Title:
Statistical methods for financial engineering
Author:
Remillard, Bruno.
ISBN:
9781439856949
Personal Author:
Publication Information:
Boca Raton, FL : CRC Press, 2013.
Physical Description:
xxxiii, 462 pages : illustrations ; 25 cm.
General Note:
Formerly CIP.
Contents:
Machine generated contents note: 1.Black-Scholes Model -- Summary -- 1.1.The Black-Scholes Model -- 1.2.Dynamic Model for an Asset -- 1.2.1.Stock Exchange Data -- 1.2.2.Continuous Time Models -- 1.2.3.Joint Distribution of Returns -- 1.2.4.Simulation of a Geometric Brownian Motion -- 1.2.5.Joint Law of Prices -- 1.3.Estimation of Parameters -- 1.4.Estimation Errors -- 1.4.1.Estimation of Parameters for Apple -- 1.5.Black-Scholes Formula -- 1.5.1.European Call Option -- 1.5.1.1.Put-Call Parity -- 1.5.1.2.Early Exercise of an American Call Option -- 1.5.2.Partial Differential Equation for Option Values -- 1.5.3.Option Value as an Expectation -- 1.5.3.1.Equivalent Martingale Measures and Pricing of Options -- 1.5.4.Dividends -- 1.5.4.1.Continuously Paid Dividends -- 1.6.Greeks -- 1.6.1.Greeks for a European Call Option -- 1.6.2.Implied Distribution -- 1.6.3.Error on the Option Value -- 1.6.4.Implied Volatility -- 1.6.4.1.Problems with Implied Volatility --
Contents note continued: 1.7.Estimation of Greeks using the Broadie-Glasserman Methodologies -- 1.7.1.Pathwise Method -- 1.7.2.Likelihood Ratio Method -- 1.7.3.Discussion -- 1.8.Suggested Reading -- 1.9.Exercises -- 1.10.Assignment Questions -- 1.A.Justification of the Black-Scholes Equation -- 1.B.Martingales -- 1.C.Proof of the Results -- 1.C.1.Proof of Proposition 1.3.1 -- 1.C.2.Proof of Proposition 1.4.1 -- 1.C.3.Proof of Proposition 1.6.1 -- Bibliography -- 2.Multivariate Black-Scholes Model -- Summary -- 2.1.Black-Scholes Model for Several Assets -- 2.1.1.Representation of a Multivariate Brownian Motion -- 2.1.2.Simulation of Correlated Geometric Brownian Motions -- 2.1.3.Volatility Vector -- 2.1.4.Joint Distribution of the Returns -- 2.2.Estimation of Parameters -- 2.2.1.Explicit Method -- 2.2.2.Numerical Method -- 2.3.Estimation Errors -- 2.3.1.Parametrization with the Correlation Matrix -- 2.3.2.Parametrization with the Volatility Vector --
Contents note continued: 2.3.3.Estimation of Parameters for Apple and Microsoft -- 2.4.Evaluation of Options on Several Assets -- 2.4.1.Partial Differential Equation for Option Values -- 2.4.2.Option Value as an Expectation -- 2.4.2.1.Vanilla Options -- 2.4.3.Exchange Option -- 2.4.4.Quanto Options -- 2.5.Greeks -- 2.5.1.Error on the Option Value -- 2.5.2.Extension of the Broadie-Glasserman Methodologies for Options on Several Assets -- 2.6.Suggested Reading -- 2.7.Exercises -- 2.8.Assignment Questions -- 2.A.Auxiliary Result -- 2.A.1.Evaluation of E {eaZ N(b + cZ)} -- 2.B.Proofs of the Results -- 2.B.1.Proof of Proposition 2.1.1 -- 2.B.2.Proof of Proposition 2.2.1 -- 2.B.3.Proof of Proposition 2.3.1 -- 2.B.4.Proof of Proposition 2.3.2 -- 2.B.5.Proof of Proposition 2.4.1 -- 2.B.6.Proof of Proposition 2.4.2 -- 2.B.7.Proof of Proposition 2.5.1 -- 2.B.8.Proof of Proposition 2.5.3 -- Bibliography -- 3.Discussion of the Black-Scholes Model -- Summary -- 3.1.Critiques of the Model --
