Advanced Derivatives Pricing and Risk Management book cover

Advanced Derivatives Pricing and Risk Management

Theory, Tools, and Hands-On Programming Applications

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.

Audience
Students in finance programs, particularly financial engineering.

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Published: September 2005

Imprint: Academic Press

ISBN: 978-0-12-047682-4

Reviews

  • “Albanese and Campolieti carefully select the most important and relevant topics in financial derivatives pricing and risk management. Their work strikes a fine balance between theory and financial practice. A dozen carefully designed numerical projects are included that serve to introduce students to actual implementation issues in pricing and risk management. The book is succinctly written, with clear and insightful descriptions of state-of-the-art financial models. The style of presentation demonstrates the authors' unique pedagogical exposition of the quantitative and financial concepts in derivative pricing and risk management. Advanced Derivatives Pricing and Risk Management is destined to be a valuable text and reference for students and practitioners in the field of financial engineering.” — Yue Kuen Kwok, Associate Professor, Department of Mathematics, Hong Kong University of Science and Technology “The set of projects on the accompanying CDROM give students and professors the opportunity to work in a simulated environment and can be used, as is the goal here, to train students in building software modules for pricing, hedging, etc. The projects enhance the understanding of the material and extend the book's usefulness by enabling students to tackle other situations not explicitly addressed in the modules provided. VBA is easy to learn and can facilitate rapid developments of real applications. In addition, the choice of Excel as the Graphic User Interface (GUI) is very appropriate. Furthermore, the existence of a built-in visual basic editor allows users to see the code, modify it to suit different needs and to experiment with it. Hence these features facilitate student learning to produce software themselves.” — Eliezer Prisman, Nigel Martin Chair in Finance, Director of the Financial Engineering Collaborative Diploma, Schulich School of Business, York University, Toronto "...provides a combination of theoretical and practical tools from which any advanced undergraduate and graduate student, professional quant and researcher will benefit. It differs from existing books in quantitative finance from the ready-to-use software and accessible theoretical tools provided as a complete package. As a bonus, the book also gives a detailed exposition on cutting-edge theoretical techniques published here for the first time." - Technical Analysis of Stocks & Commodities

