Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation
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 modelling of the dependence structure of these assets. We follow the approach of Kraft and Steffensen (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 computational 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.