Contents note continued: 3.1.1.Independence -- 3.1.2.Distribution of Returns and Goodness-of-Fit Tests of Normality -- 3.1.3.Volatility Smile -- 3.1.4.Transaction Costs -- 3.2.Some Extensions of the Black-Scholes Model -- 3.2.1.Time-Dependent Coefficients -- 3.2.1.1.Extended Black-Scholes Formula -- 3.2.2.Diffusion Processes -- 3.3.Discrete Time Hedging -- 3.3.1.Discrete Delta Hedging -- 3.4.Optimal Quadratic Mean Hedging -- 3.4.1.Offline Computations -- 3.4.2.Optimal Solution of the Hedging Problem -- 3.4.3.Relationship with Martingales -- 3.4.3.1.Market Price vs Theoretical Price -- 3.4.4.Markovian Models -- 3.4.5.Application to Geometric Random Walks -- 3.4.5.1.Illustrations -- 3.4.6.Incomplete Markovian Models -- 3.4.7.Limiting Behavior -- 3.5.Suggested Reading -- 3.6.Exercises -- 3.7.Assignment Questions -- 3.A.Tests of Serial Independence -- 3.B.Goodness-of-Fit Tests -- 3.B.1.Cramer-von Mises Test -- 3.B.1.1.Algorithms for Approximating the P-Value --
Contents note continued: 3.B.2.Lilliefors Test -- 3.C.Density Estimation -- 3.C.1.Examples of Kernels -- 3.D.Limiting Behavior of the Delta Hedging Strategy -- 3.E.Optimal Hedging for the Binomial Tree -- 3.F.A Useful Result -- Bibliography -- 4.Measures of Risk and Performance -- Summary -- 4.1.Measures of Risk -- 4.1.1.Portfolio Model -- 4.1.2.VaR -- 4.1.3.Expected Shortfall -- 4.1.4.Coherent Measures of Risk -- 4.1.4.1.Comments -- 4.1.5.Coherent Measures of Risk with Respect to a Stochastic Order -- 4.1.5.1.Simple Order -- 4.1.5.2.Hazard Rate Order -- 4.2.Estimation of Measures of Risk by Monte Carlo Methods -- 4.2.1.Methodology -- 4.2.2.Nonparametric Estimation of the Distribution Function -- 4.2.2.1.Precision of the Estimation of the Distribution Function -- 4.2.3.Nonparametric Estimation of the VaR -- 4.2.3.1.Uniform Estimation of Quantiles -- 4.2.4.Estimation of Expected Shortfall -- 4.2.5.Advantages and Disadvantages of the Monte Carlo Methodology --
Contents note continued: 4.3.Measures of Risk and the Delta-Gamma Approximation -- 4.3.1.Delta-Gamma Approximation -- 4.3.2.Delta-Gamma-Normal Approximation -- 4.3.3.Moment Generating Function and Characteristic Function of Q -- 4.3.4.Partial Monte Carlo Method -- 4.3.4.1.Advantages and Disadvantages of the Methodology -- 4.3.5.Edgeworth and Cornish-Fisher Expansions -- 4.3.5.1.Edgeworth Expansion for the Distribution Function -- 4.3.5.2.Advantages and Disadvantages of the Edge-worth Expansion -- 4.3.5.3.Cornish-Fisher Expansion -- 4.3.5.4.Advantages and Disadvantages of the Cornish-Fisher Expansion -- 4.3.6.Saddlepoint Approximation -- 4.3.6.1.Approximation of the Density -- 4.3.6.2.Approximation of the Distribution Function -- 4.3.6.3.Advantages and Disadvantages of the Methodology -- 4.3.7.Inversion of the Characteristic Function -- 4.3.7.1.Davies Approximation -- 4.3.7.2.Implementation -- 4.4.Performance Measures -- 4.4.1.Axiomatic Approach of Cherny-Madan --
