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The Homogeneous Transformation

Chapter
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Part of the Applied Quantitative Finance book series (AQF)

Abstract

In general, the number of sub-FTDs in the replication formula is a function of n, the size of the basket, and k, the order of the basket default swap. The most time-consuming step in the evaluation is the generation of the sub-FTDs, for all possible combinations. If we had a homogeneous basket, then, for a given subset size l, all the FTD instruments would have exactly the same value; and the pricing equation would simplify substantially. In particular, the number of sub-FTDs to compute, would reduce to one evaluation per l-subset, hence a total of \(N\left( k,n\right) =k\) FTD evaluations for the whole \(k{\text {th}}\)-to-default swap. The first (natural) approximation that we consider is to transform the original non-homogeneous basket to a homogeneous one while preserving some properties of the aggregate default distribution. In the approach described here, for each default order, we use the corresponding percentile of the aggregate default distribution, and we require that this quantity remains invariant with respect to the homogeneous approximation. We shall see that this transformation is exact for an FTD swap, and that, for higher-order defaults, the approximation gives very good results.

References

  1. J.P. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, New York, 2000)Google Scholar
  2. S.M. Ross, Simulation (Academic Press, Boston, 1997)Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.LondonUK

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