Rough Stochastic Volatility Models And The Applications In Option Pricing

This thesis constructs rough Heston Model with volatility of volatility(vol-of-vol)by modifying generalized nonlinear Hawkes Process which is nearly unstable.That is,vol-of-vol is not a constant but a function about time t.Then this thesis derives the model characteristic function by virtue of affine process.Moreover,this thesis proves the existence,uniqueness and regularity of the solution to nonlinear fractional Riccati Equation and it is solved by the Adams methods.Besides,Fourier-cosine and Adams are used to price and calibration,also,numerical example are used to verify the convergence of the theory and effectiveness of the model.The thesis further studies the piecewise rough Heston Model and its option pricing problem.First,under the fractional Brownian motion with the Hurst index H as piecewise constant,a piecewise rough Heston model is created.Next,utilizing Euler-Maruyama scheme and fast Euler-Maruyama scheme,and combing the Monte Carlo method for pricing.Then,this thesis optimizes and calibrates the model and numerical examples verify the effectiveness of the piecewise rough Heston model.Furthermore,this thesis studies the European option pricing under the multivariate rough Heston model.Under the multivariate rough Heston model,obtain the characteristic function of the asset log price is determined by the system nonlinear fractional Riccati equations.Then,the existence,uniqueness and regularity of solution to system nonlinear fractional Riccati equations are proved and the equations are solved by the Adams scheme,higher order scheme.And then,the Fourier-cosine methods are combined with the Adams scheme,higher order scheme to price the options.Finally,numerical examples verify the correctness of the theoretical results and the effectiveness.