Contents note continued: 4.4.2.The Sharpe Ratio -- 4.4.3.The Sortino Ratio -- 4.4.4.The Omega Ratio -- 4.4.4.1.Relationship with Expectiles -- 4.4.4.2.Gaussian Case and the Sharpe Ratio -- 4.4.4.3.Relationship with Stochastic Dominance -- 4.4.4.4.Estimation of Omega and G -- 4.5.Suggested Reading -- 4.6.Exercises -- 4.7.Assignment Questions -- 4.A.Brownian Bridge -- 4.B.Quantiles -- 4.C.Mean Excess Function -- 4.C.1.Estimation of the Mean Excess Function -- 4.D.Bootstrap Methodology -- 4.D.1.Bootstrap Algorithm -- 4.E.Simulation of QF,a,b -- 4.F.Saddlepoint Approximation of a Continuous Distribution Function -- 4.G.Complex Numbers in MATLAB -- 4.H.Gil-Pelaez Formula -- 4.1.Proofs of the Results -- 4.I.1.Proof of Proposition 4.1.1 -- 4.I.2.Proof of Proposition 4.1.3 -- 4.I.3.Proof of Proposition 4.2.1 -- 4.I.4.Proof of Proposition 4.2.2 -- 4.I.5.Proof of Proposition 4.3.1 -- 4.I.6.Proof of Proposition 4.4.1 -- 4.I.7.Proof of Proposition 4.4.2 -- 4.I.8.Proof of Proposition 4.4.4 --
Contents note continued: Bibliography -- 5.Modeling Interest Rates -- Summary -- 5.1.Introduction -- 5.1.1.Vasicek Result -- 5.2.Vasicek Model -- 5.2.1.Ornstein-Uhlenbeck Processes -- 5.2.2.Change of Measurement and Time Scales -- 5.2.3.Properties of Ornstein-Uhlenbeck Processes -- 5.2.3.1.Moments of the Ornstein-Uhlenbeck Process -- 5.2.3.2.Stationary Distribution of the Ornstein-Uhlenbeck Process -- 5.2.4.Value of Zero-Coupon Bonds under a Vasicek Model -- 5.2.4.1.Vasicek Formula for the Value of a Bond -- 5.2.4.2.Annualized Bond Yields -- 5.2.5.Estimation of the Parameters of the Vasicek Model Using Zero-Coupon Bonds -- 5.2.5.1.Measurement and Time Scales -- 5.2.5.2.Duan Approach for the Estimation of Non Observable Data -- 5.2.5.3.Joint Conditional Density of the Implied Rates -- 5.2.5.4.Change of Variables Formula -- 5.2.5.5.Application of the Change of Variable Formula to the Vasicek Model -- 5.2.5.6.Precision of the Estimation -- 5.3.Cox-Ingersoll-Ross (CIR) Model --
Contents note continued: 5.3.1.Representation of the Feller Process -- 5.3.1.1.Properties of the Feller Process -- 5.3.1.2.Measurement and Time Scales -- 5.3.2.Value of Zero-Coupon Bonds under a CIR Model -- 5.3.2.1.Formula for the Value of a Zero-Coupon Bond under the CIR Model -- 5.3.2.2.Annualized Bond Yields -- 5.3.2.3.Value of a Call Option on a Zero-Coupon Bond -- 5.3.2.4.Put-Call Parity -- 5.3.3.Parameters Estimation of the CIR Model Using Zero-Coupon Bonds -- 5.3.3.1.Measurement and Time Scales -- 5.3.3.2.Joint Conditional Density of the Implied Rates -- 5.3.3.3.Application of the Change of Variable Formula for the CIR Model -- 5.3.3.4.Precision of the Estimation -- 5.4.Other Models for the Spot Rates -- 5.4.1.Affine Models -- 5.5.Suggested Reading -- 5.6.Exercises -- 5.7.Assignment Questions -- 5.A.Interpretation of the Stochastic Integral -- 5.B.Integral of a Gaussian Process -- 5.C.Estimation Error for a Ornstein-Uhlenbeck Process -- 5.D.Proofs of the Results --
Contents note continued: 5.D.1.Proof of Proposition 5.2.1 -- 5.D.2.Proof of Proposition 5.2.2 -- 5.D.3.Proof of Proposition 5.3.1 -- 5.D.4.Proof of Proposition 5.3.2 -- 5.D.5.Proof of Proposition 5.3.3 -- Bibliography -- 6.Levy Models -- Summary -- 6.1.Complete Models -- 6.2.Stochastic Processes with Jumps -- 6.2.1.Simulation of a Poisson Process over a Fixed Time Interval -- 6.2.2.Jump-Diffusion Models -- 6.2.3.Merton Model -- 6.2.4.Kou Jump-Diffusion Model -- 6.2.5.Weighted-Symmetric Models for the Jumps -- 6.3.Levy Processes -- 6.3.1.Random Walk Representation -- 6.3.2.Characteristics -- 6.3.3.Infinitely Divisible Distributions -- 6.3.4.Sample Path Properties -- 6.3.4.1.Number of Jumps of a Levy Process -- 6.3.4.2.Finite Variation -- 6.4.Examples of Levy Processes -- 6.4.1.Gamma Process -- 6.4.2.Inverse Gaussian Process -- 6.4.2.1.Simulation of Tα,β -- 6.4.3.Generalized Inverse Gaussian Distribution -- 6.4.4.Variance Gamma Process -- 6.4.5.Levy Subordinators --
