### Browsing by Department "Matematiikan laitos"

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Item The integrated volatility implied by option prices, a Bayesian approach(Teknillinen korkeakoulu, 2008) Kaila, Ruth; Matematiikan laitosIn this thesis, we present the new concept of implied integrated volatility. When the stock price volatility is stochastic, the integrated volatility is the time-average of the stock price variance. This volatility is a fundamental quantity in option theory, as the stock price returns depend on the stock price volatility only via the integrated volatility. The implied integrated volatility is the integrated volatility implied by option Hull-White prices. It is a stochastic extension of the Black-Scholes implied volatility. Unlike the latter, however, it is independent of the strike price of options. We suggest that this volatility can be used in volatility estimation, in pricing illiquid options consistently with corresponding liquid ones, and in hedging options. Estimating the implied integrated volatility is an ill-posed inverse problem. We present methods to estimate it within a Bayesian framework. This approach provides us with not only a point estimate, but also the possibility to gauge the reliability of this estimate.Item Self-improving phenomena in the calculus of variations on metric spaces(Teknillinen korkeakoulu, 2008) Maasalo, Outi Elina; Matematiikan laitosThis dissertation studies the integrability properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and a doubling measure. The work consists of three articles in which we study the higher integrability of functions satisfying a reverse Hölder inequality, quasiminimizers of the Dirichlet integral and superharmonic functions.