# Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

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## Abstract

**:**

## 1. Introduction

#### Organization of the Paper

## 2. Fractional Adams–Bashforth–Moulton Method and Its Error Analysis

## 3. Option Pricing Models, Implied Volatility, Characteristic Functions of the Option Pricing Models

- Introduce the famous Black–Scholes pricing model and discuss about implied volatility as well as its importance in the financial market.
- Focus on one of the most famous financial model amongst the practitioners—classical Heston model and the newly developed rough Heston model.
- Display its characteristic functions and its connection to call option pricing formula using the inversion of characteristic function.

#### 3.1. Black–Scholes Model and Implied Volatility

#### 3.2. Classical Heston Model and Rough Heston Model

- The model reproduce several stylized facts of low frequency stock data, e.g., the leverage effect, time-varying volatility and fat tails.
- It generates similar shapes and dynamics for the implied volatility surface.
- Efficient computation for the classical Heston model using the explicit formula for the characteristic function of the asset log-price (we will discuss it later).

#### 3.3. Characteristic Functions and Their Connection to Call Option Pricing

## 4. Small and Long Time Expansion of Solution for the Fractional Riccati Equation

#### 4.1. Small Time Expansion on Solution of Fractional Riccati Equation

#### 4.2. Large Time Expansion on Solution of Fractional Riccati Equation

**Proposition**

**1.**

**Proof.**

## 5. Multipoint Padé Approximation Method for Fractional Riccati Equation

`solve`function in MATLAB in less than one second. Now that we have the required multipoint Padé method, it is time we test it against the general approach (fractional Adams method).

## 6. Numerical Experiment And Performances

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Riemann–Liouville Fractional Integrals and Fractional Derivatives

