Researches On The Small-time A Symptotics Simulation And Markovian Approximation Of The Rough Bergomi Model

The rough Bergomi(rBergomi)model has gained acceptance for stochastic volatility modelling due to its power-law at-the-money(ATM)volatility skew which is consistent with empirical studies and the market impact function under the no-arbitrage assumption.However,the stochastic process which characterizes this volatility model is rougher than that of a Brownian motion.In particular,the lack of Markovianity makes the classical pricing and hedging methods infeasible such as PDEs or Fourier transform,and which also makes the risk management of derivatives an intricate task.Due to this fact,the construction of a dynamically consistent model is not a trivial exercise.A large body of works presents different novel methods to occur that deficiency.In this thesis,we present the asymptotic approximation with a stochastic trace estimation(STE)of the implied volatility(IV)smiles for the short-dated rBergomi model.As for the long-dated rBergomi model,we justify the conventional 2-factor Bergomi model as the tractable aBergomi model for the Markovian approximation of the IV smiles.For the short-dated rBergomi asymptotics,the pathwise large derivation behaviour under the rBergomi dynamics is studied.Based on the IV expansions which build on the large derivations,there emerges several simulation methods,e.g.Forde-Zhang method,Karhunen-Loeve(KL)method and a0 method.However,those methods all encounter some difficulties in practice.They reveals either the inaccuracy,or the inefficiency.We implement an improvement on the KL decomposition to save the mass computation when the Hurst parameter is close to zero.Meanwhile,the rate function is still computed by the Ritz method.Our improvement is using the STE method to conquer the large demand of basis functions in the KL method.Therein,the linear-time randomized algorithm for approximating the trace of matrix functions is coupled with the Chebyshev interpolation and the Hutchinson's method.We also demonstrate our simulation method for the short-dated rBergomi IV smiles numerically.For the long-dated rBergomi approximation,its non-Markovianity brings mathematical and computational challenges for model calibration and simulation.To overcome these difficulties,we show that the rBergomi model can be approximated by the aBergomi model,which has the Markovian property.Our main theoretical result is to establish and describe the quasi-affine structure of the rBergomi model and the corresponding affine structure of our aBergomi model.We experimentally evaluate the Markovian approximation's accuracy and efficiency.