Calibration and Simulation of the Heston Model

Abstract

In this thesis, we present a comprehensive guide on how to implement and calibrate the Heston model using real market data, and on how to price American options using Monte Carlo simulations. We address the challenges that will arise in the implementation of the pricing formula and explore the prevalent solutions found in the existing literature. To enhance calibration accuracy we examine two optimization algorithms: the deterministic Nelder-Mead and the stochastic Differential Evolution. Through testing we discern that, while the former outperforms the latter in terms of calibration speed, the latter significantly surpasses the former in accuracy. We then go on to calibrate the model to S&P500 index options with European exercise using the Differential Evolution algorithm. The resulting volatility surface shows an overall very good fit to the empirical surface. However, we rediscover what has already been shown multiple times in the existing literature; that the Heston model has troubles fitting the smile for short maturities. For the purpose of pricing American options we then go on to compare several Monte Carlo simulation schemes. Exact simulation of the Heston model is both complex and very slow. Through a review of existing research in the field we find that a non-specialized approach with a log-Euler scheme for the stock price and a Milstein scheme for the variance yields a very good trade-off between computational speed and accuracy. Furthermore, we implement and evaluate several variance reduction techniques, and show, that control variates results in the highest reduction in the standard deviation of the estimates. Finally, using the parameters obtained by the calibration, the optimized discretization scheme and the control variates for variance reduction, we then price American options on the SPX index using the LSM algorithm. The results are shown to be consistent with theoretical pricing properties of American options.