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| Preface | |
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| An Introduction to Probability Theory | |
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| The Notion of a Set and a Sample Space | |
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| Sigma Algebras or Field | |
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| Probability Measure and Probability Space | |
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| Measurable Mapping | |
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| Cumulative Distribution Functions | |
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| Convergence in Distribution | |
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| Random Variables | |
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| Discrete Random Variables | |
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| Example of Discrete Random Variables: The Binomial Distribution | |
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| Hypergeometric Distribution | |
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| Poisson Distribution | |
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| Continuous Random Variables | |
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| Uniform Distribution | |
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| The Normal Distribution | |
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| Change of Variable | |
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| Exponential Distribution | |
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| Gamma Distribution | |
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| Measurable Function | |
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| Cumulative Distribution Function and Probability Density Function | |
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| Joint, Conditional and Marginal Distributions | |
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| Expected Values of Random Variables and Moments of a Distribution | |
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| The Bernoulli Law of Large Numbers | |
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| Conditional Expectations | |
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| Stochastic Processes | |
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| Stochastic Processes | |
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| Martingales Processes | |
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| Brownian Motions | |
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| Brownian Motion and the Reflection Principle | |
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| Geometric Brownian Motions | |
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| An Application of Brownian Motions | |
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| Ito Calculus and Ito Integral | |
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| Total Variation and Quadratic Variation of Differentiable Functions | |
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| Quadratic Variation of Brownian Motions | |
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| The Construction of the Ito Integral | |
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| Properties of the Ito Integral | |
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| The General Ito Stochastic Integral | |
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| Properties of the General Ito Integral | |
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| Construction of the Ito Integral with Respect to Semi-Martingale Integrators | |
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| Quadratic Variation of a General Bounded Martingale | |
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| Quadratic Variation of a General Bounded Martingale | |
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| Ito Lemma and Ito Formula | |
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| The Riemann-Stieljes Integral | |
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| The Black and Scholes Economy | |
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| Introduction | |
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| Trading Strategies and Martingale Processes | |
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| The Fundamental Theorem of Asset Pricing | |
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| Martingale Measures | |
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| Girsanov Theorem | |
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| Risk-Neutral Measures | |
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| The Randon-Nikodym Condition | |
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| Geometric Brownian Motion | |
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| The Black and Scholes Model | |
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| Introduction | |
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| The Black and Scholes Model | |
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| The Black and Scholes Formula | |
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| Black and Scholes in Practice | |
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| The Feynman-Kac Formula | |
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| The Kolmogorov Backward Equation | |
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| Change of Numeraire | |
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| Black and Scholes and the Greeks | |
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| Monte Carlo Methods | |
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| Introduction | |
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| The Data Generating Process (DGP) and the Model | |
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| Pricing European Options | |
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| Variance Reduction Techniques | |
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| Antithetic Variate Methods | |
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| Control Variate Methods | |
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| Common Random Numbers | |
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| Importance Sampling | |
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| Monte Carlo European Options | |
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| Variance Reduction Techniques - First Part | |
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| Monte Carlo 1 | |
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| Monte Carlo 2 | |
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| Monte Carlo Methods and American Options | |
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| Introduction | |
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| Pricing American Options | |
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| Dynamic Programming Approach and American Option Pricing | |
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| The Longstaff and Schwartz Least Squares Method | |
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| The Glasserman and Yu Regression Later Method | |
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| Upper and Lower Bounds and American Options | |
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| Multiassets Simulation | |
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| Pricing a Basket Option Using the Regression Methods | |
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| American Option Pricing: The Dual Approach | |
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| Introduction | |
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| A General Framework for American Option Pricing | |
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| A Simple Approach to Designing Optimal Martingales | |
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| Optimal Martingales and American Option Pricing | |
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| A Simple Algorithm for American Option Pricing | |
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| Empirical Results | |
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| Computing Upper Bounds | |
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| Empirical Results | |
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| Estimation of Greeks using Monte Carlo Methods | |