## Appendix B. Mittag–Leffler Function

**Lemma**

**A1.**

**Corollary**

**A1.**

## References

- Hull, J.; White, A. The pricing of options on assets with stochastic volatilities. J. Financ.
**1987**, 42, 281–300. [Google Scholar] [CrossRef] - Heston, S.L. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud.
**1993**, 6, 327–343. [Google Scholar] [CrossRef] [Green Version] - Renault, E.; Touzi, N. Option hedging and implied volatilities in a stochastic volatility model 1. Math. Financ.
**1996**, 6, 279–302. [Google Scholar] [CrossRef] - Lee, R.W. Implied volatility: Statics, dynamics, and probabilistic interpretation. In Recent Advances in Applied Probability; Springer: Berlin, Germany, 2005; pp. 241–268. [Google Scholar]
- Lewis, A.L. Option Valuation under Stochastic Volatility II; Finance Press: Newport Beach, CA, USA, 2009. [Google Scholar]
- Medvedev, A.; Scaillet, O. Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility. Rev. Financ. Stud.
**2007**, 20, 427–459. [Google Scholar] [CrossRef] [Green Version] - Bates, D.S. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Rev. Financ. Stud.
**1996**, 9, 69–107. [Google Scholar] [CrossRef] - Gatheral, J.; Jaisson, T.; Rosenbaum, M. Volatility is rough. Quant. Financ.
**2018**, 18, 933–949. [Google Scholar] [CrossRef] - Mandelbrot, B.B.; Van Ness, J.W. Fractional Brownian motions, fractional noises and applications. SIAM Rev.
**1968**, 10, 422–437. [Google Scholar] [CrossRef] - Fukasawa, M. Asymptotic analysis for stochastic volatility: Martingale expansion. Financ. Stoch.
**2011**, 15, 635–654. [Google Scholar] [CrossRef] [Green Version] - Alòs, E.; León, J.A.; Vives, J. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Financ. Stoch.
**2007**, 11, 571–589. [Google Scholar] [CrossRef] [Green Version] - Comte, F.; Renault, E. Long memory in continuous-time stochastic volatility models. Math. Financ.
**1998**, 8, 291–323. [Google Scholar] [CrossRef] - Cheridito, P. Mixed fractional Brownian motion. Bernoulli
**2001**, 7, 913–934. [Google Scholar] [CrossRef] - El Euch, O.; Rosenbaum, M. The characteristic function of rough Heston models. Math. Financ.
**2019**, 29, 3–38. [Google Scholar] [CrossRef] [Green Version] - El Euch, O.; Fukasawa, M.; Rosenbaum, M. The microstructural foundations of leverage effect and rough volatility. Financ. Stoch.
**2018**, 22, 241–280. [Google Scholar] [CrossRef] [Green Version] - Bennedsen, M.; Lunde, A.; Pakkanen, M.S. Hybrid scheme for Brownian semistationary processes. Financ. Stoch.
**2017**, 21, 931–965. [Google Scholar] [CrossRef] [Green Version] - McCrickerd, R.; Pakkanen, M.S. Turbocharging Monte Carlo pricing for the rough Bergomi model. Quant. Financ.
**2018**, 18, 1877–1886. [Google Scholar] [CrossRef] [Green Version] - Abi Jaber, E.; El Euch, O. Multifactor Approximation of Rough Volatility Models. SIAM J. Financ. Math.
**2019**, 10, 309–349. [Google Scholar] [CrossRef] [Green Version] - Abi Jaber, E. Lifting the Heston model. Quant. Financ.
**2019**, 19, 1995–2013. [Google Scholar] [CrossRef] - El Euch, O.; Gatheral, J.; Rosenbaum, M. Roughening Heston. Available online: https://ssrn.com/abstract=3116887 (accessed on 25 March 2019).
- Callegaro, G.; Grasselli, M.; Pages, G. Rough but not so tough: Fast hybrid schemes for fractional Riccati equations. arXiv
**2018**, arXiv:1805.12587. [Google Scholar] - Momani, S.; Shawagfeh, N. Decomposition method for solving fractional Riccati differential equations. Appl. Math. Comput.
**2006**, 182, 1083–1092. [Google Scholar] [CrossRef] - Odibat, Z.; Momani, S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul.
**2006**, 7, 27–34. [Google Scholar] [CrossRef] - Odibat, Z.; Momani, S. Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals
**2008**, 36, 167–174. [Google Scholar] [CrossRef] - Cang, J.; Tan, Y.; Xu, H.; Liao, S.J. Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals
**2009**, 40, 1–9. [Google Scholar] [CrossRef] - Jafari, H.; Tajadodi, H.; Matikolai, S.H. Homotopy perturbation pade technique for solving fractional Riccati differential equations. Int. J. Nonlinear Sci. Numer. Simul.
**2010**, 11, 271–276. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Khan, J.A.; Qureshi, I.M. A new stochastic approach for solution of Riccati differential equation of fractional order. Ann. Math. Artif. Intell.
**2010**, 60, 229–250. [Google Scholar] [CrossRef] - Diethelm, K.; Ford, N.J.; Freed, A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms
**2004**, 36, 31–52. [Google Scholar] [CrossRef] [Green Version] - Gatheral, J.; Radoicic, R. Rational approximation of the rough Heston solution. Int. J. Theor. Appl. Financ.
**2019**, 22, 1950010. [Google Scholar] [CrossRef] - Thompson, I.J. Coupled reaction channels calculations in nuclear physics. Comput. Phys. Rep.
**1988**, 7, 167–212. [Google Scholar] [CrossRef] - Lanti, E.; Dominski, J.; Brunner, S.; McMillan, B.; Villard, L. Padé Approximation of the Adiabatic Electron Contribution to the Gyrokinetic Quasi-Neutrality Equation in the ORB5 Code; Journal of Physics: Conference Series; IOP Publishing: Bristol, UK, 2016; Volume 775, p. 012006. [Google Scholar]
- Rakityansky, S.; Sofianos, S.; Elander, N. Pade approximation of the S-matrix as a way of locating quantum resonances and bound states. J. Phys. A Math. Theor.
**2007**, 40, 14857. [Google Scholar] [CrossRef] [Green Version] - Baker, G.A.; Baker, G.A., Jr.; Graves-Morris, P.; Baker, S.S. Padé Approximants; Cambridge University Press: Cambridge, UK, 1996; Volume 59. [Google Scholar]
- Lubinsky, D.S. Rogers-Ramanujan and the Baker-Gammel-Wills (padé) conjecture. Ann. Math.
**2003**, 157, 847–889. [Google Scholar] [CrossRef] - Lubinsky, D.S. Reflections on the Baker–Gammel–Wills (Padé) Conjecture. In Analytic Number Theory, Approximation Theory, and Special Functions; Springer: Berlin, Germany, 2014; pp. 561–571. [Google Scholar]
- Starovoitov, A.P.; Starovoitova, N.A. Padé approximants of the Mittag-Leffler functions. Sb. Math.
**2007**, 198, 1011. [Google Scholar] [CrossRef] - Winitzki, S. Uniform approximations for transcendental functions. In International Conference on Computational Science and Its Applications; Springer: Berlin, Germany, 2003; pp. 780–789. [Google Scholar]
- Atkinson, C.; Osseiran, A. Rational solutions for the time-fractional diffusion equation. SIAM J. Appl. Math.
**2011**, 71, 92–106. [Google Scholar] [CrossRef] - Zeng, C.; Chen, Y.Q. Global Padé approximations of the generalized Mittag-Leffler function and its inverse. Fract. Calc. Appl. Anal.
**2015**, 18, 1492–1506. [Google Scholar] [CrossRef] [Green Version] - Dumitru, B.; Kai, D.; Enrico, S. Fractional Calculus: Models and Numerical Methods; World Scientific: Singapore, 2012; Volume 3. [Google Scholar]
- Diethelm, K.; Ford, N.J.; Freed, A.D.; Luchko, Y. Algorithms for the fractional calculus: A selection of numerical methods. Comput. Methods Appl. Mech. Eng.
**2005**, 194, 743–773. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Tao, C. On the fractional Adams method. Comput. Math. Appl.
**2009**, 58, 1573–1588. [Google Scholar] [CrossRef] [Green Version] - Black, F.; Scholes, M. The pricing of options and corporate liabilities. J. Political Econ.
**1973**, 81, 637–654. [Google Scholar] [CrossRef] [Green Version] - El Euch, O.; Rosenbaum, M. Perfect hedging in rough Heston models. Ann. Appl. Probab.
**2018**, 28, 3813–3856. [Google Scholar] [CrossRef] [Green Version] - Gatheral, J. The Volatility Surface: A Practitioner’s Guide; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 357. [Google Scholar]
- Alos, E.; Gatheral, J.; Radoičić, R. Exponentiation of conditional expectations under stochastic volatility. Quant. Financ.
**2020**, 20, 13–27. [Google Scholar] [CrossRef] - Carr, P.; Madan, D. Option valuation using the fast Fourier transform. J. Comput. Financ.
**1999**, 2, 61–73. [Google Scholar] [CrossRef] [Green Version] - Lewis, A. Option Valuation under Stochastic Volatility; Technical report; Finance Press: Newport Beach, CA, USA, 2000. [Google Scholar]
- Lewis, A.L. A Simple Option Formula for General Jump-Diffusion and other Exponential Lévy processes. 2001. Available online: https://ssrn.com/abstract=282110 (accessed on 15 February 2020).
- Haubold, H.J.; Mathai, A.M.; Saxena, R.K. Mittag-Leffler functions and their applications. J. Appl. Math.
**2011**, 2011, 298628. [Google Scholar] [CrossRef] [Green Version] - Bateman, H. Higher Transcendental Functions [Volumes I-III]; McGraw-Hill Book Company: New York, NY, USA, 1953. [Google Scholar]

