Pricing credit derivatives in a 'Libor Market Model'

Contents

1 Modelling the Term Structure

1.1 Introduction

1.2 Libor Market Model

1.2.1 Motivation

1.2.2 Setup

1.2.3 Corresponding Heath-Jarrow-Morton Model

1.3 Change of Numeraire

1.3.1 Girsanov’s Theorem

1.3.2 The Tk-Forward Measure Pk

1.3.3 Dynamics under the Terminal Measure PK+1

1.4 Pricing Interest Rate Derivatives

1.4.1 Caps

1.4.2 Floors

1.5 Conclusion

2 Modelling Credit Risk

2.1 Introduction

2.2 Default Risk in the Libor Market Model

2.2.1 Discrete Tenor Setup

2.2.2 Heath-Jarrow-Morton Background

2.3 Forward Measures in the Defaultable Market Model

2.3.1 The Tk-Forward Measure Pk

2.3.2 The Tk-Survival Measure Pk

2.3.3 Change of Measure from Forward to Survival Measure

2.4 Dynamics

2.4.1 Default Free Forward Rates

2.4.2 Defaultable Forward Rates

2.4.3 Forward Spreads and Intensities

2.4.4 Dynamics of the Default Intensities under the Terminal Survival Measure

2.5 Independence vs. Correlation

2.6 Conclusion

3 Modelling Recovery

3.1 Introduction

3.2 Valuation of Recovery Payoffs

3.3 Value of Defaultable Bonds

3.4 Conclusion

4 Pricing Credit Derivatives

4.1 Credit Default Swap

4.1.1 Valuation

4.1.2 Default Swap Rate

4.2 Pricing Options on CDS

4.3 Conclusion

5 Calibration

5.1 Introduction

5.2 Extracting Information from Market Data

5.2.1 Term Structure

5.2.2 Forward Rate Volatilities

5.2.3 Recovery Rates

5.2.4 Default Intensities

5.2.5 Intensity Volatilities and Correlation

5.3 Conclusion

6 Simulation

6.1 Setup

6.1.1 Dynamics

6.1.2 Random Numbers

6.2 Results

6.2.1 Credit Default Swaps

6.2.2 Default Payment

6.2.3 Pricing Options

6.3 Conclusion

Research Objectives and Core Topics

The primary objective of this thesis is to extend the well-known Libor Market Model to incorporate credit risk in a market-consistent manner, providing a theoretical framework for pricing credit derivatives such as defaultable bonds, credit default swaps, and options on credit default swaps.

• Extension of the Libor Market Model to account for default risk.
• Development of theoretical pricing formulae for various credit derivatives.
• Practical calibration of the model to real-world market data.
• Analysis and implementation of simulation methods for complex pricing tasks.
• Comparison of analytical approximations versus simulation-based results.

Excerpt from the Book

Summary of Chapters

Modelling the Term Structure: This chapter introduces the default-free market model and fundamental financial theorems, establishing the foundation for interest rate derivative pricing.

Modelling Credit Risk: Here, credit risk is incorporated into the Libor market model using discrete tenor default intensities, introducing new survival-contingent measures.

Modelling Recovery: This chapter presents a fractional recovery of par model, defining how to value positive recovery payoffs and defaultable bonds.

Pricing Credit Derivatives: This chapter derives fair pricing formulae for credit default swaps and options on credit default swaps.

Calibration: This section discusses methods to extract essential input variables, such as forward rates and default intensities, from market data for model calibration.

Simulation: The final chapter covers the implementation of the model using Monte Carlo techniques and presents the simulation results for pricing credit derivatives.

Keywords

Libor Market Model, Credit Derivatives, Credit Default Swap, Default Intensity, Term Structure, Risk Management, Market-Consistent Pricing, Calibration, Monte Carlo Simulation, Recovery Rates, Survival Measure, Interest Rate Derivatives, Default Risk, Forward Rates, Arbitrage-free pricing

Frequently Asked Questions

What is the core subject of this thesis?

The thesis focuses on pricing credit derivatives in a market-consistent way by extending the standard Libor Market Model to include default risk.

What are the central thematic areas?

The key themes include modeling interest rate term structures, incorporating default risk through intensity-based approaches, modeling recovery, and calibrating these models to market data for practical application.

What is the primary objective?

The primary goal is to provide a robust theoretical and practical framework that allows for the fair pricing of credit derivatives by building upon observable market rates.

Which scientific methods are employed?

The author uses mathematical finance techniques, specifically stochastic calculus, change of numeraire, and measure transformation (e.g., forward and survival measures), combined with Monte Carlo simulation for verification.

What is covered in the main part of the work?

The main body covers the theoretical foundation (Part I), which details term structure modeling, credit risk, recovery, and derivative pricing, followed by the practical implementation (Part II), which discusses calibration and numerical simulation results.

Which keywords best characterize this work?

The most relevant keywords are Libor Market Model, Credit Default Swap, Default Intensity, Term Structure, and Recovery Rates.

How is the "default event" defined within this model?

The model simplifies the definition of a default event by aggregating various credit events (e.g., bankruptcy, failure to pay) into a single "default" event that occurs on pre-known tenor dates.

Why is the "survival measure" introduced?

The survival measure is introduced because it serves as an analogue to the standard forward measure, specifically used to price defaultable payoffs contingent on the survival of the reference entity.