHGoogle Scholar - 6.Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud.
**6**(2), 327–343 (1993). https://doi.org/10.1093/rfs/6.2.327 CrossRefzbMATHGoogle Scholar - 7.Ballestra, L.V., Pacelli, G., Zirilli, F.: A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model. J. Bank. Finance
**31**(11), 3420–3437 (2007). https://doi.org/10.1016/j.jbankfin.2007.04.013 CrossRefGoogle Scholar - 8.Atiya, A.F., Wall, S.: An analytic approximation of the likelihood function for the Heston model volatility estimation problem. Quant. Finance
**9**(3), 289–296 (2009). https://doi.org/10.1080/14697680802595601 MathSciNetCrossRefzbMATHGoogle Scholar - 9.Hout, K.I., Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model.
**7**(2), 303–320 (2010)MathSciNetGoogle Scholar - 10.Forde, M., Jacquier, A., Mijatović, A.: A note on essential smoothness in the Heston model. Finance Stoch.
**15**(4), 781–784 (2011). https://doi.org/10.1007/s00780-011-0162-z MathSciNetCrossRefzbMATHGoogle Scholar - 11.Hout, K.I.: Finite difference approximation of hedging quantities in the Heston model. AIP (2012). https://doi.org/10.1063/1.4756108
- 12.Lenkšas, A., Mackevičius, V.: A second-order weak approximation of Heston model by discrete random variables. Lith. Math. J.
**55**(4), 555–572 (2015). https://doi.org/10.1007/s10986-015-9298-4 MathSciNetCrossRefzbMATHGoogle Scholar - 13.Boguslavskaya, E., Muravey, D.: An explicit solution for optimal investment in Heston model. Theory Probab. Appl.
**60**(4), 679–688 (2016). https://doi.org/10.1137/s0040585x97t987946 MathSciNetCrossRefzbMATHGoogle Scholar - 14.Cui, Z., Feng, R., Anne, M.: Variable annuities with VIX-linked fee structure under a Heston-type stochastic volatility model. SSRN Electron. J.
**21**(3), 458–483 (2016). https://doi.org/10.2139/ssrn.2862657 MathSciNetCrossRefGoogle Scholar - 15.Cui, Y., del Baño Rollin, S., Germano, G.: Full and fast calibration of the Heston stochastic volatility model. Eur. J. Oper. Res.
**263**(2), 625–638 (2017). https://doi.org/10.1016/j.ejor.2017.05.018 MathSciNetCrossRefzbMATHGoogle Scholar - 16.Altmayer, M., Neuenkirch, A.: Discretising the Heston model: an analysis of the weak convergence rate. IMA J. Numer. Anal.
**37**(4), 1930–1960 (2017). https://doi.org/10.1093/imanum/drw063 MathSciNetCrossRefzbMATHGoogle Scholar - 17.Canale, A., Mininni, R.M., Rhandi, A.: Analytic approach to solve a degenerate parabolic PDE for the Heston model. Math. Methods Appl. Sci.
**40**(13), 4982–4992 (2017). https://doi.org/10.1002/mma.4363 MathSciNetCrossRefzbMATHGoogle Scholar - 18.Shreve, S.E.: Stochastic Calculus for Finance II: Continuous-time Models, vol. 11. Springer, New York (2004)zbMATHGoogle Scholar
- 19.Broadie, M., Kaya, O.: Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res.
**54**(2), 217–231 (2006). https://doi.org/10.1287/opre.1050.0247 MathSciNetCrossRefzbMATHGoogle Scholar - 20.Andersen, L.: Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance
**11**(3), 1–42 (2008). https://doi.org/10.21314/jcf.2008.189 CrossRefGoogle Scholar - 21.Alfonsi, A.: High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput.
**79**(269), 209–209 (2010). https://doi.org/10.1090/s0025-5718-09-02252-2 MathSciNetCrossRefzbMATHGoogle Scholar - 22.Kahl, C., Gunther, M., Rossberg, T.: Structure preserving stochastic integration schemes in interest rate derivative modeling. Appl. Numer. Math.
