Calibration and Hedging in Finance
Abstract: This thesis treats aspects of two fundamental problems in applied financial mathematics: calibration of a given stochastic process to observed marketprices on financial instruments (which is the topic of the first paper) and strategies for hedging options in financial markets that are possibly incomplete (which is the topic of the second paper).Calibration in finance means choosing the parameters in a stochastic process so as to make the prices on financial instruments generated by the process replicate observed market prices. We deal with the so called local volatility model which is one of the most widely used models in option pricing across all asset classes. The calibration of a local volatility surface to option marketprices is an ill-posed inverse problem as a result of the relatively small number of observable market prices and the unsmooth nature of these prices in strike and maturity. We adopt the practice advanced by some authors to formulate this inverse problem as a least squares optimization under the constraint that option prices follow Dupire’s partial differential equation. We develop two algorithms for performing the optimization: one based on techniques from optimal control theory and another in which a numerical quasi-Newton algorithmis directly applied to the objective function. Regularization of the problem enters easily in both problem formulations. The methods are tested on three months of daily option market quotes on two major equity indices.The resulting local volatility surfaces from both methods yield excellent replications of the observed market prices.Hedging is the practice of offsetting the risk in a financial instrument by taking positions in one or several other tradable assets. Quadratic hedging is a well developed theory for hedging contingent claims in incomplete markets by minimizing the replication error in a suitable L2-norm. This theory, though, is not widely used among market practitioners and relatively few scientific papers evaluate how well quadratic hedging works on real marketdata. We construct a framework for comparing hedging strategies, and use it to empirically test the performance of quadratic hedging of European call options on the Euro Stoxx 50 index modeled with an affine stochastic volatility model with and without jumps. As comparison, we use hedging in the standard Black-Scholes model. We show that quadratic hedging strategies significantly outperform hedging in the Black-Scholes model for out of the money options and options near the money of short maturity when only spot is used in the hedge. When in addition another option is used for hedging, quadratic hedging outperforms Black-Scholes hedging also for medium dated options near the money.
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