Libor Market Mode - Theory and Practice

Chair of Investments, Portfolio Management and Pension Finance
Faculty of Economics and Business Administration
Johann Wolfgang Goethe - University
Frankfurt am Main

Libor Market Model: Theory and Practice

Thesis
Submitted in partial fulfilment of the requirements for the degree
of Diplom-Kauffrau (Diploma in Business Administration)

by: Irina Götsch
January 25, 2006

Contents

List of Figures ... IV
List of Tables ... V
List of Abbreviations ... VI
List of Notations ... VII

1. Introduction ... 1

2. Comparison with Alternative Interest Rate Models ... 2

3. General Option Pricing ... 3
3.1. Fundamentals of Derivatives Valuation ... 3
3.2. Change-of-Numeraire Theorem ... 6
3.3. Girsanov´s Theorem ... 7

4. Libor Market Model Theory: Arbitrage-free Forward Libor Rate Dynamics ... 9
4.1. Forward Libor Rate Process as a Martingale ... 9
4.2. Dynamics of Forward Libor Rates under the Forward Measure ... 11
4.3. Extension to Several Factors ... 15

5. Obtaining the Data Input for the Libor Market Model ... 18
5.1. Derivation of Time Zero Forward Libor Rates ... 18
5.2. Calibration of Volatility Parameters to Cap Prices ... 20
5.3. Calibration to Swaption Prices ... 22
5.3.1. Calibration of Correlation and Volatility Parameters to Swaptions ... 22
5.3.2. Cascade Calibration ... 25

6. Forward Libor Rates Volatility Modeling ... 29
6.1. The Term Structure of Volatility ... 29
6.2. Constant Volatility Structure ... 29
6.3. Piecewise-Constant Volatility Structure ... 30
6.4. Parametric Volatility Structure ... 33
6.5. Determining of Volatility Parameters with the Two-step Approach ... 34

7. Forward Libor Rates Correlation Modeling ... 35
7.1. Specifications of Forward Rate Correlation ... 35
7.1.1. Full Rank Specification with Reduced Number of Parameters ... 35
7.1.2. Reduced –Rank Correlation Specifications ... 37
7.2. Obtaining an Exogenous Correlation Matrix for Cascade Calibration ... 40
7.2.1. Step 1: Historical Estimation of Correlation Matrix ... 40
7.2.2. Step 2 and 3: Fitting Historically Estimated Correlation Matrix to a Parametric Form and Reducing the Rank ... 40

8. Hedging ... 41

9. The Libor Market Modell: Practice ... 42
9.1. Implementation Steps with Monte Carlo Simulations ... 42
9.2. Implementation of the LMM: Results ... 50
9.2.1. Study 1: Valuation of Caplets and Caps ... 50
9.2.2. Study 2: Valuation of Discrete Barrier Caps ... 53
9.2.3. Study 3: Cascade Calibration ... 55
9.2.4. Study 4:Valuation of European Swaptions ... 63
9.2.5. Study 5: Valuation of Ratchets ... 64

10. Summary and conclusion ... 65

References ... 66
Appendix A: Recovering of Black´s Formula for Caplets in the Libor Market Model. ... 68
Appendix B: Stochastic Euler Scheme ... 72
Appendix C: Source Code ... 73

1. Introduction

The Libor market model was introduced by Miltersen, Sandmann and Sondermann (1997)¹ and Brace, Gatarek and Musiela(1997). ² This model is called BGM after the authors of one of the first papers where it was introduced. It is a model to price and hedge standard and exotic interest rate derivatives whose payoff can be decomposed into a set of forward Libor rates. Therefore another name of the model is the Libor market model (LMM).³ This model is constructed by forming a process for stochastic evolution of forward Libor rates⁴ of various maturities. Libor rate (London Interbank Offered Rate) is a short-term interest rate⁵ offered by banks on deposits from other banks in Eurocurrency markets and is usually used by traders as a proxy for risk-free rate when valuing derivatives.

