The over-the-counter (OTC) interest rate derivative market is large and rapidly developing. InMarch 2005, the Bank for International Settlements published its “Triennial Central Bank Survey”which examined the derivative market activity in 2004 (http://www.bis.org/publ/rpfx05.htm).The reported total gross market value of OTC derivatives stood at $6.4 trillion at the end of June2004. The gross market value of interest rate derivatives comprised a massive 71.7% of the total,followed by foreign exchange derivatives (17.5%) and equity derivatives (5%). Further, the dailyturnover in interest rate option trading increased from 5.9% (of the total daily turnover in theinterest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success ofthe interest rate derivative market has resulted in the introduction of exotic interest rate productsand the ongoing search for accurate and efficient pricing and hedging techniques for them.Interest rate caps and (European) swaptions form the largest and the most liquid part of theinterest rate option market. These vanilla instruments depend only on the level of the yield curve.The market standard for pricing them is the Black (1976) model. Caps and swaptions are typicallyused by traders of interest rate derivatives to gamma and vega hedge complex products. Thus animportant feature of an interest rate model is not only its ability to recover an arbitrary input yieldcurve, but also an ability to calibrate to the implied at-the-money cap and swaption volatilities.The LIBOR market model developed out of the market’s need to price and hedge exotic interestrate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertationis this popular class of interest rate models.The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolutionof their values, assuming that the underlying movements are continuous, is driven by a finitenumber of Brownian motions. The traditional approach to modelling the term structure of interestrates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in theLIBOR market model, the discrete forward rates are modelled directly. The additional assumptionimposed is that the volatility function of the discrete forward rates is a deterministic function oftime. In Chapter 2 we provide a brief overview of the history of interest rate modelling which ledto the LIBOR market model. The general theory of derivative pricing is presented, followed bya exposition and derivation of the stochastic differential equations governing the forward LIBORrates.The LIBOR market model framework only truly becomes a model once the volatility functionsof the discrete forward rates are specified. The information provided by the yield curve, the cap andthe swaption markets does not imply a unique form for these functions. In Chapter 3, we examinevarious specifications of the LIBOR market model. Once the model is specified, it is calibratedto the above mentioned market data. An advantage of the LIBOR market model is the ability tocalibrate to a large set of liquid market instruments while generating a realistic evolution of theforward rate volatility structure (Piterbarg 2004). We examine some of the practical problems thatarise when calibrating the market model and present an example calibration in the UK market.The necessity, in general, of pricing derivatives in the LIBOR market model using Monte Carlosimulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulationapproaches are presented, together with an examination of the various discretizations of the forwardrate stochastic differential equations. The chapter concludes with some numerical results comparingthe performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performanceof the discretization approaches.In the final chapter we discuss numerical techniques based on Monte Carlo simulation for pricing American derivatives. We present the primal and dual American option pricing problemformulations, followed by an overview of the two main numerical techniques for pricing Americanoptions using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class ofinterest rate derivatives that have early exercise provisions (Bermudan style) to exercise into variousunderlying interest rate products. A popular approach for valuing these instruments in the LIBORmarket model is to estimate the continuation value of the option using parametric regression and,subsequently, to estimate the option value using backward induction. This approach relies on thechoice of relevant, i.e. problem specific predictor variables and also on the functional form of theregression function. It is certainly not a “black-box” type of approach.Instead of choosing the relevant predictor variables, we present the sliced inverse regressiontechnique. Sliced inverse regression is a statistical technique that aims to capture the main featuresof the data with a few low-dimensional projections. In particular, we use the sliced inverse regressiontechnique to identify the low-dimensional projections of the forward LIBOR rates and then weestimate the continuation value of the option using nonparametric regression techniques. Theresults for a Bermudan swaption in a two-factor LIBOR market model are compared to those inAndersen (2000).
