## Modelling and Simulation of Stochastic Volatility in FinanceThe famous Black-Scholes model was the starting point of a new financial industry and has been a very important pillar of all options trading since. One of its core assumptions is that the volatility of the underlying asset is constant. It was realised early that one has to specify a dynamic on the volatility itself to get closer to market behaviour. There are mainly two aspects making this fact apparent. Considering historical evolution of volatility by analysing time series data one observes erratic behaviour over time. Secondly, backing out implied volatility from daily traded plain vanilla options, the volatility changes with strike. The most common realisations of this phenomenon are the implied volatility smile or skew. The natural question arises how to extend the Black-Scholes model appropriately. Within this book the concept of stochastic volatility is analysed and discussed with special regard to the numerical problems occurring either in calibrating the model to the market implied volatility surface or in the numerical simulation of the two-dimensional system of stochastic differential equations required to price non-vanilla financial derivatives. We introduce a new stochastic volatility model, the so-called Hyp-Hyp model, and use Watanabe's calculus to find an analytical approximation to the model implied volatility. Further, the class of affine diffusion models, such as Heston, is analysed in view of using the characteristic function and Fourier inversion techniques to value European derivatives. |

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### Contents

1 | |

5 | |

6 | |

22 Affine diffusion stochastic volatility models | 13 |

Monte Carlo methods | 29 |

32 QuasiMonte Carlo | 32 |

33 Path construction methods | 34 |

34 Fractional Fourier transformation for spectral path construction | 46 |

53 Optimal choice of alpha | 97 |

54 Numerical results | 109 |

Numerical integration schemes for stochastic volatility models | 113 |

61 Numerical integration of meanreverting CEV processes | 115 |

62 Numerical integration of stochastic volatility models | 127 |

63 Multidimensional stochastic volatility models | 146 |

Conclusion | 161 |

Balanced Milstein Methods for ordinary SDEs | 163 |

### Common terms and phrases

5th-order aﬃne diﬀusion analytical asymptotic expansion at-the-money Balanced Milstein Method Black-Scholes model Brownian bridge Brownian motion calculate characteristic function complex logarithm conditional expectations conditional Monte Carlo control variate covariance matrix deﬁned derive diﬀerent Diffusion interpolation Drift discretisation Drift interpolation eﬀective eﬃcient error European option exponential ﬁgure ﬁnite ﬁrst fractional Fourier transformation Gaussian given Heston model hyperbolic transformation implied volatility implied volatility surface integrand inverse Jšackel Kahl Lemma linear local volatility log-Euler mean-reverting CEV process numerical integration numerical integration schemes obtain optimal option pricing formula Ornstein-Uhlenbeck process path construction methods Pathwise Adapted Linearisation positive deﬁnite problem Proof saddlepoint approximation semi-analytical simulation Skew solution spectral decomposition spectral path construction stepsize stochastic diﬀerential equation stochastic volatility models Strong convergence term theorem transformation function transformed Ornstein-Uhlenbeck process underlying asset variance Watanabe’s Wiener process x-axis y-axis