Contents

  • I Pricing Theory and Risk Management 111 Pricing Theory 131.1 Single Period, Finite Financial Models . . . . . . . . . . . . . . . . . 161.2 Continuous state spaces . . . . . . . . . . . . . . . . . . 241.3 Multivariate Continuous Distributions: Basic Tools . . . . . . . . . . 281.4 Brownian Motion, Martingales and Stochastic Integrals . . . . . . . . 351.5 Stochastic Differential Equations and Ito’s formula . . . . . . . . . . 461.6 Geometric Brownian Motion . . .521.7 Forwards and European Calls and Puts . . . . . . . . . . . . . . . . . 611.8 Static Hedging and Replication of Exotic Payoffs . . . . . . . . . . . 681.9 Continuous Time Financial Models . . . . . . . . . . . . . . . . . . . 771.10 Dynamic Hedging and Derivative Asset Pricing in Continuous Time . 841.11 Hedging with Forwards and Futures . . . . . . . . . . . . . . . . . . 901.12 Pricing formulas of the Black-Scholes type . . . . . . . . . . . . . . 961.13 Partial Differential Equations for Pricing Functions and Kernels . . . 1081.14 American Options . . . . . . . . . . . . . . . . . . . . 1141.14.1 Arbitrage-Free Pricing and Optimal Stopping Time Formulation 1141.14.2 Perpetual American Options . . . . . . . . . . . . . . . . . . 1251.14.3 Properties of the Early-Exercise Boundary . . . . . . . . . . . 1271.14.4 The PDE and Integral Equation Formulation . . . . . . . . . 1292 Fixed Income Instruments 1352.1 Bonds, Futures, Forwards and Swaps . . . . . . . . . . . . . . . . . . 1352.1.1 Bonds . . . . . . . . . . . . . . . . . . . . . 1352.1.2 Forward rate agreements . . . . . . . . . . . . . . . . . . . 1382.1.3 Floating rate notes . . . . . . . . . . . . . . . . . . . . . 1392.1.4 Plain-Vanilla Swaps . . . . . . . . . . . . . . . . . . . . . 1402.1.5 Constructing the discount curve . . . . . . . . . . . . . . . . 1412.2 Pricing measures and Black-Scholes formulas . . . . . . . . . . . . . 1432.2.1 Stock options with stochastic interest rates. . . . . . . . . . . 1442.2.2 Swaptions. . .. . . . . . . . . . . . . . . . . 1452.2.3 Caplets. . . . . . . . . . . . . . . . . . . . . 1462.2.4 Options on Bonds. . . . . . . . . . . . . . . . . . . . . . 1472.2.5 Futures-forward price spread . . . . . . . . . . . . . . . . . . 1472.2.6 Bond futures options . . . . . . . . . . .. . . . . . . . . . 1492.3 One-factor models for the short rate . . . . . . . . . . . . . . . . . . 1512.3.1 Bond pricing equation . . . . . . . . . . . . . . . . . . . . 1512.3.2 Hull-White, Ho-Lee and Vasicek Models . . . . . . . . . . . 1522.3.3 Cox-Ingersoll-Ross model . . . . . . . . . . . . . . . . . . . 1582.3.4 Flesaker-Hughston model . . . . . . . . . . . . . . . . . . . 1632.4 Multifactor models . . . . . . . . . . . . . . . . . . . . . 1662.4.1 HJM with no-arbitrage constraints . . . . . . . . . . . . . . . 1672.4.2 BGMJ with no-arbitrage constraints . . . . . . . . . . . . . . 1692.5 Real World Interest Rate Models . . . . . . . . . . . . . . . . . . . . 1713 Advanced Topics in Pricing Theory: Exotic Options and State DependentModels 1753.1 Introduction to Barrier Options . . . . . . . . . . . . . . . . . . . . 1773.2 Single-Barrier Kernels for the Simplest Model: The Wiener Process . 1793.2.1 Driftless Case . . . . . . . . . . . . . . . . . . . . . . 1793.2.2 Brownian Motion with Drift . . . . . . . . . . . . . . . . . . 1853.3 Pricing Kernels and European Barrier Option Formulas for GeometricBrownian Motion . . . . . . . . . . . . . . . . . . . . . 1873.4 First Passage Time . . . . . . . . . . . . . . . . . . . . . . 1963.5 Pricing Kernels and Barrier Option Formulas for Linear and QuadraticVolatility Models . . . . . . . . . . . . . . . . . . . . . 2003.5.1 Linear Volatility Models Revisited . . . . . . . . . . . . . . 2003.5.2 Quadratic Volatility Models . . . . . . . . . . . . . . . . . . 2083.6 Green’s Functions Method for Diffusion Kernels . . . . . . . . . . . 2193.6.1 Eigenfunction Expansions for the Green’s Function and theTransition Density . . . . . . . . . . . . . . . . . . . . 2283.7 Kernels for the Bessel Process . . . . . . . . . . . . . . . . . . . . 