From Volatility Smile to Rough Volatility: a Comparative Analysis of Option Pricing Models.

This dissertation aims to analyze the evolution of the main volatility models used in quantitative finance, providing a theoretical framework and a natural empirical extension in Python. The starting point is the Black-Scholes-Merton model, taken as the theoretical baseline, highlighting its limitations in covering the presence of a volatility smile, skewness, and volatility clustering. This leads to Local Volatility models introduction, followed by Stochastic Volatility ones, with particular reference to the Heston framework. The analysis then shifts to rough volatility, motivated by empirical evidence showing that the regulaity of the Brownian Motion is not enough in current market scenarios. In this context, the Rough Heston model and several approximations aimed at reducing its computational complexity, such as the Poor Man’s Heston Approximation, are examined in depth. Next, the thesis considers a computationally innovative extension based on Stochastic Volterra Equations and convolution kernels, in which the memory component is modeled through a non-Markovian structure that can be transformed into a more tractable Markovian diffusion. This preserves the rough behavior of volatility while improving numerical simulation, especially in the fractional case. Building on the Volterra-type approach to memory in volatility dynamics, the final theoretical section introduces Path-Dependent Volatility models as a recent and tractable alternative. The focus is on a four-factor specification where volatility is driven by observable memory factors constructed from past returns, with the aim of improving the calibration of the option smiles. The models are tested empirically on real S&P 500 ETF option data. The overall objective is to show how volatility modeling has evolved from simplified and regular assumptions to increasingly flexible structures, capable of better capturing the irregularity, memory, and complexity of financial markets.