Iterating cancelable snowballs and related exotics in a many-factor Libor model

We propose a valuation method for exotic cancelable and callable structures in a multi-factor Libor model which are path-dependent in the sense that after canceling or calling, one cancels a sequence of cash-flows or receives a sequence of cash-flows in the future, respectively. The method, which is based on a Monte Carlo procedure for standard Bermudans recently developed and further extended in Kolodko and Schoenmakers [8], Bender and Schoenmakers [2], is compared to and also combined with popular known approaches by Andersen [1], Longstaff and Schwartz [9], and Piterbarg [10]. As a main example we consider the (cancelable) snowball swap, a highly sensitive interest rate product with growing popularity, in a full-blown Libor market model. From the treatment of this example it will be clear how to design Monte Carlo valuation algorithms for related cancelable or callable path-dependent products. For our example it turns out that price lower bounds obtained by the regression approach using explanatory variables as in [10] may be significantly off. Even an enhancement of this approach by an Andersen like modification of the corresponding exercise boundary does not lead to an acceptable small gap between the lower price and an upper bound price obtained via the dual method by Rogers [12], Haugh and Kogan [5] (see [7] for an alternative dual and [3] for upper bounds via consumption processes). However, via improving the stopping rule entailed by this enhancement using the iteration procedure in [8] we end up with acceptable prices. The proposed approach is quite generic, as in principle it only requires a Monte Carlo simulation mechanism for an underlying Markovian system, for instance a Markovian system of SDEs. In particular, it can be used to improve upon popular methods, such as [1], [9], and [10] to get satisfactorily accurate target results (in this respect the method outperforms the standard policy iteration described in [11], see [8] for details). Straightforward application of the policy iteration procedure based on [8] requires just like the duality approach a nested Monte Carlo simulation and is thus rather slow. Therefore we include a method of variance reduction which has a flavor of stratified sampling. Moreover, we underline that the improved stopping rule which is important for the buyer of the product for instance, can already be obtained at the cost of a standard (not nested) Monte Carlo simulation.