Two simulation schemes for the rough Heston model: a comparison

Since the publication of the pioneering article Volatility is Rough, it is commonly accepted the fact that the logarithm of the market volatility behaves like a fractional Brownian motion with Hurst index H < 1/2 . Because of this, rough volatility models are very popular nowadays. In particular we want to focus our attention on the so called rough Heston model: for this kind of model the volatility is described by a one dimensional, affine Volterra process. Even if this choice is coherent with some market features and the characteristic function of the log-price is known in semi-closed form, the drawback is that the variance process is not a Markov process and so some difficulties arises in the simulation. Indeed, for the rough Heston model, the only one simulation method known up to now is the Hybrid Quadratic Exponential (HQE) scheme. On the other hand, one other possibility, is to approximate the rough Heston model with the lifted Heston model which is based on the substitution of the fractional kernel, which appears in the original model, with the sum of an appropriate number of exponential kernels: this choice preserves the Markovian structure of the variance process and simplifies the simulation. Recently, one new simulation scheme which deals with the lifted Heston model. Thus the goal of this thesis is to compare this new scheme with the HQE: in particular we will perform different numerical tests in order to check the level of accuracy of these schemes in reproducing some market features.