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| Finite Difference Approximations | |
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| Pathwise Derivatives Estimation | |
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| Likelihood Ratio Method | |
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| Discussion | |
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| Pathwise Greeks using Monte Carlo | |
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| Exotic Options | |
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| Introduction | |
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| Digital Options | |
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| Asian Options | |
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| Forward Start Options | |
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| Barrier Options | |
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| Hedging Barrier Options | |
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| Digital Options | |
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| Pricing and Hedging Exotic Options | |
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| Introduction | |
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| Monte Carlo Simulations and Asian Options | |
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| Simulation of Greeks for Exotic Options | |
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| Monte Carlo Simulations and Forward Start Options | |
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| Simulation of the Greeks for Exotic Options | |
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| Monte Carlo Simulations and Barrier Options | |
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| The Price and the Delta of Forward Start Options | |
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| The Price of Barrier Options Using Importance Sampling | |
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| Stochastic Volatility Models | |
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| Introduction | |
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| The Model | |
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| Square Root Diffusion Process | |
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| The Heston Stochastic Volatility Model (HSVM) | |
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| Processes with Jumps | |
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| Application of the Euler Method to Solve SDEs | |
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| Exact Simulation Under SV | |
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| Exact Simulation of Greeks Under SV | |
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| Stochastic Volatility Using the Heston Model | |
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| Implied Volatility Models | |
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| Introduction | |
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| Modelling Implied Volatility | |
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| Examples | |
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| Local Volatility Models | |
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| An Overview | |
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| The Model | |
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| Numerical Methods | |
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| An Introduction to Interest Rate Modelling | |
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| A General Framework | |
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| Affine Models (AMs) | |
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| The Vasicek Model | |
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| The Cox, Ingersoll and Ross (CIR) Model | |
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| The Hull and White (HW) Model | |
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| The Black Formula and Bond Options | |
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| Interest Rate Modelling | |
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| Some Preliminary Definitions | |
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| Interest Rate Caplets and Floorlets | |
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| Forward Rates and Numeraire | |
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| Libor Futures Contracts | |
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| Martingale Measure | |
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| Binomial and Finite Difference Methods | |
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| The Binomial Model | |
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| Expected Value and Variance in the Black and Scholes and Binomial Models | |
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| The Cox-Ross-Rubinstein Model | |
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| Finite Difference Methods | |
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| The Binomial Method | |
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| An Introduction to MATLAB | |
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| What is MATLAB? | |
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| Starting MATLAB | |
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| Main Operations in MATLAB | |
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| Vectors and Matrices | |
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| Basic Matrix Operations | |
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| Linear Algebra | |
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| Basics of Polynomial Evaluations | |
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| Graphing in MATLAB | |
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| Several Graphs on One Plot | |
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| Programming in MATLAB: Basic Loops | |
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| M-File Functions | |
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| MATLAB Applications in Risk Management | |
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| MATLAB Programming: Application in Financial Economics | |
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| Mortgage Backed Securities | |
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| Introduction | |
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| The Mortgage Industry | |
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| The Mortgage Backed Security (MBS) Model | |
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| The Term Structure Model | |
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| Preliminary Numerical Example | |
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| Dynamic Option Adjusted Spread | |
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| Numerical Example | |
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| Practical Numerical Examples | |
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| Empirical Results | |
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| The Pre-Payment Model | |
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| Value at Risk | |
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| Introduction | |
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| Value at Risk (VaR) | |
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| The Main Parameters of a VaR | |
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| VaR Methodology | |
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| Historical Simulations | |
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| Variance-Covariance Method | |
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| Monte Carlo Method | |
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| Empirical Applications | |
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| Historical Simulations | |
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| Variance-Covariance Method | |
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| Fat Tails and VaR | |
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| Generalized Extreme Value and the Pareto Distribution | |
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| Bibliography | |
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| References | |
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| Index | |