Risk sensitive utility indifference pricing of perpetual American options under fixed transaction costs
Friday, 5.12.14, 11:30-12:30, Raum 404, Eckerstr. 1
The problems of risk sensitive portfolio optimisation under transaction costs have taken a considerable attention in the recent literature on mathematical finance. We study the associated problems of risk sensitive utility indifference pricing for perpetual American options with fixed transaction costs in the classical model of financial market with two tradable assets. Assume that the investors trading in the market must pay transaction costs equal to a fixed fraction of the entire portfolio wealth each time they trade. The objective is to maximise the asymptotic (risk null and risk adjusted) exponential growth rates based on the expected logarithmic or power utility of the difference between the terminal portfolio wealth and a certain amount of the option payoffs. It is shown that the optimal trading policy keeps the number of shares held in the assets unchanged between the transactions. In order to determine the optimal trading times and sizes, we reduce the initial problems to the appropriate (discounted) time-inhomogeneous optimal stopping problems for a one-dimensional diffusion process representing the fraction of the portfolio wealth held by the investor in the risky asset. The optimal trading and exercise times are proved to be the first times at which the risky fraction process exits certain regions restricted by two time-dependent boundaries. Then, certain amounts of assets should be bought or sold or the options should be exercised whenever the risky fraction process hits either the lower or the upper time-dependent curve. The latter are characterised as unique solutions of the associated parabolic-type free-boundary problems for the value functions satisfying the smooth-fit conditions at the curved boundaries. The optimal asymptotic growth rates and trading sizes are specified as parameters maximising the value functions of the resulting optimal stopping problems. We illustrate these results on the examples of the perpetual American call and put as well as the asset-or-nothing options, for which we obtain the utility indifference prices as well as the optimal trading and exercise boundaries in a closed form.
Statistical analysis of modern sequencing data – quality control, modelling and interpretation
Friday, 23.1.15, 11:30-12:30, Raum 404, Eckerstr. 1
Causal Discovery From Bivariate Relationships
Friday, 6.2.15, 11:30-12:30, Raum 404, Eckerstr. 1
In Causal Discovery, we ask which models from a certain causal model class\n(e.g., DAGs) would be consistent with a given dataset. Many causal discovery\nalgorithms are based on conditional independence testing. However, conditional\nindependence is difficult to test, especially when parametric assumptions like\nnormality cannot be made. Hence, we ask to what extent causal discovery is still\npossible when we restrict our attention to only pairwise relationships, for\nwhich a wide variety of both parametric and non-parametric statistical\nindependence tests are available. Suprisingly, we find that the entire class of\nedge-maximal DAGs that are consistent with a given set of pairwise dependencies\ncan be described by a single graph, which can be constructed by a rather simple\nalgorithm. Furthermore, we give a precise characterization of how much\ndiscrimination power we lose by not looking at conditional independencies.\nFinally, we empirically investigate the failed discovery rate of the pairwise\napproach -- assuming a correct DAG exists, how often is it rejected? -- and\ncompare the results to those the partial correlation based PC algorithm.\n
Parameter selection for nonlinear modeling using L1 regularization
Friday, 13.2.15, 11:30-12:30, Raum 404, Eckerstr. 1
A major goal in systems biology is to reveal potential drug targets for cancer therapy. A common property of cancer cells is the alteration of signaling pathways triggering cell-fate decisions resulting in uncontrolled proliferation and tumor growth. However, addressing cancer-specific alterations experimentally by investigating each node in the signaling network one after the other is difficult or even not possible at all. Here, we combine quantitative time-resolved data from different cell lines with non-linear modeling under L1 regularization, which is capable of detecting cell-type specific parameters. To adapt the least-squares numerical optimization routine to L1 regularization, sub-gradient strategies as well as truncation of proposed optimization steps were implemented. Likelihood-ratio tests were used to determine the optimal penalization strength resulting in a sparse solution in terms of a minimal number of cell-type specific parameters that is in agreement with the data. The uniqueness of the solution is investigated using the profile likelihood. Based on the minimal set of cell-type specific parameters experiments were designed for improving identifiability and to validate the model. The approach constitutes a general method to infer an overarching model with a minimum number of individual parameters for the particular models.