Contents note continued: 6.5.Change of Distribution -- 6.5.1.Esscher Transforms -- 6.5.2.Examples of Application -- 6.5.2.1.Merton Model -- 6.5.2.2.Kou Model -- 6.5.2.3.Variance Gamma Process -- 6.5.2.4.Normal Inverse Gaussian Process -- 6.5.3.Application to Option Pricing -- 6.5.4.General Change of Measure -- 6.5.5.Incompleteness -- 6.6.Model Implementation and Estimation of Parameters -- 6.6.1.Distributional Properties -- 6.6.1.1.Serial Independence -- 6.6.1.2.Levy Process vs Brownian Motion -- 6.6.2.Estimation Based on the Cumulants -- 6.6.2.1.Estimation of the Cumulants -- 6.6.2.2.Application -- 6.6.2.3.Discussion -- 6.6.3.Estimation Based on the Maximum Likelihood Method -- 6.7.Suggested Reading -- 6.8.Exercises -- 6.9.Assignment Questions -- 6.A.Modified Bessel Functions of the Second Kind -- 6.B.Asymptotic Behavior of the Cumulants -- 6.C.Proofs of the Results -- 6.C.1.Proof of Lemma 6.5.1 -- 6.C.2.Proof of Corollary 6.5.2 -- 6.C.3.Proof of Proposition 6.6.1 --
Contents note continued: 6.C.4.Proof of Proposition 6.4.1 -- Bibliography -- 7.Stochastic Volatility Models -- Summary -- 7.1.Garch Models -- 7.1.1.Garch(1,1) -- 7.1.2.Garch(p,q) -- 7.1.3.Egarch -- 7.1.4.Ngarch -- 7.1.5.Gjr-Garch -- 7.1.6.Augmented Garch -- 7.2.Estimation of Parameters -- 7.2.1.Application for GARCH(p,q) Models -- 7.2.2.Tests -- 7.2.3.Goodness-of-Fit and Pseudo-Observations -- 7.2.4.Estimation and Goodness-of-Fit When the Innovations Are Not Gaussian -- 7.3.Duan Methodology of Option Pricing -- 7.3.1.Lrnvr Criterion -- 7.3.2.Continuous Time Limit -- 7.3.2.1.A New Parametrization -- 7.4.Stochastic Volatility Model of Hull-White -- 7.4.1.Market Price of Volatility Risk -- 7.4.2.Expectations vs Partial Differential Equations -- 7.4.3.Option Price as an Expectation -- 7.4.4.Approximation of Expectations -- 7.4.4.1.Monte Carlo Methods -- 7.4.4.2.Taylor Series Expansion -- 7.4.4.3.Edgeworth and Gram-Charlier Expansions -- 7.4.4.4.Approximate Distribution --
Contents note continued: 7.5.Stochastic Volatility Model of Heston -- 7.6.Suggested Reading -- 7.7.Exercises -- 7.8.Assignment Questions -- 7.A.Khmaladze Transform -- 7.A.1.Implementation Issues -- 7.B.Proofs of the Results -- 7.B.1.Proof of Proposition 7.1.1 -- 7.B.2.Proof of Proposition 7.4.1 -- 7.B.3.Proof of Proposition 7.4.2 -- Bibliography -- 8.Copulas and Applications -- Summary -- 8.1.Weak Replication of Hedge Funds -- 8.1.1.Computation of g -- 8.2.Default Risk -- 8.2.1.n-th to Default Swap -- 8.2.2.Simple Model for Default Time -- 8.2.3.Joint Dynamics of Xi and Yi -- 8.2.4.Simultaneous Evolution of Several Markov Chains -- 8.2.4.1.Credit Metrics -- 8.2.5.Continuous Time Model -- 8.2.5.1.Modeling the Default Time of a Firm -- 8.2.6.Modeling Dependence Between Several Default Times -- 8.3.Modeling Dependence -- 8.3.1.An Image is Worth a Thousand Words -- 8.3.2.Joint Distribution, Margins and Copulas -- 8.3.3.Visualizing Dependence -- 8.4.Bivariate Copulas --