2303.7.1 The Barrier-free Kernel: No Absorption . . . . . . . . . . . . 2313.7.2 The Case of Two Finite Barriers with Absorption . . . . . . . 2343.7.3 The Case of a Single Upper Finite Barrier with Absorption . . 2383.7.4 The Case of a Single Lower Finite Barrier with Absorption . . 2413.8 New Families of Analytical Pricing Formulas: “From x-Space to FSpace”. . . . .. . . . . . . . . . . . . . . . . . . . 2423.8.1 Transformation Reduction Methodology . . . . . . . . . . . . 2433.8.2 Bessel Families of State Dependent Volatility Models . . . . . 2493.8.3 The 4-Parameter Sub-Family of Bessel Models . . . . . . . . 2523.8.3.1 Recovering the CEV Model . . . . . . . . . . . . . 2563.8.3.2 Recovering Quadratic Models . . . . . . . . . . . . 2593.8.4 Conditions for Absorption or Probability Conservation . . . . 2613.8.5 Barrier Pricing Formulas for Multi-Parameter Volatility Models 2643.9 Appendix A: Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . 2683.10 Appendix B: Alternative Proof of Theorem 3.1 . . . . . . . . . . . . 2703.11 Appendix C: Some Properties of Bessel Functions . . . . . . . . . . . 272CONTENTS 74 Numerical Methods for Value-at-Risk 2754.1 Risk Factor Models . . . . . . . . . . . . . . . . . . . . . 2794.1.1 The lognormal model . . . . . . . . . . . . . . . . . . . . 2794.1.2 The asymmetric Student’s t model . . . . . . . . . . . . . . . 2804.1.3 The Parzen model . . . . . . . . . . . . . . . . . . . . . 2824.1.4 Multivariate models . . . . . . . . . . . . . . . . . . . . . 2844.2 Portfolio Models . . . . . . . . . . . . . . . . . . . . . 2864.2.1 _-approximation . . . . . . . .. . . . . . . . . . 2874.2.2 __-approximation . . . . .. . . . . . . . . . . . . 2894.3 Statistical estimations for __-portfolios . . . . . . . . . . . . . . . . 2914.3.1 Portfolio decomposition and portfolio dependent estimation . 2914.3.2 Testing independence . . . . . . . . . . . . . . . . . . 2934.3.3 A few implementation issues . . . . . . . . . . . . . . . . . . 2954.4 Numerical methods for __-portfolios . . . . . . . . . . . . . . . . . 2974.4.1 Monte Carlo methods and variance reduction . . . . . . . . . 2974.4.2 Moment methods . . . . . . . . . . . . .. . . . . . . . 3004.4.3 Fourier Transform of the Moment Generating Function . . . . 3034.5 The fast convolution method . . . . . . . . . . . . . . . . . . . 3054.5.1 The pdf of a quadratic random variable . . . . . . . . . . . . 3064.5.2 Discretization . . . . . . . . . . . . . . . . . 3074.5.3 Accuracy and convergence . . . . . . . . . . . . . . . . . . 3084.5.4 The computational details . . . . . . . . . . . . . . . . . . . 3084.5.5 Convolution with the fast Fourier transform . . . . . . . . . . 3084.5.6 Computing value-at-risk . . . . . . . . . . . . . . . . . . . . 3144.5.7 Richardson’s extrapolation improves accuracy . . . . . . . . . 3154.5.8 Computational complexity . . . . . . . . . . . . . . . . . . . 3174.6 Examples . . . . . . . . . . . . . . 3184.6.1 Fat-tails and value-at-risk . . . . . . . . . . . . . . . . . . . . 3184.6.2 So which result can we trust? . . . . . . . . . . . . . . . . . . 3194.6.3 Computing the gradient of value-at-risk . . . . . . . . . . . . 3194.6.4 The value-at-risk gradient and portfolio composition . . . . . 3204.6.5 Computing the gradient . . . . . . . . . . . . . . . . . . . . 3214.6.6 Sensitivity analysis and the linear approximation . . . . . . . 3234.6.7 Hedging with value-at-risk . . . . . . . . . . . . . . . . . . . 3244.6.8 Adding stochastic volatility . . . . . . . . . . . . . . . . . . 3254.7 Risk factor aggregation and dimension reduction . . . . . . . . . . . 3264.7.1 Method 1: reduction with small mean square error . . . . . . 3274.7.2 Method 2: reduction by low-rank approximation . . . . . . . 3294.7.3 Absolute versus relative value-at-risk . . . . . . . . . . . . . 3324.7.4 Example: a comparative experiment . . . . . . . . . . . . . . 3324.7.5 Example: dimension reduction and optimization . . . . . . . 3334.8 Perturbation theory . . . . . . . .. . . . . . . . . . 3344.8.1 When is value-at-risk well-posed? . . . . . . . . . . . . . . . 