High dimensional wavelet methods for structured financial products

Within the last decade structured financial products have become an important tool for the risk management of financial institutions. These products had a major role in the financial crisis of 2007, partly due to oversimplifications of the mathematical models and, as a result, the market for these products strongly declined. However, a major part of these products is still on the balance sheets of companies worldwide and nowadays, the market for structured financial products is back on the rise. This thesis investigates a new numerical approach to value structured financial products. The main difficulty of the pricing these products is the inclusion of the portfolio dependency structure and the integration of the vast number of states of the portfolio. As a starting point for the numerical valuation, the model of Kraft and Steffensen (2006) has been chosen, which describes the portfolio states by a Markov chain. As the number of states increases exponentially with the number of assets in the portfolio, this model is mainly of theoretical importance. The price of a structured financial product in this model is described by a coupled system of partial differential equation, describing the value of the portfolio for each state of the Markov chain depending on the time and macroeconomic state variables. A typical portfolio of 125 assets leads to a system of 2125 coupled parabolic partial differential equations. To solve this enormous number of equations numerically with wavelet methods, two tools are essential. First, the construction of a L2-orthonormal wavelet bases is introduced and, second, an implementation of the hierarchical Tucker format, a linear algebra format for the efficient handling of extremely large matrices and vectors is described. The L2orthonormal wavelet basis serves primarily to allow for several macroeconomic state variables without increasing the condition of the associated linear system, whereas the implementation of the hierarchical Tucker format contributes to the manageability of the large amount of states in the Markov chain. A numerical experiment shows that combining these two tools, the pricing of structured financial products is possible without the so far necessary simplification of the underlying portfolio. Furthermore, an algorithm for the valuation of portfolio tranches is suggested, based on the efficient search of certain, affected states of the Markov chain.

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