Extracting volatility smiles from historical spot data

Abstract: The Black-Scholes model has been the fundamental framework for option pricing since its publication 1973, but it is known to have shortcomings. To correct for this, plenty of research in option pricing theory has been focused on calibrating a stochastic process to match asset behavior in the financial markets better than the geometric Brownian motion that Black-Scholes assume describe asset behaviour justly. A model that has gained popularity in the industry is the SABR volatility model. In this thesis we develop a numerical option pricing algorithm using the Hedged Monte Carlo method, for which we explore various modifications and additions. Due to its numerical nature, it can be used to price options without assuming a statistical process for the underlying asset. Instead, it estimates option prices based solely on historical data. We evaluate the algorithm with simulated data from the classic Black-Scholes framework and the SABR volatility model to see that the price estimates from our algorithm matches the theoretically correct values. Having validated the algorithm, we apply it on historical FX spot data and obtain empirical volatility smiles that lie close to the smiles observed in the current market.