Pricing of Interest Rate Derivatives with the LIBOR Market Model
by Linus Kajsajuntti

Abstract
In the beginning of the 90's Heath, Jarrow and Morton (HJM) presented a revolutionary approach to interest rate modelling. Instead of modelling the instantaneous spot rate, as in the then popular short rate models, the whole instantaneous forward rate curve was modelled. However, since the instantaneous spot and forward rates are non-existing in the market, a satisfying calibration of both short rate models and the HJM framework against the cap or swaption markets is very hard to obtain.  In 1997, Brace, Gatarek and Musiela (BGM) published a work which took the HJM framework to a new level. Modelling discretely tenored forward rates instead of instantaneous forward rates implied a possibility to perfectly recover the cap market.  Since the BGM model has been widely accepted by both academics and professionals as the benchmark model for pricing and hedging LIBOR derivatives it has acquired
the name the LIBOR market model.

This thesis deals with pricing exotic derivatives with the LIBOR market model. In addition to a perfect recovery of the cap market an accurate approximation formula for eective calibration to swaptions is implemented. Much eort is put on assuring a stable and accurate evolution of the forward rate structure and it is shown how to design an evolution scheme that suits a given derivative. Pricing schemes with fast convergence is developed by the use of quasi-Monte Carlo integration based on a highdimensional
Sobol low-discrepancy sequence. It is shown that a clever implementation of the quasi-Monte Carlo integration implies at least a factor 10 faster convergence and that this, in contrast with theoretical results, continues to hold in very high dimensions.

CONTENTS
1 Introduction 5
2 Monte Carlo and quasi-Monte Carlo methods 8
2.1 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Pseudo-random numbers . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Quasi-Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Problem dimensionality . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Sobol number generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Path construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Incremental path construction . . . . . . . . . . . . . . . . . . . 15
2.4.2 Brownian Bridge path construction . . . . . . . . . . . . . . . . . 16
2.4.3 Several Brownian motions from Sobol sequences . . . . . . . . . 16
2.5 Implementation of the Sobol sequence generator . . . . . . . . . . . . . 17
3 The LIBOR market model 18
3.1 Forward rate dynamics in the LMM . . . . . . . . . . . . . . . . . . . . 19
3.2 The drift function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Discretising the forward rate equation . . . . . . . . . . . . . . . . . . . 22
3.3.1 The short step method . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 The long step method . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 A swap rate based market model . . . . . . . . . . . . . . . . . . . . . . 27
4 Characterising and pricing LIBOR derivatives 28
4.1 Characterising LIBOR derivatives classes . . . . . . . . . . . . . . . . . 28
4.2 Eective pricing schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 The short step method pricing scheme . . . . . . . . . . . . . . . 31
4.2.2 The long step method pricing scheme . . . . . . . . . . . . . . . 32
4.2.3 A hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Calibrating the LMM 35
5.1 Specifying the inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 The instantaneous volatility function . . . . . . . . . . . . . . . . 35
5.1.2 The instantaneous correlation function . . . . . . . . . . . . . . . 37
5.2 Calibrating the LMM to caplets . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Calibrating the LMM to both caplets and swaptions . . . . . . . . . . . 40
5.3.1 Swaption pricing in a forward rate based LMM . . . . . . . . . . 40
5.3.2 Simultaneous calibration to both cap and swaption markets . . . 42
6 Results 44
6.1 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.1.1 Correlation calibration . . . . . . . . . . . . . . . . . . . . . . . . 44
6.1.2 Calibrating to caplets . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1.3 Calibrating to swaptions . . . . . . . . . . . . . . . . . . . . . . . 46
6.1.4 Calibrating to both caps and swaptions . . . . . . . . . . . . . . 48
6.2 Monte Carlo pricing results . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2.1 Caplet pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2.2 Swaption pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 Case study: Pricing a spread option . . . . . . . . . . . . . . . . . . . . 53
6.3.1 Sobol vs Mersenne Twister . . . . . . . . . . . . . . . . . . . . . 54
6.3.2 Calculating the greeks . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3.3 Correlation dependency . . . . . . . . . . . . . . . . . . . . . . . 57
7 Discussion and possible further developments 58
7.1 Discussion of obtained results . . . . . . . . . . . . . . . . . . . . . . . . 58
7.1.1 Monte Carlo vs quasi-Monte Carlo integration . . . . . . . . . . 58
7.1.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.1.3 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . 59
A Arbitrage, martingales and various mathematical tools 61
A.1 Arbitrage and martingale pricing . . . . . . . . . . . . . . . . . . . . . . 61
A.1.1 Probability and stochastic processes . . . . . . . . . . . . . . . . 61
A.1.2 Arbitrage pricing by replicating portfolio . . . . . . . . . . . . . 63
A.1.3 Equivalent martingale measure and martingale pricing . . . . . . 64
A.1.4 Some useful stochastic calculus . . . . . . . . . . . . . . . . . . . 65
B Interest rate markets dynamics 68
B.1 The basic bond and rate processes . . . . . . . . . . . . . . . . . . . . . 68
B.1.1 The zero-coupon bond process . . . . . . . . . . . . . . . . . . . 68
B.1.2 Spot and forward rates . . . . . . . . . . . . . . . . . . . . . . . . 69
B.1.3 Interest rate swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.2 Plain vanilla options on the basic instruments . . . . . . . . . . . . . . . 73
B.2.1 Plain vanilla options on FRAs: caps and
oors . . . . . . . . . . 73
B.2.2 Plain vanilla options on swaps: swaptions . . . . . . . . . . . . . 74
B.3 The HJM forward rate dynamics . . . . . . . . . . . . . . . . . . . . . . 75
References 77