Da Black e Scholes ai modelli a Volatilità Locale: analisi della volatilità implicita

In this work we deal with the problem of the implied volatility dynamics. Our aim is to find a model consistent with market quotes of implied volatility and a model that reproduces the volatility smile observed in market data. In 1973, Black and Scholes introduced the famous Black-Scholes model for derivatives pricing. They derived a pricing PDE that defines the evolution of the option price over time. In the traditional Black-Scholes framework a constant volatility value is assumed. The implied volatility of an option is the volatility value that reproduces the market price in the Black-Scholes model. If we compute the implied volatilities from market data, we find different volatilities for different maturities and strike values. So the Black-Scholes model is not able to reproduce the skewness of the implied volatility curve that is present in the market data. During the last twenty years several models have been proposed to improve the classic Black-Scholes framework for equity derivatives pricing. In particular, two main strands of research have been widely developed and used: Local Volatility and Stochastic Volatility. Both these approaches relaxed the Black-Scholes hypothesis of a constant volatility. In fact, Local Volatility models assume volatility to be a deterministic function of the underlying asset and time, whereas Stochastic Volatility models consider volatility as a random process itself. While the former models are able to be well calibrated to traded vanilla options, the latter can reproduce a more realistic dynamics of implied volatility. In this thesis, we study the Local Volatility models. In 1994 Dupire introduced a non-parametric Local Volatility model which is consistent with the volatility smile. Dupire was able to demonstrate the existence and uniqueness of the local volatility function under arbritrage-free hypothesis and furthermore this model is as complete as the Black-Scholes one. The model can be calibrated exactly to any given set of arbitrage-free implied volatilities. However, building this arbitrage-free implied volatility surface turns out to be a challenging task. The structure of this thesis is as follows: we start out in Chapter 1 with a short overview of preliminary mathematical knowledge. We present mathematical definitions and results of, among others, options, arbitrage, risk-neutral measure, discrete time models and the Fundamental Theorems of Asset Pricing. Then we handle continuous time models, Itô Calculus and Stochastic Integral theory. In Chapter 2 we briefly introduce the Black-Scholes model, outlining its strengths and weaknesses. Then we focus on the implied volatility problem and on the volatility stylized facts. Here we provide an example using market data on the Dow Jones Industrial Average Index. Chapter 3 is the main part of this work and it deals with Local Volatility models. First, a distinction is made between parametric and non-parametric models, after which several derivations of Dupire’s formula are proposed. We derive an expression for the local volatility model in terms of European call option prices and in terms of implied volatilities. Because of that, we introduce Dupire’s equation which is a sort of Fokker-Plack equation. We also discuss the Constant Elasticity of Variance model, a parametric Local Volatility model. Then we handle the SVI parameterization that we will use as input for Dupire’s local volatility model. We made also a briefly digression on arbitrage-free conditions. We derive conditions to guarantee the absence of arbitrage in the call option price surface and equivalent conditions for the total variance surface. In the last Chapter we report all the results we have carried out in the previous parts, adding many implementational details. The outcomes are presented with the aid of graphs and tabs.