The Libor Market model is one of the most popular interest rate models. The broad acceptance is due to its consistence with the standard market formula - Black´s cap (floor) formula. Cap (floor) market is one of the main markets in the interest rate derivatives market and is very liquid. The reproduction of the standard pricing formula for the standard instruments is a very important feature of a pricing model. The pricing model needs to successfully extract the information about the probability distribution of the future values of underlyings from the liquid instruments available in the market. For this purpose the pricing models must be able to explain the observed prices of the underlying instruments.

The goal of this thesis is to examine the LMM theoretically and apply practically to derivatives pricing. The input data structuring and calibration to market and historical data, implementing and pricing issues will be specifically investigated.

This work begins with the comparison of the LMM to alternative interest rate models in chapter 2. A review of basic theory of the valuation of derivatives, which will be used in the next chapters, is presented in chapter 3. Theoretical description of the LMM is pre- sented in the next chapter. Chapter 5 investigates several methods of calibrating directly to market cap and swaption prices. The way of obtaining the initial Libor yield curve is also summarized. In chapter 6 and 7 modeling of forward Libor rates volatility and correlation is presented. Hedging issues are to find in chapter 8. Chapter 9 covers pricing with the LMM by Monte Carlo simulations. This chapter presents the results of implementing the cascade calibration and of valuation of derivatives to illustrate the performance of the LMM. Finally the last chapter summarises and concludes the thesis.
 

2. Comparison with Alternative Interest Rate Models

Interest rate derivatives are more difficult to value than equity derivatives. The reason is the double role of interest rates: the determination of discounting and the payoff of the derivative. The specific feature of the valuation of many interest rate derivatives is the need to develop the behavior of the entire zero-coupon yield curve with different volatilities at the different points of the yield curve.⁶

The reason why other interest rate models were developed are the limitations of the standard market Black’s formula. The Black model does not show how the interest rates develop through time. Consequently, it cannot be used to value american-style or path dependent interest rate derivatives and other non-standard interest rate derivatives. The traditional term structure models in finance are equilibrium models. The disadvantage of these models is that today’s term structure of interest rates is an output. When valuing derivatives it is very important to use the model where today’s term structure of interest rates is an input, and thus the model is consistent with the term structure observed in the market. No-arbitrage models were designed for this purpose. They take the initial term structure as given and define the evolution of the interest rates. The Libor Market Model is a no-arbitrage model. The LMM and the Heath, Jarrow and Morton Model have an advantage over other no-arbitrage models like Ho-Lee or Hull-White models. These include the ability to use several factors and to have more flexibility in choosing volatility term structure.⁷ The disadvantage of the HJM model is its expression in terms of instantaneous forward rates, which are not directly observable in the market and must be estimated from the traded coupon bonds.⁸ In addition, the calibration of this model to the prices of actively traded instruments is difficult. In order to overcome these problems the LMM⁹ was proposed. Its derivation is based on the HJM framework.¹⁰ The LMM is expressed in terms of discrete forward Libor rates, which are directly observable from the market. The compatibility of the LMM with the market standard Black formula simplifies the calibration because the quoted prices for standard interest rate derivatives can be directly used as an input for the LMM.

[...]

1 Miltersen, K.R., Sandmann, K., Sondermann, D.(1997).

2 Brace, A., Gatarek, D., Musiela, M.(1997).

3 In this thesis the model name, which will be used, is Libor Market model or LMM.

4 In this thesis forward Libor rates will also called forward rates or Libors.

5 LIBOR is usually quoted on 1, 3 or 6-month term.

6 Cf. Pelsser,A.(2000), p.3.

7 Cf. Hull, J.(2005),28 and 29 chapter.

8 Cf. Branger, N., Schlag, C.(2004), p.149.

9 Another market model is the swap market model, which will not be considered here.

10 Cf. Brigo, D., Mercurio, F. (2001), p.187.