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Advanced Derivatives Pricing and Risk Management,

Edition 1 Theory, Tools, and Hands-On Programming Applications

Editors: By Claudio Albanese and Giuseppe Campolieti

Publication Date: 08 Sep 2005

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Advanced Derivatives Pricing and Risk Management covers the most important and cutting-edge topics in financial derivatives pricing and risk management, striking a fine balance between theory and practice. 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.

The book is designed for students in finance programs, particularly financial engineering.

Key Features

*Includes easy-to-implement VB/VBA numerical software libraries*Proceeds from simple to complex in approaching pricing and risk management problems*Provides analytical methods to derive cutting-edge pricing formulas for equity derivatives

About the author

By Claudio Albanese, Professor of Mathematical Finance, Imperial College, London, UK and Giuseppe Campolieti, Associate Professor of Mathematics, SHARCNET Chair in Financial Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

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

ISBN: 9780120476824

Page Count: 426

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