**58**(3), 284–295 (2008). https://doi.org/10.1016/j.apnum.2006.11.013 MathSciNetCrossRefzbMATHGoogle Scholar - 23.Dabiri, A., Butcher, E.A.: Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl. Math. Model.
**56**, 424–448 (2018). https://doi.org/10.1016/j.apm.2017.12.012 MathSciNetCrossRefGoogle Scholar - 24.Al-Khaled, K., Alquran, M.: An approximate solution for a fractional model of generalized Harry Dym equation. Math. Sci.
**8**(4), 125–130 (2014). https://doi.org/10.1007/s40096-015-0137-x MathSciNetCrossRefzbMATHGoogle Scholar - 25.Arshed, S.: Quintic B-spline method for time-fractional superdiffusion fourth-order differential equation. Math. Sci.
**11**(1), 17–26 (2016). https://doi.org/10.1007/s40096-016-0200-2 MathSciNetCrossRefzbMATHGoogle Scholar - 26.Bhrawy, A.H., Zaky, M.A.: Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn.
**85**(3), 1815–1823 (2016). https://doi.org/10.1007/s11071-016-2797-y MathSciNetCrossRefzbMATHGoogle Scholar - 27.Li, X., Yang, X.: Error estimates of finite element methods for stochastic fractional differential equations. J. Comput. Math.
**35**(3), 346–362 (2017). https://doi.org/10.4208/jcm.1607-m2015-0329 MathSciNetCrossRefzbMATHGoogle Scholar - 28.Ahmadi, N., Vahidi, A.R., Allahviranloo, T.: An efficient approach based on radial basis functions for solving stochastic fractional differential equations. Math. Sci.
**11**(2), 113–118 (2017). https://doi.org/10.1007/s40096-017-0211-7 MathSciNetCrossRefzbMATHGoogle Scholar - 29.Zaky, M.A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput. Appl. Math.
**37**, 1–14 (2017). https://doi.org/10.1007/s40314-017-0530-1 MathSciNetCrossRefGoogle Scholar - 30.Dabiri, A., Moghaddam, B.P., Machado, J.A.T.: Optimal variable-order fractional PID controllers for dynamical systems. J. Comput. Appl. Math.
**339**, 40–48 (2018). https://doi.org/10.1016/j.cam.2018.02.029 MathSciNetCrossRefzbMATHGoogle Scholar - 31.Keshi, F.K., Moghaddam, B.P., Aghili, A.: A numerical approach for solving a class of variable-order fractional functional integral equations. Comput. Appl. Math.
**37**(4), 4821–4834 (2018). https://doi.org/10.1007/s40314-018-0604-8 MathSciNetCrossRefzbMATHGoogle Scholar - 32.Zaky, M.A., Doha, E.H., Taha, T.M., Baleanu, D.: New recursive approximations for variable-order fractional operators with applications. Math. Model. Anal.
**23**(2), 227–239 (2018). https://doi.org/10.3846/mma.2018.015 MathSciNetCrossRefGoogle Scholar - 33.Machado, J.A.T., Moghaddam, B.P.: A robust algorithm for nonlinear variable-order fractional control systems with delay. Int. J. Nonlinear Sci. Numer. Simul.
**19**(3–4), 231–238 (2018). https://doi.org/10.1515/ijnsns-2016-0094 CrossRefzbMATHGoogle Scholar - 34.Dabiri, A., Butcher, E.A., Nazari, M.: Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. J. Sound Vib.
