Stochastic volatility models: present, past and future

Abstract

[en] In Chapter 1, we will introduce the Black-Scholes model and a brief introduction to quantitative finance concepts related to this model. In Chapter 2, we will talk about implied volatility and how to calculate it by numerical methods. In Chapter 3 we will introduce the stochastic volatility models and the jump volatility models studied by Hull and White in [12], Fouque, Papanicolau and Sircar in [8] and by Merton in [19]. In Chapter 4, we will introduce the statics and dynamics of implied volatility based on Lee’s paper [16]. In addition, we will plot the volatility smile and volatility skew based on models introduced in Chapter 3. In Chapter 5 we will introduce fractional Brownian motion, which has an important role in many fields, as meteorology, finance, telecommunications and hydrology, the last is because Hurst observed that Nile river water had a consistent cyclical behavior, which for seven consecutive years the water level increased and was greater than in the following seven years, which in turn created a cycle of seven years of abundance and seven years of scarcity. Until then, it was thought that there was no depending on the behavior of the increase in water between one year and another. In addition, we will introduce some concepts on Malliavin calculus to introduce the fractional volatility model studied by Alòs, León and Vives in [2].