Contents note continued: 8.4.1.Examples of Copulas -- 8.4.2.Sklar Theorem in the Bivariate Case -- 8.4.3.Applications for Simulation -- 8.4.4.Simulation of (U1, U2) ~ C -- 8.4.5.Modeling Dependence with Copulas -- 8.4.6.Positive Quadrant Dependence (PQD) Order -- 8.5.Measures of Dependence -- 8.5.1.Estimation of a Bivariate Copula -- 8.5.1.1.Precision of the Estimation of the Empirical Copula -- 8.5.1.2.Tests of Independence Based on the Empirical Copula -- 8.5.2.Kendall Function -- 8.5.2.1.Estimation of Kendall Function -- 8.5.2.2.Precision of the Estimation of the Kendall Function -- 8.5.2.3.Tests of Independence Based on the Empirical Kendall Function -- 8.5.3.Kendall Tau -- 8.5.3.1.Estimation of Kendall Tau -- 8.5.3.2.Precision of the Estimation of Kendall Tau -- 8.5.4.Spearman Rho -- 8.5.4.1.Estimation of Spearman Rho -- 8.5.4.2.Precision of the Estimation of Spearman Rho -- 8.5.5.van der Waerden Rho -- 8.5.5.1.Estimation of van der Waerden Rho --
Contents note continued: 8.5.5.2.Precision of the Estimation of van der Waerden Rho -- 8.5.6.Other Measures of Dependence -- 8.5.6.1.Estimation of ρ(J) -- 8.5.6.2.Precision of the Estimation of ρ(J) -- 8.5.7.Serial Dependence -- 8.6.Multivariate Copulas -- 8.6.1.Kendall Function -- 8.6.2.Conditional Distributions -- 8.6.2.1.Applications of Theorem 8.6.2 -- 8.6.3.Stochastic Orders for Dependence -- 8.6.3.1.Frechet-Hoeffding Bounds -- 8.6.3.2.Application -- 8.6.3.3.Supermodular Order -- 8.7.Families of Copulas -- 8.7.1.Independence Copula -- 8.7.2.Elliptical Copulas -- 8.7.2.1.Estimation of ρ -- 8.7.3.Gaussian Copula -- 8.7.3.1.Simulation of Observations from a Gaussian Copula -- 8.7.4.Student Copula -- 8.7.4.1.Simulation of Observations from a Student Copula -- 8.7.5.Other Elliptical Copulas -- 8.7.6.Archimedean Copulas -- 8.7.6.1.Financial Modeling -- 8.7.6.2.Recursive Formulas -- 8.7.6.3.Conjecture -- 8.7.6.4.Kendall Tau for Archimedean Copulas --
Contents note continued: 8.7.6.5.Simulation of Observations from an Archimedean Copula -- 8.7.7.Clayton Family -- 8.7.7.1.Simulation of Observations from a Clayton Copula -- 8.7.8.Gumbel Family -- 8.7.8.1.Simulation of Observations from a Gumbel Copula -- 8.7.9.Frank Family -- 8.7.9.1.Simulation of Observations from a Frank Copula -- 8.7.10.Ali-Mikhail-Haq Family -- 8.7.10.1.Simulation of Observations from an Ali-Mikhail-Haq Copula -- 8.7.11.PQD Order for Archimedean Copula Families -- 8.7.12.Farlie-Gumbel-Morgenstern Family -- 8.7.13.Plackett Family -- 8.7.14.Other Copula Families -- 8.8.Estimation of the Parameters of Copula Models -- 8.8.1.Considering Serial Dependence -- 8.8.2.Estimation of Parameters: The Parametric Approach -- 8.8.2.1.Advantages and Disadvantages -- 8.8.3.Estimation of Parameters: The Semiparametric Approach -- 8.8.3.1.Advantages and Disadvantages -- 8.8.4.Estimation of ρ for the Gaussian Copula --
