Stochastic volatility and option pricing with long-memory in discrete and continuous time

It is commonly accepted that certain financial data exhibit long-range dependence. We consider a continuous-time stochastic volatility model in which the stock price is Geometric Brownian Motion with volatility described by a fractional Ornstein--Uhlenbeck process. We also study two discrete-time models: a discretization of the continuous model via a Euler scheme and a discrete model in which the returns are a zero mean i.i.d. sequence where the volatility is a fractional ARIMA process. We implement a particle filtering algorithm to estimate the empirical distribution of the unobserved volatility, which we then use in the construction of a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long-memory parameter we compute an implied value by calibrating the model with real data. We compare the performance of the three models using simulated data and we price options on the S&P 500 index.