Option pricing with stochastic volatility and jump diffusion processes: comparing Heston, Merton, Bates

Introduction Black, Scholes and Merton in 1973 derived a closed-form solution for evaluating the price of a European call and put option written on a non- dividend paying stock. Their model, nowadays everywhere known as the Black-Scholes model, has been a real revolution in the finance world since it provided the first solution to the significant problem of option pricing. The main breakthrough has consisted in proposing a closed-form formula easily evaluable without using numerical methods. Of course the Black-Scholes framework has also several drawbacks which are direct consequences of those assumptions that make it so attractive. One of them comes out from the hypothesis that the stock price evolves according to a dynamic whose diffusion term is constant. Literature has demonstrated that this is not true in the real world. The implied volatility is not flat, but it depends on the maturity and the strike considered, and typically shows smiles (for currency options) or skews (for equity options). This has mispricing consequences in the valuation process, that is a systematic error in evaluating in-the-money and out-of- the-money options if the implied volatility of the at-the-money ones is used. From here a lot of different frameworks have been formulated, most of them trying to modify the Black-Scholes model in such a way to take into account the empirical evidence. In 1975 one of the first attempts had been made by Merton: he added an additional source of risk (a compound Poisson process) in the stock price dynamic leaving constant the diffusion term. With this introduction the stock price is affected not only by the classic Wiener process, but also by a jump process representing the arrival of new information which makes the price evolve according to non-continuous 11