In the process of modeling, the agent has to make a number of decisions. Therefore two agents can set up different models. This thesis compares the influence of different model choices to stochastic control theory and option pricing. On the one hand it quantifies analytically error bounds in terms of key parameters. On the other hand it checks the implications of model choices empirically.
Lévy processes are popular in financial modeling since they are able to explain many of the stylized facts of asset prices. In particular, some processes like the normal inverse Gaussian (NIG) or the hyperbolic Lévy process have become particularly relevant since they are able to capture the return distribution of most asset prices. These Lévy processes are pure-jump, and therefore give distinctively different paths of the asset prices compared to a Brownian motion with continuous paths. In empirical analysis of financial price data, one may detect big jumps, however, the small jumps are very hard to separate from the observations of a Brownian motion. Thus, it is not a simple task to decide whether a Lévy process with jumps or a Brownian motion is governing the small variations in a stock price, say. First Merton’s portfolio optimization problem is considered. We aim for a mathematical quantification of the difference of the optimal investment strategies and find error bounds that are proportional to the variation of the small jumps. Then option pricing is considered, where there are two underlying assets that are dependent. Here the error bounds turn out to be of the same type as in Article 1.
The second part of the thesis considers the pricing of options on forwards in energy markets. Many models for the electricity spot price divide the price evolutions into a short-term and a long-term component. Electricity markets are well-known for their large price variations and rare, big spikes, which are captured by the short-term component. We examine the influence of the short-term and long-term factor on the spot and prove that the short-term factor is insignificant for pricing options in many relevant cases.
The electricity spot is not a tradable asset and the no-arbitrage argumentation that is based on cost and carry strategies cannot be applied. Therefore one can use different measures to price the futures and the option. The goal of this paper is to examine empirically wether or not the traded options are priced under the same pricing measure as the futures. Before doing so, we need to specify a model for spot and futures and fit it to data. The results indicate that one should use a different measure to price the option.
List of articles.
Article 1 / Chapter 2: Benth, F. E., and Schmeck, M. D.: Stability of Merton’s portfolio optimization problem for Lévy models. Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports Volume 85, Issue 5, 2013, pages 833-858. This is an Author's Original Manuscript of an article whose final and definitive form, the Version of Record, has been published in the Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports 25 Apr 2012 copyright Taylor & Francis, available online at: doi:10.1080/17442508.2012.665056
Article 2 / Chapter 4: Benth, F. E., and Schmeck, M. D. (2012). Pricing and Hedging Options in Energy Markets by Black-76. This is the pre-peer reviewed version of the following article: Pricing and hedging options in energy markets using Black-76 (2014), with F.E. Benth. The Journal of Energy Markets. 7(2) pp. 35–69 risk.net
Article 3 / Chapter 5: Benth, F. E., and Schmeck, M. D. (2012). Pricing Futures and Options in Electricity Markets. Submitted version, published as: Pricing futures and options in electricity markets (2014), with F.E. Benth. In Sofia Ramos & Helena Veiga (ed.), The Interrelationship Between Financial and Energy Markets. Springer Publishing Company. pp. 233–260