Aplikace f-divergence ve financích

• Název práce v češtině: Aplikace f-divergence ve financích
• Název v anglickém jazyce: f-divergence applications in finance
• Klíčová slova: f-divergence|relativní entropie
• Klíčová slova anglicky: f-divergence|relative entropy
• Akademický rok vypsání: 2024/2025
• Typ práce: diplomová práce
• Ústav: Katedra pravděpodobnosti a matematické statistiky (32-KPMS)
• Vedoucí / školitel: doc. RNDr. Jan Večeř, Ph.D.

Zásady pro vypracování

The aim of this thesis is to link the concept of an f-divergence used in information theory to the various performance statistics used in finance. The f-divergence measures the difference between two probability measures in terms of an expectation of the likelihood ratio of the two measures evaluated in a convex function f with f(1) = 0. The most prominent example of f-divergence is the Kullback-Leibler divergence known as relative entropy. An important observation is that the prices in finance can be expressed in terms the likelihood ratios of the respective state price densities of the assets and thus the f-divergence can be understood as a type of an option payoff. It turns out that many examples of f-divergencies are fully used in finance in terms of various contracts and performance measures, but there are still several examples that are not explored in practice. f-divergencies that act symmetrically on the individual densities can be related to performance benchmarking with respect to an index. The fact that finance allows for performance measures involving multiple assets (more than two), the question is whether the concept of f-divergence to be generalized to multiple measures. A natural generalization is possible for total variance that can be linked to the maximum likelihood estimator.

The ultimate goal of the student is to explore the links between the f-divergencies and finance, illustrate the properties, suggest generalizations to multiple assets and compute the exact values in different probabilistic models. A special attention should be given to the total variance and Kullback-Leibler divergence.

Seznam odborné literatury

Csiszár, I. (1967). "Information-type measures of difference of probability distributions and indirect observation". Studia Scientiarum Mathematicarum Hungarica. 2: 229–318.

Csiszár, I.; Shields, P. (2004). "Information Theory and Statistics: A Tutorial" Foundations and Trends in Communications and Information Theory. 1 (4): 417–528.

Liese, F.; Vajda, I. (2006). "On divergences and informations in statistics and information theory". IEEE Transactions on Information Theory. 52 (10): 4394–4412.

Kullback, Solomon (1959), Information Theory and Statistics, John Wiley & Sons. Republished by Dover Publications in 1968; reprinted in 1978: ISBN 0-8446-5625-9.