Abstract
We introduce a new numerical approach to value structured financial products. These financial products typically feature a large number of underlying assets and require the explicit modeling of the dependence structure of these assets. We follow the approach of Kraft and Steffensen (Rev Finance 11:209–252, 2006), who explicitly describe the possible value combinations of the assets via a Markov chain with a portfolio state space. As the number of states increases exponentially with the number of assets in the portfolio, this model so far has been – despite its theoretical appeal – not computationally tractable. The price of a structured financial product in this model is determined by a coupled system of parabolic PDEs, 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 n assets leads to a system of N = 2n coupled parabolic partial differential equations. It is shown that this high number of PDEs can be solved by combining an adaptive multiwavelet method with the Hierarchical Tucker Format. We present numerical results for n = 128.
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Notes
- 1.
With a slight abuse of notation, we set \(\mathbb{R}^{\mathcal{I}} =\ell _{2}(\mathcal{I})\) for any countable (possibly infinite) set \(\mathcal{I}\) as well as \(\mathbb{R}^{\mathcal{I}\times \mathcal{I}}\) as the set of linear operators from \(\ell_{2}(\mathcal{I})\) into \(\ell_{2}(\mathcal{I})\).
- 2.
The indexation here is adapted to our problem at hand and thus differs from the standard literature on the Hierarchical Tucker Format.
- 3.
By span(A) we denote the linear span of the column vectors of a matrix A.
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Kiesel, R., Rupp, A., Urban, K. (2014). Valuation of Structured Financial Products by Adaptive Multiwavelet Methods in High Dimensions. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_16
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