Contents Critique 26. portfolio optimization 32

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1 Contents Preface vii 1 Financial problems and numerical methods MATLAB environment Why MATLAB? Fixed-income securities: analysis and portfolio immunization Basic valuation of xed-income securities Interest rate sensitivity and bond portfolio immunization MATLAB functions to deal with xedincome securities Critique Portfolio optimization Basics of mean-variance portfolio optimization MATLAB functions to deal with meanvariance portfolio optimization Critique Derivatives Modeling the dynamics of asset prices 46 i

2 ii Black-Scholes model Black-Scholes model in MATLAB Pricing American options by binomial lattices Option pricing by Monte Carlo simulation Value-at-risk 66 S1.1 Stochastic di erential equations and Ito's lemma 70 References 72 2 Basics of numerical analysis Nature of numerical computation Working with a nite precision arithmetic Number representation, rounding, and truncation Error propagation and instability Order of convergence and computational complexity Solving systems of linear equations Condition number for a matrix Direct methods for solving systems of linear equations Tridiagonal matrices Iterative methods for solving systems of linear equations Function approximation and interpolation Solving nonlinear equations Bisection method Newton's method Solving nonlinear equations in MATLAB Numerical integration 115 References Optimization methods Classi cation of optimization problems Finite- vs. in nite-dimensional problems Unconstrained vs. constrained problems Convex vs. nonconvex problems Linear vs. nonlinear problems Continuous vs. discrete problems 130

3 iii Deterministic vs. stochastic problems Numerical methodsfor unconstrained optimization Steepest descent method The subgradient method Newton and the trust region methods No-derivatives algorithms: quasi-newton method and simplex search Unconstrained optimization in MATLAB Methods for constrained optimization Penalty function approach Kuhn-Tucker conditions Duality theory Kelley's cutting plane algorithm Active set method Linear programming Geometric and algebraic features of linear programming Simplex method Duality in linear programming Interior point methods Linear programming in MATLAB Branch and bound methods for nonconvex optimization LP-based branch and bound for MILP models Heuristic methods for nonconvex optimization L-shaped method for two-stage linear stochastic programming 192 S3.1 Elements of convex analysis 195 S3.1.1 Convexity in optimization 195 S3.1.2 Convex polyhedra and polytopes 199 References Principles of Monte Carlo simulation Monte Carlo integration Generating pseudorandom variates Generating pseudorandom numbers Inverse transform method 210

4 iv Acceptance-rejection method Generating normal variates by the polar approach Setting the number of replications Variance reduction techniques Antithetic variates Common random numbers Control variates Variance reduction by conditioning Strati ed sampling Importance sampling Quasi-Monte Carlo simulation Generating Halton's low-discrepancy sequences Generating Sobol's low-discrepancy sequences Integrating simulation and optimization 244 References Finite di erence methods for partial di erential equations Introduction and classi cation of PDEs Numerical solution by nite di erence methods Bad example of a nite di erence scheme Instability in a nite di erence scheme Explicit and implicit methods for second-order PDEs Solving the heat equation by an explicit method Solving the heat equation by an implicit method Solving the heat equation by the Crank- Nicolson method Convergence, consistency, and stability 273 S5.1 Classi cation of second-order PDEs and characteristic curves 275 References Optimization models for portfolio management Mixed-integer programming models Multistage stochastic programming models 285

5 v Split-variable formulation Compact formulation Sample asset and liability management model formulation Scenario generation for multistage stochastic programming Fixed-mix model based on global optimization 305 References Option valuation by Monte Carlo simulation Simulating asset price dynamics Pricing a vanilla European option by Monte Carlo simulation Using antithetic variatesto price a vanilla European option Using antithetic variates to price a European option with truncated payo Using control variates to price a vanilla European option Using Halton low-discrepancy sequences to price a vanilla European option Introduction to exotic and path-dependent options Barrier options Asian options Lookback options Pricing a down-and-out put Pricing an Asian option 336 References Option valuation by nite di erence methods Applying nite di erence methods to the Black- Scholes equation Pricing a vanilla European option by an explicit method Financial interpretation of the instability of the explicit method Pricing a vanilla European option by a fully implicit method Pricing a barrier option by the Crank-Nicolson method 353

6 vi 8.5 Dealing with American options 354 References 360 Appendix A Introduction to MATLAB programming 361 A.1 MATLAB environment 361 A.2 MATLAB graphics 368 A.3 MATLAB programming 369 Appendix B Refresher on probability theory 373 B.1 Sample space, events, and probability 373 B.2 Random variables, expectation, and variance 375 B.2.1 Common continuous random variables 377 B.3 Jointly distributed random variables 380 B.4 Independence, covariance, and conditional expectation 382 B.5 Parameter estimation 385 References 389 Index 391