3344.8.2 Perturbations of the return model . . . . . . . . . . . . . . . 3364.8.3 Proof of a first-order perturbation property . . . . . . . . . . 3364.8.4 Error bounds and the condition number . . . . . . . . . . . . 3378 CONTENTS4.8.5 Example: mixture model . . . . . . . . . . . . . . . . . . . . 339II Numerical Projects in Pricing and Risk Management 3535 Project: Arbitrage Theory 3555.1 Basic Terminology and Concepts: Asset Prices, States, Returns andPayoffs . . . . . . . . . . . . . . . . . . . . 3555.2 Arbitrage Portfolios and The Arbitrage Theorem . . . . . . . . . . . 3575.3 An example of single period asset pricing: Risk-Neutral Probabilitiesand Arbitrage . .. . . . . . . . . . . . . . . . . 3585.4 Arbitrage detection and the formation of arbitrage portfolios in the Ndimensionalcase . . . . . . . . . . .. . . . . . . . . . . . . . 3606 Project: The Black-Scholes (Lognormal) Model 3616.1 Black-Scholes pricing formula . . . . . . . . . . . . . . . . . . . . 3616.2 Black-Scholes sensitivity analysis . . . . . . . . . . . . . . . . . . . 3657 Project: Quantile-quantile plots 3677.1 Log-returns and standardization . . . . . . . . . . . . . . . . 3677.2 Quantile-Quantile plots . . . . . . . . . . . . . . . . . . . . . 3688 Project: Monte Carlo Pricer 3718.1 Scenario Generation . . . . . . . . . . . . . . . . . . 3718.2 Calibration . . . . . . . . . . . . . . . . . . 3728.3 Pricing Equity Basket Options . . . . . . . . . . . . . . . . . . . . 3749 Project: The Binomial Lattice Model 3779.1 Building the Lattice . . . . . . . . . . . . . . . . . . . . 3779.2 Lattice Calibration and Pricing . . . . . . . . . . . . . . . . . . . . 37910 Project: The Trinomial Lattice Model 38310.1 Building the Lattice . . . . . . . . . . . . . . . . . . 38310.2 Pricing procedure . . . . . . . . . . . . . . . . . . . 38610.3 Calibration . . . . . . . . . . . . . . . 38810.4 Pricing barrier options . . . . . . . .. . . . . . . . . . . . . 38910.5 Put-call parity in trinomial lattices . . . . . . . . . . . . . . . . . . . 39010.6 Computing the sensitivities . . . . . . . . . . . . . . . . . 39111 Project: Crank-Nicolson option pricer 39311.1 The Lattice for the Crank-Nicolson pricer . . . . . . . . . . . . . . . 39311.2 Pricing with Crank-Nicolson . . . . . . . . . . . . . . . . 39411.3 Calibration . . . . . . . . . . . . . . . . . . 39611.4 Pricing barrier options . . . . . . . . . . . . . . . . . . 396CONTENTS 912 Project: Static Hedging of Barrier Options 39912.1 Analytical Pricing Formulas for Barrier Options . . . . . . . . . . . . 39912.2 Replication of up-and-out barrier options . . . . . . . . . . . . . . . . 40212.3 Replication of down-and-out barrier options . . . . . . . . . . . . . . 40513 Project: Variance Swaps 40913.1 The logarithmic payoff . . . . . . . . . . . . . . . . . . . . 40913.2 Static Hedging: replication of a logarithmic payoff . . . . . . . . . . 41014 Project: Monte Carlo VaR for Delta-Gamma Portfolios 41514.1 Multivariate Normal Distribution . . . . . . . . . . . . . . . 41514.2 Multivariate Student-t Distributions . . . . . . . . . . . .. . . . . 41815 Project: Covariance estimation and scenario generation in VaR 42115.1 Generating covariance matrices of a given spectrum . . . . . . . . . . 42115.2 Re-estimating the covariance matrix and the spectral shift . . . . . . . 42216 Project: Interest Rate Trees: Calibration and Pricing 42516.1 Background Theory . . . . .. . . . . . . . . . . . . . . 42516.2 Binomial Lattice Calibration for Discount Bonds . . . . . . . . . . . 42716.3 Binomial pricing of FRAs, Swaps, Caplets, Floorlets, Swaptions andother derivatives . . . . . . . . . . . . . . . . . . 43116.4 Trinomial Lattice Calibration and Pricing in the Hull-White model . . 43716.4.1 The First Stage: The Lattice with zero drift . . . . . . . . . . 43716.4.2 The Second Stage: Lattice calibration with drift and reversion 44116.4.3 Pricing options . . . . . . . .. . . . . . . . . . . 44516.5 Calibration and pricing within the Black-Karasinski model . . . . . . 446

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