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| Preface to the Second Edition | |
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| From the Preface to the First Edition | |
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| Background | |
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| Motivation | |
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| Need for numerical methods | |
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| Need for numerical computing environments: why MATLAB? | |
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| Need for theory | |
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| For further reading | |
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| References | |
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| Financial Theory | |
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| Modeling uncertainty | |
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| Basic financial assets and related issues | |
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| Bonds | |
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| Stocks | |
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| Derivatives | |
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| Asset pricing, portfolio optimization, and risk management | |
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| Fixed-income securities: analysis and portfolio immunization | |
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| Basic theory of interest rates: compounding and present value | |
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| Basic pricing of fixed-income securities | |
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| Interest rate sensitivity and bond portfolio immunization | |
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| MATLAB functions to deal with fixed-income securities | |
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| Critique | |
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| Stock portfolio optimization | |
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| Utility theory | |
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| Mean-variance portfolio optimization | |
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| MATLAB functions to deal with mean-variance portfolio optimization | |
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| Critical remarks | |
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| Alternative risk measures: Value at Risk and quantile-based measures | |
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| Modeling the dynamics of asset prices | |
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| From discrete to continuous time | |
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| Standard Wiener process | |
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| Stochastic integrals and stochastic differential equations | |
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| Ito's lemma | |
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| Generalizations | |
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| Derivatives pricing | |
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| Simple binomial model for option pricing | |
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| Black-Scholes model | |
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| Risk-neutral expectation and Feynman-Kac formula | |
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| Black-Scholes model in MATLAB | |
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| A few remarks on Black-Scholes formula | |
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| Pricing American options | |
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| Introduction to exotic and path-dependent options | |
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| Barrier options | |
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| Asian options | |
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| Lookback options | |
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| An outlook on interest-rate derivatives | |
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| Modeling interest-rate dynamics | |
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| Incomplete markets and the market price of risk | |
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| For further reading | |
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| References | |
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| Numerical Methods | |
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| Basics of Numerical Analysis | |
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| Nature of numerical computation | |
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| Number representation, rounding, and truncation | |
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| Error propagation, conditioning, and instability | |
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| Order of convergence and computational complexity | |
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| Solving systems of linear equations | |
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| Vector and matrix norms | |
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| Condition number for a matrix | |
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| Direct methods for solving systems of linear equations | |
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| Tridiagonal matrices | |
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| Iterative methods for solving systems of linear equations | |
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| Function approximation and interpolation | |
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| Ad hoc approximation | |
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| Elementary polynomial interpolation | |
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| Interpolation by cubic splines | |
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| Theory of function approximation by least squares | |
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| Solving non-linear equations | |
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| Bisection method | |
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| Newton's method | |
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| Optimization-based solution of non-linear equations | |
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| Putting two things together: solving a functional equation by a collocation method | |
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| Homotopy continuation methods | |
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| For further reading | |
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| References | |
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| Numerical Integration: Deterministic and Monte Carlo Methods | |
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| Deterministic quadrature | |
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| Classical interpolatory formulas | |
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| Gaussian quadrature | |
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| Extensions and product rules | |
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| Numerical integration in MATLAB | |
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| Monte Carlo integration | |
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| Generating pseudorandom variates | |
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| Generating pseudorandom numbers | |
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| Inverse transform method | |
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| Acceptance-rejection method | |
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| Generating normal variates by the polar approach | |
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| Setting the number of replications | |
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| Variance reduction techniques | |
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| Antithetic sampling | |
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| Common random numbers | |
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| Control variates | |
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| Variance reduction by conditioning | |
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| Stratified sampling | |
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| Importance sampling | |
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| Quasi-Monte Carlo simulation | |
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| Generating Halton low-discrepancy sequences | |
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| Generating Sobol low-discrepancy sequences | |
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| For further reading | |
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| References | |
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| Finite Difference Methods for Partial Differential Equations | |
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| Introduction and classification of PDEs | |
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| Numerical solution by finite difference methods | |
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| Bad example of a finite difference scheme | |
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| Instability in a finite difference scheme | |
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| Explicit and implicit methods for the heat equation | |
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| Solving the heat equation by an explicit method | |
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| Solving the heat equation by a fully implicit method | |
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| Solving the heat equation by the Crank-Nicolson method | |
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| Solving the bidimensional heat equation | |
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| Convergence, consistency, and stability | |
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| For further reading | |
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| References | |
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| Convex Optimization | |
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| Classification of optimization problems | |
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| Finite- vs. infinite-dimensional problems | |
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| Unconstrained vs. constrained problems | |
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| Convex vs. non-convex problems | |
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| Linear vs. non-linear problems | |
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| Continuous vs. discrete problems | |
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| Deterministic vs. stochastic problems | |
