Pricing callable libor exotics using forward libor models

Abstract

Callable Libor exotics are among the most challenging interest rate derivatives to price and risk-manage. These derivative contracts are loosely defined by the provision that the holder has a Bermuda-style (i.e. multiple-exercise) option to exercise into various underlying interest rate instruments. The instruments into which one can exercise can be, for instance, interest rate swaps (for Bermuda swaptions), interest rate caps (for captions, callable capped floaters, callable inverse floaters), collections of digital call and put options on Libor rates (callable range accruals), collections of options on spreads between various CMS rates (callable CMS coupon diffs), and so on. From a modeling prospective, callable Libor exotics are difficult to handle. Simple, “first generation” interest rate models like Ho-Lee, Hull-White, Black-Karasinsky cannot be used because of their inability to calibrate to a rich enough set of market instruments. One has to use “second generation” models with richer, more flexible volatility structures. Among the latter, forward Libor models (also known as Libor Market and BGM models) are arguably the best suited for the job. Building a pricing and risk management framework for callable Libor exotics based on forward Libor models is a formidable task. Conceptual and technical issues abound. Calibration, valuation and risk sensitivities calculations all present unique challenges. This paper takes a look at these problems and possible ways that may be used to address these challenges. We aim to present a comprehensive review of problems one has to deal with when developing the callable Libor exotics capability for forward Libor models. To get an idea of the scope of the paper, one just needs to ask why using forward Libor models for callable Libor exotics (CLE for short) is so hard? Problems start with volatility calibration. Multiple types of optionality embedded in CLEs mean that they depend on volatilities of many different rates. What market instruments do we calibrate the model to? Matching today’s prices of market instruments is just one part of the solution. Choices affecting the dynamics of the volatility structure have a significant impact on CLE prices. Questions arising out of this situation though in the domain of general interest rate modeling, have profound importance for callable Libor exotics. After calibration comes pricing. Pricing must be done using Monte-Carlo, as it is the only viable numerical method available for forward Libor models. Successful pricing of Bermuda-style options in Monte-Carlo hinges on the ability to formulate good rules for choosing exercise strategies. For instruments as complex as callable Libor exotics, what are they? How do we make sure we are not significantly underpricing CLE’s because our exercise boundaries are not optimal? For pricing, the issues of speed and accuracy must also be addressed. A Monte-Carlo valuation is typically quite slow. What methods do we use to speed valuation up and/or make it more accurate? What variance reduction techniques work and what do not? As we move into the realm of Greeks calculations, problems become really hard. Obtaining good, clean and robust risk sensitivities from aMonte Carlo-based model is one of the hardest practical problems. First, why are the numerical properties of Greeks so much worse for CLEs than for other, seemingly related instruments? What methods do we use to obtain good deltas? What is a usable definition of a vega in a forward Libor model? How vegas can be computed?