Risk Sensitivities of Bermuda Swaptions

Abstract

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. Market Model; Cheyette model. Int. J. Theor. Appl. Finan. 2004.07:465-509. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 02/03/15. For personal use only.466 V. V. Piterbargare called deltas. Typically, they are calculated and presented in the following way. A time line is split into time intervals (buckets) of common length, for example three months. Forward interest rates that correspond to all buckets are shocked in turn. A shocked interest rate curve is fed into the model, and the Bermuda swaption is revalued. The sensitivity of the Bermuda swaption value to this shock (often referred to as a "bucketed" delta) is then computed by subtracting a base value of the Bermuda swaption from its shocked value, and normalizing the difference by the size of the bump.Second-order derivatives to the changes in rates are also common, and usually called bucketed gammas.Bermuda swaptions are sensitive to changes in interest rate volatility as well. This sensitivity is also usually bucketed, in the sense that the Bermuda swaption sensitivity to various interest rate volatilities is computed. Bermuda swaptions are related to European swaptions. In most markets, European swaptions are actively traded and their implied volatilities are readily observable. For that reason, interest rate volatility sensitivities, called vegas, of Bermuda swaptions are computed with respect to shocks to European swaption volatilities.The Greeks calculation scheme, as presented above, requires a new value of a Bermuda swaption to be obtained for each bucketed Greek (sometimes even two new valuations are required if two-sided numerical derivatives are used). Thus, computing bucketed deltas to individual shocks of all 3-month tenor rates for a 40 year Bermuda swaption will require about 120 calls to the valuation model. Each valuation requires solving a PDE equation or a rollback on a lattice/tree. Adding gammas and vegas increases the number of valuations further. Multiplied by the size of a typical portfolio of Bermuda swap...

Numerical Methods in Finance

“…This result was then extended in [9], see also [37], to Bermudan options in terms of the Malliavin derivative process of X , without ellipticity condition. Here, we state it in terms of the first variation process rX , compare with Corollary 5.1 and see (5.3) in [9].…”

“…[15,26,37,58,59]) Assume X 2 L p 0 .˝; A; P / with a distribution P X . (a) ASYMPTOTIC ERROR BOUND (ZADOR'S THEOREM) (see e.g.…”

“…Particle filtering methods have been studied in the context of option pricing in [37] and [47]. Posterior inference on these unobservables is accomplished using sequential Monte Carlo (or particle filtering) methodology.…”

Pathwise derivative methods on single-asset American option sensitivity estimation

“…In the literature, two papers are closely related with the main theme of this paper. Piterbarg (2004a) and Piterbarg (2004b) present sensitivity estimators for American-style swaptions, which are options based on interest rate swaps. However, the approach in these two papers is very informal and not rigorous.…”