Contents
1 Introduction and Reading Guide
2.1 Equities and Basic Options
2.3 The Multiperiod Binomial Model
2.5 Strengths and Weaknesses of Binomial Trees
2.5.4 Exotic Options and Trees
2.5.5 Greeks and Binomial Trees
2.5.6 Grid Adaptivity and Trees
3 Finite Differences and the Black-Scholes PDE
3.1 A Continuous Time Model for Equity Prices
3.2 Black-Scholes Model: From the SDE to the PDE
3.6 Finite Differences and the Heat Equation
4 Mean Reversion and Trinomial Trees
4.1.1 Interest Rates and Compounding
4.1.2 Libor Rates and Vanilla Interest Rate Swaps
4.2 Black76 for Caps and Swaptions
4.3 One-Factor Short Rate Models
4.3.1 Prominent Short Rate Models
4.4 The Hull-White Model in More Detail
5 Upwinding Techniques for Short Rate Models
5.1 Derivation of a PDE for Short Rate Models
5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model
5.3.4 An Algorithm in Pseudocode
6 Boundary, Terminal and Interface Conditions and their Influence
6.1 Terminal Conditions for Equity Options
6.2 Terminal Conditions for Fixed Income Instruments
6.3 Callability and Bermudan Options
6.6.1 Double Barrier Options and Dirichlet Boundary Conditions
6.6.2 Artificial Boundary Conditions and the Neumann Case
7.1.1 Weighted Residual Methods
7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems
7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model
7.6 Appendix: Higher Order Elements
7.6.2 Local and Natural Coordinates
8 Solving Systems of Linear Equations
9.1 The Principles of Monte Carlo Integration
9.2 Pricing Derivatives with Monte Carlo Methods
9.2.1 Discretizing the Stochastic Differential Equation
9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model
9.3 An Introduction to the Libor Market Model
9.4.1 Properties of a Random Number Generator
9.4.4 Recent Developments in Random Number Generation
9.4.6 Random Number Generation for Commonly Used Distributions
10 Advanced Monte Carlo Techniques
10.1 Variance Reduction Techniques
10.1.4 Additional Techniques for Variance Reduction
10.2.1 Low-Discrepancy Sequences
10.3 Brownian Bridge Technique
10.3.1 A Steepener under a Libor Market Model
11.1 Pricing American options using the Longstaff and Schwartz algorithm
11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments
11.2.1 Algorithm: Extended LSMC Method for Bermudan Options
11.2.2 Notes on Basis Functions and Regression
11.3.1 A Bermudan Callable Floater under Different Short-rate Models
11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model
11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework
12 Characteristic Function Methods for Option Pricing
12.1.3 Infinite Activity Models
12.2.1 Fast Fourier Transform Methods
12.2.2 Fourier-Cosine Expansion Methods
13 Numerical Methods for the Solution of PIDEs
13.2 Numerical Solution of the PIDE
13.2.1 Discretization of the Spatial Domain
13.2.2 Discretization of the Time Domain
13.2.3 A European Option under the Kou Jump Diffusion Model
13.3 Appendix: Numerical Integration via Newton-Cotes Formulae
14 Copulas and the Pitfalls of Correlation
14.2.2 Important Copula Functions
14.2.3 Parameter estimation and sampling
14.2.4 Default Probabilities for Credit Derivatives
15 Parameter Calibration and Inverse Problems
15.1 Implied Black-Scholes Volatilities
15.2 Calibration Problems for Yield Curves
15.3 Reversion Speed and Volatility
15.4.1 Dupire’s Inversion Formula
15.4.2 Identifying Local Volatility
15.5 Identifying Parameters in Volatility Models
15.5.1 Model Calibration for the FTSE-100
16.1 Model Calibration and Optimization
16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems
16.2 Heuristically Inspired Algorithms
16.3 A Hybrid Algorithm for Heston Model Calibration
17.1 Value at Risk and Expected Shortfall
17.1.4 Individual and Contribution VaR
17.2 Principal Component Analysis
17.2.1 Principal Component Analysis for Non-scalar Risk Factors
17.2.2 Principal Components for Fast Valuation
18 Quantitative Finance on Parallel Architectures
18.1 A Short Introduction to Parallel Computing
18.2 Different Levels of Parallelization
18.4 Parallelization of Single Instrument Valuations using (Q)MC
18.5 Parallelization of Hybrid Calibration Algorithms
19 Building Large Software Systems for the Financial Industry