Informational markets: Risk modelling, risk management and portfolio analysis

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An informational market is a market where there are two flows of information. One flow denoted by F represents the ``public" information, which is available to all agents, and a larger flow G that contains F and additional information about events that might affect the market. Thus, our starting point in this project is the initial market model described by the pair (S,F), to which we add several random times. Here S represents the stocks' price process and F is the filtration that models the ``public" of information. The random times can model the death times of insured agents for life insurance, or the defaults times of firms for credit risk theory, or more generally the occurrence times of events that might impact the market. Given that death times can not be seen before they occur, the progressively enlargement approach for modelling the connection between F, G and the random times, seems tailored-fit for these aforementioned domains. In this setting, our main objectives reside in understanding the different types of risks induced by the uncertainties in the random times alone and/or jointly with the filtration F. Precisely, we are interested in answering the following questions: Can we single out the risk(s) that can be classified as ``pure random times' risk"? How many are there of such risk(s)? How these risk(s) interplay with the risk(s) coming from the initial market (S,F)? Above all, can we define a risk-basis for the larger flow G, and every risk with respect to this filtration can be decomposed uniquely with respect to this basis? Via detailed answers to these questions, we will deeply understand how risks build up as soon as we add more uncertainties to a model, we provide a cutting- edge mathematical modelling and quantification for the risk. This will allow us to efficiently design risk management tools afterwards. Besides these, the projected applications of our study are numerous and multifold. First of all, we will be able to tell fully how far the additional uncertainties affect the fundamental financial and/or economic concepts of arbitrage theory and market's viability that are intimately related to market's liquidity. Furthermore, when these concepts fail for (S,G), we should be able to suggest market's regulation(s) policies in order to repair the damage induced by the ``extra" information. Secondly, we want to describe the dynamics and the stochastic structures for mortality and/or longevity derivatives, such as longevity bonds, which are highly important for the mortality/longevity securitization. We are also interested in describing completely and fully the set of all deflators under G, which is interesting in itself besides its vital role in the portfolio analysis. Finally, we are will investigate the impact of the random times on optimal portfolios (for various optimization criterion), the numeraire portfolio, and on the existing pricing rules as well.