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| Numerical methods for unconstrained optimization | |
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| Steepest descent method | |
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| The subgradient method | |
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| Newton and the trust region methods | |
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| No-derivatives algorithms: quasi-Newton method and simplex search | |
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| Unconstrained optimization in MATLAB | |
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| Methods for constrained optimization | |
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| Penalty function approach | |
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| Kuhn-Tucker conditions | |
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| Duality theory | |
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| Kelley's cutting plane algorithm | |
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| Active set method | |
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| Linear programming | |
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| Geometric and algebraic features of linear programming | |
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| Simplex method | |
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| Duality in linear programming | |
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| Interior point methods | |
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| Constrained optimization in MATLAB | |
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| Linear programming in MATLAB | |
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| A trivial LP model for bond portfolio management | |
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| Using quadratic programming to trace efficient portfolio frontier | |
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| Non-linear programming in MATLAB | |
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| Integrating simulation and optimization | |
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| Elements of convex analysis | |
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| Convexity in optimization | |
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| Convex polyhedra and polytopes | |
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| For further reading | |
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| References | |
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| Pricing Equity Options | |
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| Option Pricing by Binomial and Trinomial Lattices | |
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| Pricing by binomial lattices | |
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| Calibrating a binomial lattice | |
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| Putting two things together: pricing a pay-later option | |
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| An improved implementation of binomial lattices | |
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| Pricing American options by binomial lattices | |
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| Pricing bidimensional options by binomial lattices | |
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| Pricing by trinomial lattices | |
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| Summary | |
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| For further reading | |
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| References | |
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| Option Pricing by Monte Carlo Methods | |
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| Path generation | |
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| Simulating geometric Brownian motion | |
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| Simulating hedging strategies | |
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| Brownian bridge | |
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| Pricing an exchange option | |
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| Pricing a down-and-out put option | |
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| Crude Monte Carlo | |
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| Conditional Monte Carlo | |
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| Importance sampling | |
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| Pricing an arithmetic average Asian option | |
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| Control variates | |
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| Using Halton sequences | |
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| Estimating Greeks by Monte Carlo sampling | |
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| For further reading | |
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| References | |
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| Option Pricing by Finite Difference Methods | |
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| Applying finite difference methods to the Black-Scholes equation | |
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| Pricing a vanilla European option by an explicit method | |
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| Financial interpretation of the instability of the explicit method | |
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| Pricing a vanilla European option by a fully implicit method | |
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| Pricing a barrier option by the Crank-Nicolson method | |
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| Dealing with American options | |
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| For further reading | |
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| References | |
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| Advanced Optimization Models and Methods | |
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| Dynamic Programming | |
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| The shortest path problem | |
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| Sequential decision processes | |
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| The optimality principle and solving the functional equation | |
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| Solving stochastic decision problems by dynamic programming | |
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| American option pricing by Monte Carlo simulation | |
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| A MATLAB implementation of the least squares approach | |
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| Some remarks and alternative approaches | |
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| For further reading | |
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| References | |
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| Linear Stochastic Programming Models with Recourse | |
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| Linear stochastic programming models | |
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| Multistage stochastic programming models for portfolio management | |
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| Split-variable model formulation | |
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| Compact model formulation | |
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| Asset and liability management with transaction costs | |
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| Scenario generation for multistage stochastic programming | |
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| Sampling for scenario tree generation | |
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| Arbitrage free scenario generation | |
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| L-shaped method for two-stage linear stochastic programming | |
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| A comparison with dynamic programming | |
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| For further reading | |
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| References | |
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| Non-Convex Optimization | |
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| Mixed-integer programming models | |
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| Modeling with logical variables | |
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| Mixed-integer portfolio optimization models | |
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| Fixed-mix model based on global optimization | |
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| Branch and bound methods for non-convex optimization | |
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| LP-based branch and bound for MILP models | |
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| Heuristic methods for non-convex optimization | |
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| For further reading | |
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| References | |
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| Appendices | |
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| Introduction to MATLAB Programming | |
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| MATLAB environment | |
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| MATLAB graphics | |
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| MATLAB programming | |
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| Refresher on Probability Theory and Statistics | |
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| Sample space, events, and probability | |
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| Random variables, expectation, and variance | |
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| Common continuous random variables | |
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| Jointly distributed random variables | |
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| Independence, covariance, and conditional expectation | |
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| Parameter estimation | |
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| Linear regression | |
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| For further reading | |
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| References | |
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| Introduction to AMPL | |
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| Running optimization models in AMPL | |
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| Mean variance efficient portfolios in AMPL | |
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| The knapsack model in AMPL | |
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| Cash flow matching | |
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| For further reading | |
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| References | |
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| Index | |