[Cuchulainn]

PDE Valuation of Interest Rate Derivatives
Peter Kohl-Landgraf

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Table Of Contents

1. Foundations
Stochastic Processes
SDE’s and Probability Distributions
Changing Probability Measures: Girsanov’s Theorem
Connection to PDEs: The Feynman-Kac Theorem
Applications in Finance

2. Fixed Income Markets
The Yield Curve
Interest Rate Securities
Interest Rate Derivatives
General Modeling Approach

3. Models of the Yield Curve
A Summary of Short Rate Models
The Heath-Jarrow-Morton Framework
The Libor Market Model - Direct Derviation from HJM

4. Markovian Representations of the Yield Curve
Separable Volatility: The Cheyette Model
The Analytical Bond Price
The Valuation PDE
The Case of Constant Parameters - Connections to Hull-White
Multi-Factor Volatility

5. Numerical Solution
Discretization of Differential Opterators
Finite Difference Schemes in Multiple Spatial Dimensions
Consistency, Stability and Convergence
Alternating Direction Implicit Schemes (ADI)
Treatment of Boundary Conditions (v.Neumann, Dirichlet, Generic..)

6. Practical Considerations
Early Exercise Products and Optimal Control Problems
Local and Stochastic Volatility Specifications (CEV, Displaced Diffusion)
True Stochastic Volatility
Calibration to Market Data

7. Design Issues and C++ Implementation
Components of the Finite Difference Scheme
The Valuation Model
PDE Product Valuation Routines ( e.g. Bermudan Swaption )