PhD Defense ved Martin A. Jönsson

Title: Essays in Quantitative Finance

The first two papers concern option pricing from two different perspectives: First, we study stochastic volatility models subject to parameter uncertainty in order to approach worst-case option prices based on a control theoretic approach. We explore a formulation with backward stochastic differential equations and detail several numerical methods for their solution. In an empirical study, we then compare the conservative model-derived prices with real market data of European call options. In the second paper, we study the delta-hedge for options in a setting where the hedger employs an erroneously specified volatility. We derive results for the incurred profit-and-loss from the hedge portfolio and explore two special cases: hedging with option-market implied volatility and hedging in accordance with the “true” dynamics. The theoretical implications invite the hedger to arbitrage opportunities; we scrutinize on their applicability in an empirical study.

 The third paper considers a practical hedging situation commonly faced by retailers in electricity markets. We look at the problem of price-quantity risk and study static hedging strategies for fixed-price-agreement contracts. We propose a bivariate model for this purpose and employ risk-minimization techniques. The approach is then empirically tested on market data from the Nordic power market and compared to industry standards. In the forth paper we look at the problem of optimal investment in a bond-stock-option economy driven by a stochastic volatility model. Based on martingale methods, we derive explicit formulas for the optimal portfolio strategy when the market option is plain vanilla and we set the suggested plan to work in an empirical study. In the final paper of the thesis we ask the simple question of what is a good model of volatility. In contrast to the usual measure of model-to-market fit for option prices, we focus on the volatility process itself and how well stochastic volatility models match its distributional properties. We suggest a goodness-of-fit analysis for this purpose and perform an extensive empirical study based on a market index and several individual stocks.

Supervisor: Prof Rolf Poulsen, Math, University of Copenhagen

Assessment committee:

Prof. Mogens Steffensen (chairman), MATH, University of Copenhagen

Prof. Antje Mahayni, Universität Duisburg-Essen

Prof. Claus Munk, Copenhagen Business School