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The aim of this diploma thesis is to present the theory as well as the practical applications of affine interest rate models. On the basis of the general theory established by Duffie and Kan, we put emphasis on affine models whose state variables have - in contrast to their theoretical abstract definition - a rea- sonable economic interpretation. Starting from the very first term structure models, namely the Vasicek and the Cox-Ingersoll-Ross model, we describe in sequel two- and more-factor models that have appeared in literature. By means of the Vasicek model we exemplify the calibration to market yields as well as to market cap volatilities.

However, our main focus are affine yield factor models developed by Duffie and Kan, which allow to relate the state variables to yields with different maturities. We show how to calibrate a two-factor version of this model to market data. The results are promising since the model fits the market yields from different dates very well while the parameters remain nearly constant.

Content
1 Introduction to Interest Rate Theory

1.1 Definitions and Notations
1.1.1 Short-Term Interest Rate
1.1.2 Zero-Coupon Bonds and Spot Interest Rates
1.1.3 Forward Rates
1.1.4 Interest Rate Swaps

1.2 No-Arbitrage Pricing
1.3 Factor Models of the Term Structure

1.3.1 Dynamics under P?
1.3.2 The Bond Price as Solution of a PDE

2 Affine Models

2.1 Theory of Affine Factor Models
2.1.1 Specification of the State Variable Process
2.1.2 Affine Stochastic Differential Equations
2.1.3 Ricatti Equations

2.2 Types of Affine Models

2.2.1 Gaussian Affine Models
2.2.2 CIR Affine Models
2.2.3 The Three-Factor Affine family

2.3 Classification of Affine Models

2.3.1 A Canonical Representation
2.3.2 Invariant Transformations and Equivalent Models

3 Examples of Affine Models

3.1 Examples of One-Factor Affine Models
3.1.1 The Extended Vasicek Model
3.1.2 The Extended CIR Model

3.2 Examples of Multi-Factor Affine Models

3.2.1 The Longstaff and Schwartz Two-Factor Model
3.2.2 The Central Tendency as Second Factor

3.3 Economic Models

3.3.1 The General Framework
3.3.2 IS - LM Framework

3.4 Non-Affine Models - Consol Models
3.5 Criteria for Model Selection

4 Calibration and Estimation

4.1 Obtaining a Data Set
4.1.1 Market Data for the Current Yield Curve
4.1.2 Market Data for Bond Options
4.1.3 Which Market Rate should be used for the Short-Term Rate?

4.2 Calibration to Current Market Data

4.2.1 Calibrating the Vasicek Model to the Current Term Structure
4.2.2 Calibrating the Vasicek Model to Cap Volatilities
4.2.3 Calibrating the Hull-White Extended Vasicek Model

4.3 Historical Estimation

4.3.1 Maximum Likelihood Method
4.3.2 General Method of Moments

5 Affine Yield-Factor Models

5.1 General Affine Yield-Factor Model
5.2 A Two-Factor Affine Model of the “Long”- and the Short-Term Rate
5.2.1 Deterministic Volatility
5.2.2 Calibrating the Deterministic Volatility Model to the Current Term Structure
5.2.3 Stochastic Volatility
5.2.4 Calibrating the Stochastic Volatility Model to the Current Term Structure
5.2.5 Conclusion

A Numerical Methods for Calibration

A.1 Trust-Region Methods for Nonlinear Minimization
A.1.1 Box Constraints
A.1.2 Nonlinear Least-Squares

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