Written by leading academics and practitioners in the field of financial mathematics,
the purpose of this book is to provide a unique combination of some of the most important
and relevant theoretical and practical tools from which any advanced undergraduate and
graduate student, professional quant and researcher will benefit. This book stands out
from all other existing books in quantitative finance from the sheer impressive range of
ready-to-use software and accessible theoretical tools that are provided as a complete
package. By proceeding from simple to complex, the authors cover core topics in derivative
pricing and risk management in a style that is engaging, accessible and
self-instructional. The book contains a wide spectrum of problems, worked-out solutions,
detailed methodologies and applied mathematical techniques for which anyone planning to
make a serious career in quantitative finance must master. In fact, core portions of the
book's material originated and evolved after years of classroom lectures and computer
laboratory courses taught in a world-renowned professional Master's program in
mathematical finance. As a bonus to the reader, the book also gives a detailed exposition
on new cutting-edge theoretical techniques with many results in pricing theory that are
published here for the first time.
Table of Contents
I Pricing Theory and Risk Management
1 Pricing Theory
1.1 Single Period, Finite Financial Models
1.2 Continuous state spaces
1.3 Multivariate Continuous Distributions: Basic Tools
1.4 Brownian Motion, Martingales and Stochastic Integrals
1.5 Stochastic Differential Equations and Ito?s formula
1.6 Geometric Brownian Motion
1.7 Forwards and European Calls and Puts
1.8 Static Hedging and Replication of Exotic Payoffs
1.9 Continuous Time Financial Models
1.10 Dynamic Hedging and Derivative Asset Pricing in Continuous Time .
1.11 Hedging with Forwards and Futures
1.12 Pricing formulas of the Black-Scholes type
1.13 Partial Differential Equations for Pricing Functions and Kernels
1.14 American Options
1.14.1 Arbitrage-Free Pricing and Optimal Stopping Time Formulation
1.14.2 Perpetual American Options
1.14.3 Properties of the Early-Exercise Boundary
1.14.4 The PDE and Integral Equation Formulation
2 Fixed Income Instruments
2.1 Bonds, Futures, Forwards and Swaps
2.1.1 Bonds
2.1.2 Forward rate agreements
2.1.3 Floating rate notes
2.1.4 Plain-Vanilla Swaps
2.1.5 Constructing the discount curve
2.2 Pricing measures and Black-Scholes formulas
2.2.1 Stock options with stochastic interest rates
2.2.2 Swaptions.
2.2.3 Caplets.
2.2.4 Options on Bonds.
2.2.5 Futures-forward price spread .
2.2.6 Bond futures options
2.3 One-factor models for the short rate
2.3.1 Bond pricing equation
2.3.2 Hull-White, Ho-Lee and Vasicek Models
2.3.3 Cox-Ingersoll-Ross model
2.3.4 Flesaker-Hughston model
2.4 Multifactor models
2.4.1 HJM with no-arbitrage constraints
2.4.2 BGMJ with no-arbitrage constraints
2.5 Real World Interest Rate Models
3 Advanced Topics in Pricing Theory: Exotic Options and State Dependent Models
3.1 Introduction to Barrier Options
3.2 Single-Barrier Kernels for the Simplest Model: The Wiener Process .
3.2.1 Driftless Case
3.2.2 Brownian Motion with Drift
3.3 Pricing Kernels and European Barrier Option Formulas for Geometric Brownian Motion
3.4 First Passage Time
3.5 Pricing Kernels and Barrier Option Formulas for Linear and Quadratic Volatility Models
3.5.1 Linear Volatility Models Revisited
3.5.2 Quadratic Volatility Models
3.6 Green?s Functions Method for Diffusion Kernels
3.6.1 Eigenfunction Expansions for the Green?s Function and the Transition Density
3.7 Kernels for the Bessel Process
3.7.1 The Barrier-free Kernel: No Absorption
3.7.2 The Case of Two Finite Barriers with Absorption
3.7.3 The Case of a Single Upper Finite Barrier with Absorption
3.7.4 The Case of a Single Lower Finite Barrier with Absorption
3.8 New Families of Analytical Pricing Formulas: ?From x-Space to FSpace?
3.8.1 Transformation Reduction Methodology
3.8.2 Bessel Families of State Dependent Volatility Models
3.8.3 The 4-Parameter Sub-Family of Bessel Models
3.8.3.1 Recovering the CEV Model
3.8.3.2 Recovering Quadratic Models
3.8.4 Conditions for Absorption or Probability Conservation
3.8.5 Barrier Pricing Formulas for Multi-Parameter Volatility Models
3.9 Appendix A: Proof of Lemma 3.1
3.10 Appendix B: Alternative Proof of Theorem
3.11 Appendix C: Some Properties of Bessel Functions
4 Numerical Methods for Value-at-Risk
4.1 Risk Factor Models
4.1.1 The lognormal model
4.1.2 The asymmetric Student?s t model
4.1.3 The Parzen model
4.1.4 Multivariate models
4.2 Portfolio Models
4.2.1 approximation
4.2.2 approximation
4.3 Statistical estimations for portfolios
4.3.1 Portfolio decomposition and portfolio dependent estimation .
4.3.2 Testing independence
4.3.3 A few implementation issues
4.4 Numerical methods for portfolios
4.4.1 Monte Carlo methods and variance reduction
4.4.2 Moment methods
4.4.3 Fourier Transform of the Moment Generating Function
4.5 The fast convolution method
4.5.1 The pdf of a quadratic random variable
4.5.2 Discretization
4.5.3 Accuracy and convergence
4.5.4 The computational details
4.5.5 Convolution with the fast Fourier transform
4.5.6 Computing value-at-risk
4.5.7 Richardson?s extrapolation improves accuracy
4.5.8 Computational complexity
4.6 Examples
4.6.1 Fat-tails and value-at-risk
4.6.2 So which result can we trust?
4.6.3 Computing the gradient of value-at-risk
4.6.4 The value-at-risk gradient and portfolio composition
.....
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