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