**Figure 1.**Re$\left[{D}^{\alpha}h(3-\frac{1}{2}\mathrm{i},x)\right]$ for $H=0.05$ and $H=0.499$. The long red dashed line is produced using fractional Adams method with $\mathrm{20,000}$ time steps, whereas the short green dashed line is produced using the multipoint Padé method.

**Figure 2.**Im$\left[{D}^{\alpha}h(3-\frac{1}{2}\mathrm{i},x)\right]$ for $H=0.001,0.1,0.2,0.3,0.4,0.499$. The long red dashed line is produced using fractional Adams method with $\mathrm{20,000}$ time steps, whereas the short green dashed line is produced using the multipoint Padé method.

u | ${\mathit{\chi}}_{\mathbf{Re}}$ | ${\mathsf{\Psi}}_{\mathbf{Re}}$ | ${\mathit{\chi}}_{\mathbf{Im}}$ | ${\mathsf{\Psi}}_{\mathbf{Im}}$ |
---|---|---|---|---|

0.001 | 0.0000 | 0.76% | 0.0000 | 0.47% |

1 | 0.0001 | 5.17% | 0.0001 | 2.53% |

2 | 0.0003 | 155.02% | 0.0004 | 10.88% |

3 | 0.0005 | 36.48% | 0.0010 | 22.30% |

4 | 0.0007 | 117.71% | 0.0016 | 37.03% |

5 | 0.0010 | 114.03% | 0.0023 | 53.30% |

6 | 0.0013 | 193.29% | 0.0030 | 64.30% |

7 | 0.0016 | 39.20% | 0.0038 | 69.46% |

8 | 0.0019 | 24.26% | 0.0047 | 71.08% |

9 | 0.0022 | 19.85% | 0.0056 | 70.74% |

10 | 0.0026 | 17.41% | 0.0066 | 69.36% |

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Jeng, S.W.; Kilicman, A.
Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods. *Symmetry* **2020**, *12*, 959.
https://doi.org/10.3390/sym12060959

**AMA Style**

Jeng SW, Kilicman A.
Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods. *Symmetry*. 2020; 12(6):959.
https://doi.org/10.3390/sym12060959

**Chicago/Turabian Style**

Jeng, Siow W., and Adem Kilicman.
2020. "Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods" *Symmetry* 12, no. 6: 959.
https://doi.org/10.3390/sym12060959