Pricing credit derivatives in a 'Libor Market Model'

Diplomarbeit
Institut für Gesellschafts- und Wirtschaftswissenschaften, Statistische Abteilung
Rheinische Friedrich-Wilhelms-Universität Bonn

Pricing Credit Derivatives in a Libor Market Model

by Hanno M. Damm
07.12.2002

Abstract

The growing importance of credit derivatives creates the need to price them in a market consistent manner. In this thesis the well known and accepted Libor Market Model is extended following Schönbucher (2000). The thesis consists of two main parts: one describing and explaining the theoretical framework that will yield the pricing formulae for credit derivatives, and a second part explaining how to practically implement and calibrate the model. The second part also reports results of our implementation. We show that approximations introduced by Schönbucher (2000) hold and that the model can be used to price defaultable bonds, credit default swaps as well as options on credit default swaps.
The thesis has been written at the Department of Statistics, University of Bonn in cooperation with Deutsche Postbank AG Credit Risk Management.

Contents

Introduction 1

I Theoretical Foundation

1 Modelling the Term Structure ... 3
1.1 Introduction ... 3
1.2 LiborMarketModel ...  3
1.2.1 Motivation ... 3
1.2.2 Setup ...4
1.2.3 Corresponding Heath-Jarrow-Morton Model ... 5
1.3 Change of Numeraire ... 6
1.3.1 Girsanov’s Theorem ... 7
1.3.2 The Tk-Forward Measure Pk ... 8
1.3.3 Dynamics under the Terminal Measure PK+1 ... 9
1.4 Pricing Interest Rate Derivatives ... 10
1.4.1 Caps ... 10
1.4.2 Floors ... 11
1.5 Conclusion ... 11

2 Modelling Credit Risk ... 12
2.1 Introduction ... 12
2.2 Default Risk in the LiborMarketModel ... 13
2.2.1 Discrete Tenor Setup ... 13
2.2.2 Heath-Jarrow-Morton Background ... 15
2.3 ForwardMeasures in the DefaultableMarketModel ... 16
2.3.1 The Tk-Forward Measure Pk ... 16
2.3.2 The Tk-Survival Measure Pk ... 17
2.3.3 Change ofMeasure fromForward to SurvivalMeasure ... 19
2.4 Dynamics ... 19
2.4.1 Default Free Forward Rates ... 20
2.4.2 Defaultable Forward Rates ... 20
2.4.3 Forward Spreads and Intensities ... 20
2.4.4 Dynamics of the Default Intensities under the Terminal Survival Measure ... 20
2.5 Independence vs. Correlation ... 21
2.6 Conclusion ... 22

3 Modelling Recovery ... 23
3.1 Introduction ... 23
3.2 Valuation of Recovery Payoffs ... 24
3.3 Value of Defaultable Bonds ... 25
3.4 Conclusion ... 25

4 Pricing Credit Derivatives ... 26
4.1 Credit Default Swap ... 26
4.1.1 Valuation ... 26
4.1.2 Default Swap Rate ... 27
4.2 Pricing Options on CDS ... 27
4.3 Conclusion ... 29

II Practical Implementation

5 Calibration ... 30
5.1 Introduction ... 30
5.2 Extracting Information from Market Data ... 30
5.2.1 TermStructure ... 30
5.2.2 Forward Rate Volatilities ... 31
5.2.3 Recovery Rates ... 31
5.2.4 Default Intensities ... 31
5.2.5 Intensity Volatilities and Correlation ... 32
5.3 Conclusion ... 32

6 Simulation ... 34
6.1 Setup ... 34
6.1.1 Dynamics ... 34
6.1.2 RandomNumbers ... 35
6.2 Results ... 36
6.2.1 Credit Default Swaps ... 36
6.2.2 Default Payment ... 37
6.2.3 Pricing Options ... 39
6.3 Conclusion ... 39

Final Remarks and Outlook ... 40

Bibliography ... 41

Appendix

A Market Data and Calibration Results ... 45
A.1 TermStructure ... 45
A.2 Default Intensities and CDS Quotes ... 47

B Simulation Results ... 51
B.1 Normal Distribution ... 51
B.2 Simulating of the default payoff ek ... 52
B.3 Options on CDS ... 57
B.3.1 Input Data ... 57
B.3.2 Simulation Results ... 58

