Portfolio Risk with Selected Revaluation

Calculation of portfolio risk distributions, e.g. as used in the calculation of a "value at risk", may involve a huge number of valuations for every single financial product in a portfolio. The usual situation is that a distribution of market scenarios is given by a very large set (a sample) of market data scenario X[j] (e.g. taken from a historical simulation) and that every product V[i] has to be valued under every scenarios X[j]. One application would be some stress-value-at-risk where X[j] are scenarios generated by augmenting the current market situation with market movements from a stress period.The brute force approach of valuing every product under every scenario results in an exorbitant amount of computation. This approach may be not feasible, especially when valuing complex products which in themselves require a computationally intensive numerical valuation.While many of the scenarios X[j] may involve only small changes in market data compared to the current market data, it is not admissible to assume smallness in general. In the application of tail risk the larger scenario moves will determine the results. In the case of stress-value-at-risk in particular we are dealing with large movements. Hence, a sensitivity approximation (Taylor expansion) is not feasible. On the other hand a sensitivity approximation can be performed at much smaller computational cost and may be well suited if either the movement is small or the product is almost linear.Furthermore, these calculations have to be performed daily.In this note we present a framework which allows one to keep the computational cost low at the same time using approximation with small approximation error even for scenarios representing large market data moves. The main idea is to reuse and adjust calculations from previous days