| Summary: | Student Number : 0003819T -
MSc dissertation -
School of Computational and Applied Mathematics -
Faculty of Science === The over-the-counter (OTC) interest rate derivative market is large and rapidly developing. In
March 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 June
2004. 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 daily
turnover in interest rate option trading increased from 5.9% (of the total daily turnover in the
interest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success of
the interest rate derivative market has resulted in the introduction of exotic interest rate products
and 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 the
interest 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 typically
used by traders of interest rate derivatives to gamma and vega hedge complex products. Thus an
important feature of an interest rate model is not only its ability to recover an arbitrary input yield
curve, 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 interest
rate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertation
is this popular class of interest rate models.
The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolution
of their values, assuming that the underlying movements are continuous, is driven by a finite
number of Brownian motions. The traditional approach to modelling the term structure of interest
rates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in the
LIBOR market model, the discrete forward rates are modelled directly. The additional assumption
imposed is that the volatility function of the discrete forward rates is a deterministic function of
time. In Chapter 2 we provide a brief overview of the history of interest rate modelling which led
to the LIBOR market model. The general theory of derivative pricing is presented, followed by
a exposition and derivation of the stochastic differential equations governing the forward LIBOR
rates.
The LIBOR market model framework only truly becomes a model once the volatility functions
of the discrete forward rates are specified. The information provided by the yield curve, the cap and
the swaption markets does not imply a unique form for these functions. In Chapter 3, we examine
various specifications of the LIBOR market model. Once the model is specified, it is calibrated
to the above mentioned market data. An advantage of the LIBOR market model is the ability to
calibrate to a large set of liquid market instruments while generating a realistic evolution of the
forward rate volatility structure (Piterbarg 2004). We examine some of the practical problems that
arise 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 Carlo
simulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulation
approaches are presented, together with an examination of the various discretizations of the forward
rate stochastic differential equations. The chapter concludes with some numerical results comparing
the performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performance
of 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 problem
formulations, followed by an overview of the two main numerical techniques for pricing American
options using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class of
interest rate derivatives that have early exercise provisions (Bermudan style) to exercise into various
underlying interest rate products. A popular approach for valuing these instruments in the LIBOR
market 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 the
choice of relevant, i.e. problem specific predictor variables and also on the functional form of the
regression function. It is certainly not a “black-box” type of approach.
Instead of choosing the relevant predictor variables, we present the sliced inverse regression
technique. Sliced inverse regression is a statistical technique that aims to capture the main features
of the data with a few low-dimensional projections. In particular, we use the sliced inverse regression
technique to identify the low-dimensional projections of the forward LIBOR rates and then we
estimate the continuation value of the option using nonparametric regression techniques. The
results for a Bermudan swaption in a two-factor LIBOR market model are compared to those in
Andersen (2000).
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