**388**, 230–244 (2017). https://doi.org/10.1016/j.jsv.2016.10.013 CrossRefGoogle Scholar - 35.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. https://doi.org/10.1109/bife.2010.48
- 36.Ballestra, L.V., Pacelli, G., Radi, D.: A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion. Chaos Solitons Fractals
**87**, 240–248 (2016). https://doi.org/10.1016/j.chaos.2016.04.008 MathSciNetCrossRefzbMATHGoogle Scholar - 37.Bondarenko, V., Bondarenko, V., Truskovskyi, K.: Forecasting of time data with using fractional Brownian motion. Chaos Solitons Fractals
**97**, 44–50 (2017). https://doi.org/10.1016/j.chaos.2017.01.013 MathSciNetCrossRefzbMATHGoogle Scholar - 38.Panov, V.: Modern Problems of Stochastic Analysis and Statistics: Selected Contributions in Honor of Valentin Konakov, vol. 208. Springer, Berlin (2017)CrossRefGoogle Scholar
- 39.Mostaghim, Z.S., Moghaddam, B.P., Haghgozar, H.S.: Numerical simulation of fractional-order dynamical systems in noisy environments. Comput. Appl. Math.
**133**, 1–15 (2018). https://doi.org/10.1007/s40314-018-0698-z CrossRefGoogle Scholar - 40.Lorenzo, C., Hartley, T.: Variable order and distributed order fractional operators. Nonlinear Dyn.
**29**(1–4), 57–98 (2002)MathSciNetCrossRefGoogle Scholar - 41.Zaky, M.A., Baleanu, D., Alzaidy, J.F., Hashemizadeh, E.: Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection-diffusion equation. Adv. Differ. Equa.
**2018**(1), 102 (2018). https://doi.org/10.1186/s13662-018-1561-7 MathSciNetCrossRefGoogle Scholar - 42.Moghaddam, B.P., Machado, J.A.T.: SM-algorithms for approximating the variable-order fractional derivative of high order. Fundam. Inf.
**151**(1–4), 293–311 (2017). https://doi.org/10.3233/fi-2017-1493 MathSciNetCrossRefzbMATHGoogle Scholar - 43.Moghaddam, B.P., Machado, J.A.T.: A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract. Calc. Appl. Anal.
**20**(4), 1023–1042 (2017). https://doi.org/10.1515/fca-2017-0053 MathSciNetCrossRefzbMATHGoogle Scholar - 44.Zaky, M.A.: A research note on the nonstandard finite difference method for solving variable-order fractional optimal control problems. J. Vib. Control
**24**(11), 2109–2111 (2018). https://doi.org/10.1177/1077546318761443 MathSciNetCrossRefzbMATHGoogle Scholar - 45.Moghaddam, B.P., Machado, J.A.T., Babaei, A.: A computationally efficient method for tempered fractional differential equations with application. Comput. Appl. Math.
**37**(3), 3657–3671 (2017). https://doi.org/10.1007/s40314-017-0522-1 MathSciNetCrossRefzbMATHGoogle Scholar - 46.Moghaddam, B.P., Machado, J.A.T.: Extended algorithms for approximating variable order fractional derivatives with applications. J. Sci. Comput.
**71**(3), 1351–1374 (2016). https://doi.org/10.1007/s10915-016-0343-1 MathSciNetCrossRefzbMATHGoogle Scholar - 47.Hossein-Zadeh, N.G.: Application of growth models to describe the lactation curves for test-day milk production in Holstein cows. J. Appl. Animal Res.
**45**(1), 145–151 (2016). https://doi.org/10.1080/09712119.2015.1124336 CrossRefGoogle Scholar - 48.Guthery, F.S., Burnham, K.P., Anderson, D.R.: Model selection and multimodel inference: a practical information-theoretic approach. J. Wildl. Manag.
**67**(3), 655 (2003). https://doi.org/10.2307/3802723 CrossRefGoogle Scholar - 49.Wang, X., He, X., Zhao, Y., Zuo, Z.: Parameter estimations of Heston model based on consistent extended Kalman filter. IFAC PapersOnLine
**50**(1), 14100–14105 (2017). https://doi.org/10.1016/j.ifacol.2017.08.1850 CrossRefGoogle Scholar - 50.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)Google Scholar

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