Contents note continued: 8.8.5.Estimation of ρ and ν for the Student Copula -- 8.8.6.Estimation for an Archimedean Copula Family -- 8.8.7.Nonparametric Estimation of a Copula -- 8.8.8.Nonparametric Estimation of Kendall Function -- 8.9.Tests of Independence -- 8.9.1.Test of Independence Based on the Copula -- 8.10.Tests of Goodness-of-Fit -- 8.10.1.Computation of P-Values -- 8.10.2.Using the Rosenblatt Transform for Goodness-of-Fit Tests -- 8.10.2.1.Computation of P-Values -- 8.11.Example of Implementation of a Copula Model -- 8.11.1.Change Point Tests -- 8.11.2.Serial Independence -- 8.11.3.Modeling Serial Dependence -- 8.11.3.1.Change Point Tests for the Residuals -- 8.11.3.2.Goodness-of-Fit for the Distribution of Innovations -- 8.11.4.Modeling Dependence Between Innovations -- 8.11.4.1.Test of Independence for the Innovations -- 8.11.4.2.Goodness-of-Fit for the Copula of the Innovations -- 8.12.Suggested Reading -- 8.13.Exercises -- 8.14.Assignment Questions --
Contents note continued: 8.A.Continuous Time Markov Chains -- 8.B.Tests of Independence -- 8.C.Polynomials Related to the Gumbel Copula -- 8.D.Polynomials Related to the Frank Copula -- 8.E.Change Point Tests -- 8.E.1.Change Point Test for the Copula -- 8.F.Auxiliary Results -- 8.G.Proofs of the Results -- 8.G.1.Proof of Proposition 8.4.1 -- 8.G.2.Proof of Proposition 8.4.2 -- 8.G.3.Proof of Proposition 8.5.1 -- 8.G.4.Proof of Theorem 8.7.1 -- Bibliography -- 9.Filtering -- Summary -- 9.1.Description of the Filtering Problem -- 9.2.Kalman Filter -- 9.2.1.Model -- 9.2.2.Filter Initialization -- 9.2.3.Estimation of Parameters -- 9.2.4.Implementation of the Kalman Filter -- 9.2.4.1.Solution -- 9.2.5.The Kalman Filter for General Linear Models -- 9.3.IMM Filter -- 9.3.1.IMM Algorithm -- 9.3.2.Implementation of the IMM Filter -- 9.4.General Filtering Problem -- 9.4.1.Kallianpur-Striebel Formula -- 9.4.2.Recursivity -- 9.4.3.Implementing the Recursive Zakai Equation --
Contents note continued: 9.4.4.Solving the Filtering Problem -- 9.5.Computation of the Conditional Densities -- 9.5.1.Convolution Method -- 9.5.2.Kolmogorov Equation -- 9.6.Particle Filters -- 9.6.1.Implementation of a Particle Filter -- 9.6.2.Implementation of an Auxiliary Sampling/Importance Resampling (ASIR) Particle Filter -- 9.6.2.1.ASIR0 -- 9.6.2.2.ASIR1 -- 9.6.2.3.ASIR2 -- 9.6.3.Estimation of Parameters -- 9.6.3.1.Smoothed Likelihood -- 9.7.Suggested Reading -- 9.8.Exercises -- 9.9.Assignment Questions -- 9.A.Schwartz Model -- 9.B.Auxiliary Results -- 9.C.Fourier Transform -- 9.D.Proofs of the Results -- 9.D.1.Proof of Proposition 9.2.1 -- Bibliography -- 10.Applications of Filtering -- Summary -- 10.1.Estimation of ARMA Models -- 10.1.1.AR(p) Processes -- 10.1.1.1.MA(q) Processes -- 10.1.2.MA Representation -- 10.1.3.ARMA Processes and Filtering -- 10.1.3.1.Implementation of the Kalman Filter in the Gaussian Case -- 10.1.4.Estimation of Parameters of ARMA Models --
Contents note continued: 10.2.Regime-Switching Markov Models -- 10.2.1.Serial Dependence -- 10.2.2.Prediction of the Regimes -- 10.2.3.Conditional Densities and Predictions -- 10.2.4.Estimation of the Parameters -- 10.2.4.1.Implementation of the E-step -- 10.2.5.M-step in the Gaussian Case -- 10.2.6.Tests of Goodness-of-Fit -- 10.2.7.Continuous Time Regime-Switching Markov Processes -- 10.3.Replication of Hedge Funds -- 10.3.0.1.Measurement of Errors -- 10.3.1.Replication by Regression -- 10.3.2.Replication by Kalman Filter -- 10.3.3.Example of Application -- 10.4.Suggested Reading -- 10.5.Exercises -- 10.6.Assignment Questions -- 10.A.EM Algorithm -- 10.B.Sampling Moments vs Theoretical Moments -- 10.C.Rosenblatt Transform for the Regime-Switching Model -- 10.D.Proofs of the Results -- 10.D.1.Proof of Proposition 10.1.1 -- 10.D.2.Proof of Proposition 10.1.2 -- Bibliography -- A.Probability Distributions -- Summary -- A.1.Introduction --
