Narrow Results

A new approach to the problem of computing risk sensitivities of Bermuda swaptions in a lattice, or PDE, framework is presented. The algorithms developed perform the task much faster and more accurately that the traditional approach in which the Greeks are computed numerically by shocking the appropriate inputs and revaluing the instrument. The time needed to execute the tradition scheme grows linearly with the number of Greeks required, whereas our approach computes any number of Greeks for a Bermuda swaption in nearly constant time. The new method explores symmetries in the structure of Bermuda swaptions to derive recursive relations between different Greeks, and is essentially model-independent. These recursive relations allow us to represent risk sensitivities in a number of ways, in particular as integrals over the "survival" density. The survival density is obtained as a solution to a forward Kolmogorov equation. This representation is the basis for practical applications of our approach.

We show how Markovian projection together with some clever parameter freezing can be used to reduce a full-fledged local volatility interest rate model — such as Cheyette [1] — to a “minimal” form in which the swap rate evolves essentially like a dividend-paying stock. Using a number of numerical examples we compare such a minimal “poor man’s” model to a full-fledged Cheyette local volatility model and the market benchmark Hull–White one-factor model. Numerical tests demonstrate that the “poor man’s” model is in fact sufficient to price Bermudan interest rate swaptions. The main practical implication of this finding is that — once local volatility, dividend and short rate parameters are properly stripped from the volatility surface and interest rate curve — one can readily use the widely popular equity derivatives software for pricing exotic interest rate options such as Bermudans.