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 )