Contents note continued: A.2.Discrete Distributions and Densities -- A.2.1.Expected Value and Moments of Discrete Distributions -- A.3.Absolutely Continuous Distributions and Densities -- A.3.1.Expected Value and Moments of Absolutely Continuous Distributions -- A.4.Characteristic Functions -- A.4.1.Inversion Formula -- A.5.Moments Generating Functions and Laplace Transform -- A.5.1.Cumulants -- A.5.1.1.Extension -- A.6.Families of Distributions -- A.6.1.Bernoulli Distribution -- A.6.2.Binomial Distribution -- A.6.3.Poisson Distribution -- A.6.4.Geometric Distribution -- A.6.5.Negative Binomial Distribution -- A.6.6.Uniform Distribution -- A.6.7.Gaussian Distribution -- A.6.8.Log-Normal Distribution -- A.6.9.Exponential Distribution -- A.6.10.Gamma Distribution -- A.6.10.1.Properties of the Gamma Function -- A.6.11.Chi-Square Distribution -- A.6.12.Non-Central Chi-Square Distribution -- A.6.12.1.Simulation of Non-Central Chi-Square Variables -- A.6.13.Student Distribution --
Contents note continued: A.6.14.Johnson SU Type Distributions -- A.6.15.Beta Distribution -- A.6.16.Cauchy Distribution -- A.6.17.Generalized Error Distribution -- A.6.18.Multivariate Gaussian Distribution -- A.6.18.1.Representation of a Random Gaussian Vector -- A.6.19.Multivariate Student Distribution -- A.6.20.Elliptical Distributions -- A.6.21.Simulation of an Elliptic Distribution -- A.7.Conditional Densities and Joint Distributions -- A.7.1.Multiplication Formula -- A.7.2.Conditional Distribution in the Markovian Case -- A.7.3.Rosenblatt Transform -- A.8.Functions of Random Vectors -- A.9.Exercises -- Bibliography -- B.Estimation of Parameters -- Summary -- B.1.Maximum Likelihood Principle -- B.2.Precision of Estimators -- B.2.1.Confidence Intervals and Confidence Regions -- B.2.2.Nonparametric Prediction Interval -- B.3.Properties of Estimators -- B.3.1.Almost Sure Convergence -- B.3.2.Convergence in Probability -- B.3.3.Convergence in Mean Square --
Contents note continued: B.3.4.Convergence in Law -- B.3.4.1.Delta Method -- B.3.5.Bias and Consistency -- B.4.Central Limit Theorem for Independent Observations -- B.4.1.Consistency of the Empirical Mean -- B.4.2.Consistency of the Empirical Coefficients of Skewness and Kurtosis -- B.4.3.Confidence Intervals I -- B.4.4.Confidence Ellipsoids -- B.4.5.Confidence Intervals II -- B.5.Precision of Maximum Likelihood Estimator for Serially Independent Observations -- B.5.1.Estimation of Fisher Information Matrix -- B.6.Convergence in Probability and the Central Limit Theorem for Serially Dependent Observations -- B.7.Precision of Maximum Likelihood Estimator for Serially Dependent Observations -- B.8.Method of Moments -- B.9.Combining the Maximum Likelihood Method and the Method of Moments -- B.10.M-estimators -- B.11.Suggested Reading -- B.12.Exercises -- Bibliography.