C Calculations ... 64
C.1 Default FreeModel ... 64
C.1.1 Proof of Equation (1.4) ... 64
C.1.2 Proof of Proposition 1 ... 64
C.2 MarketModel with Default Risk ... 65
C.2.1 Proof of Equation (2.8) ... 65
C.2.2 Proof of Equation (2.9) ... 65
C.2.3 Proof of Proposition 2 ... 65
C.2.4 Proof of Equation (2.21) ... 66
C.2.5 Proof of Equation (2.30) ... 66
C.3 RecoveryModel - Proof of Proposition 3 ... 67

Introduction

Since the early 1990s, credit derivatives have become increasingly popular and instrumental as a means of risk management. Davies, Hewer, and Rivett (2001) expect various factors to drive the global credit derivatives market to reach US-$ 1.6 trillion of notional value by 2002. Banks can trade credit risk, thus lowering the regulatory capital bound. This effect could become stronger under the new Basel Capital Accord. The accord stipulates the amount of regulatory capital bound, due to credit exposure, is dependent upon the credit standing of the debtor.¹ Credit risk protection through credit derivatives therefore can lower the relevant exposure. While under the current Accord, only credit derivatives sold by other banks and regulated security firms give lower risk weights, under the new accord this treatment is being extended to protection provided by high-quality non-banks, such as insurance companies.²
Another important application of credit derivatives is that they allow shorting credit risk. Theoretically it is possible to short corporate bonds on the spot market by means of repurchase agreements (repos). However, this market is virtually non-existent and if it exists, long maturities are not available according to Houweling and Vorst (2001). Moreover, repo transactions on the spot market expose the investor to the risk of change in the repo rate. Credit derivatives on the other hand eliminate this risk, as payments (other than payments due to default) are being agreed upon when specifying the contract. Also very long maturities of up to ten years are available (although liquidity is small at the long end). The efficiency of portfolio management thus has been improved.
Today more than thirty banks have global credit derivatives trading operations. According to Davies et al. (2001), market practitioners believe that all major banks will trade credit derivatives in the future. This relatively new market survived its ”baptism of fire” recently, when Enron defaulted. Enron was a major player in the credit derivatives market as it actively traded derivatives, as well as being a reference credit for numerous contracts. Apparently the market does not suffer from a sustainable distortion due to the Enron default.³ Therefore, it is assumed that the exponential growth of the credit derivatives market will continue.
The growing importance of credit derivatives creates the need to price them in a market consistent manner. In this thesis the well known and accepted Libor Market Model⁴ is extended following Schönbucher (2000). The thesis consists of two main parts: one describing and explaining the theoretical framework that will yield the pricing formulae for credit derivatives, and a second part explaining how to practically implement the model. The second part also reports results of our implementation.
The first part begins with a description of the default free Libor market model byMiltersen, Sandmann, and Sondermann (1997). Various mathematical tools, such as the change of measure and several fundamental theorems are introduced. The chapter ends with an presentation of closed form solutions for interest rate derivative prices. The second chapter shows how credit risk is incorporated into the model by Schönbucher (2000). The dynamics of various variables under different measures are derived, thus providing a framework that allows pricing credit derivatives. The third chapter is dedicated to issues connected with recovery. The approach by Schönbucher (2000) to model recovery is presented. In the last chapter of part one credit derivatives are described and theoretical pricing formulae are given.
Part two is split into chapters five and six. In chapter five calibration issues are discussed. In particular, methods to extract the necessary data from market data are described. In chapter six our implementation of the model and simulations results are presented. We show, that approximations introduced by Schönbucher (2000) are suffi- ciently close to simulated results and can be used when applying the closed form formulae to price credit derivatives. The thesis ends with final remarks and an outlook.
This thesis has been written for the Department of Statistics of the University of Bonn in co-operation with the Postbank Credit Risk Management Group that provided the necessary data. I would like to thank Professor Dr. Sondermann and Dr. Philipp Schönbucher of the Department of Statistics as well as Thomas Holtorf of the Postbank for giving me the opportunity to write this thesis.

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1 See Basel Committee on Banking Supervision (2001a,b,c).
2 See Rule (2001, p. 126).
3 See Papon (2002).
4 The model is build upon observable market rates, such as the London Interbank Offered Rate (LIBOR).