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El. knyga: Statistical Methods for Financial Engineering

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(HEC Montreal, Quebec, Canada)
  • Formatas: 496 pages
  • Išleidimo metai: 19-Apr-2016
  • Leidėjas: CRC Press Inc
  • Kalba: eng
  • ISBN-13: 9781439856956
Kitos knygos pagal šią temą:
Statistical Methods for Financial EngineeringDidesnis paveikslėlis
  • Formatas: 496 pages
  • Išleidimo metai: 19-Apr-2016
  • Leidėjas: CRC Press Inc
  • Kalba: eng
  • ISBN-13: 9781439856956
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While many financial engineering books are available, the statistical aspects behind the implementation of stochastic models used in the field are often overlooked or restricted to a few well-known cases. Statistical Methods for Financial Engineering guides current and future practitioners on implementing the most useful stochastic models used in financial engineering.



After introducing properties of univariate and multivariate models for asset dynamics as well as estimation techniques, the book discusses limits of the Black-Scholes model, statistical tests to verify some of its assumptions, and the challenges of dynamic hedging in discrete time. It then covers the estimation of risk and performance measures, the foundations of spot interest rate modeling, Lévy processes and their financial applications, the properties and parameter estimation of GARCH models, and the importance of dependence models in hedge fund replication and other applications. It concludes with the topic of filtering and its financial applications.

This self-contained book offers a basic presentation of stochastic models and addresses issues related to their implementation in the financial industry. Each chapter introduces powerful and practical statistical tools necessary to implement the models. The author not only shows how to estimate parameters efficiently, but he also demonstrates, whenever possible, how to test the validity of the proposed models. Throughout the text, examples using MATLAB® illustrate the application of the techniques to solve real-world financial problems. MATLAB and R programs are available on the authors website.

Recenzijos

" an interesting book with many features that are not easily found elsewhere. libraries will certainly want to acquire a copy. there are plenty of points at which even experts will pick up new ideas." J. Michael Steele, Journal of the American Statistical Association, September 2014, Vol. 109

" a successful attempt to cover the main statistical tools and methods used for practical purposes in financial engineering. In contrast to those few existing books on the implementation of stochastic models in financial markets, this monograph covers a vast number of topics from mathematical finance can be used by practitioners as a reference book, but also it can serve as an excellent textbook for training quantitative analysts " Mathematical Reviews, September 2014

Preface xxi
List of Figures
xxv
List of Tables
xxix
1 Black-Scholes Model
1(32)
Summary
1(1)
1.1 The Black-Scholes Model
1(1)
1.2 Dynamic Model for an Asset
2(3)
1.2.1 Stock Exchange Data
2(1)
1.2.2 Continuous Time Models
2(2)
1.2.3 Joint Distribution of Returns
4(1)
1.2.4 Simulation of a Geometric Brownian Motion
4(1)
1.2.5 Joint Law of Prices
5(1)
1.3 Estimation of Parameters
5(1)
1.4 Estimation Errors
6(3)
1.4.1 Estimation of Parameters for Apple
7(2)
1.5 Black-Scholes Formula
9(5)
1.5.1 European Call Option
9(1)
1.5.1.1 Put-Call Parity
10(1)
1.5.1.2 Early Exercise of an American Call Option
10(1)
1.5.2 Partial Differential Equation for Option Values
11(1)
1.5.3 Option Value as an Expectation
11(1)
1.5.3.1 Equivalent Martingale Measures and Pricing of Options
12(1)
1.5.4 Dividends
13(1)
1.5.4.1 Continuously Paid Dividends
13(1)
1.6 Greeks
14(6)
1.6.1 Greeks for a European Call Option
15(1)
1.6.2 Implied Distribution
16(1)
1.6.3 Error on the Option Value
16(3)
1.6.4 Implied Volatility
19(1)
1.6.4.1 Problems with Implied Volatility
20(1)
1.7 Estimation of Greeks using the Broadie-Glasserman Methodologies
20(4)
1.7.1 Pathwise Method
21(2)
1.7.2 Likelihood Ratio Method
23(1)
1.7.3 Discussion
23(1)
1.8 Suggested Reading
24(1)
1.9 Exercises
24(3)
1.10 Assignment Questions
27(1)
1.A Justification of the Black-Scholes Equation
27(1)
1.B Martingales
28(1)
1.C Proof of the Results
29(4)
1.C.1 Proof of Proposition 1.3.1
29(1)
1.C.2 Proof of Proposition 1.4.1
30(1)
1.C.3 Proof of Proposition 1.6.1
30(1)
Bibliography
30(3)
2 Multivariate Black-Scholes Model
33(30)
Summary
33(1)
2.1 Black-Scholes Model for Several Assets
33(3)
2.1.1 Representation of a Multivariate Brownian Motion
34(1)
2.1.2 Simulation of Correlated Geometric Brownian Motions
34(1)
2.1.3 Volatility Vector
35(1)
2.1.4 Joint Distribution of the Returns
35(1)
2.2 Estimation of Parameters
36(1)
2.2.1 Explicit Method
36(1)
2.2.2 Numerical Method
37(1)
2.3 Estimation Errors
37(4)
2.3.1 Parametrization with the Correlation Matrix
38(1)
2.3.2 Parametrization with the Volatility Vector
38(2)
2.3.3 Estimation of Parameters for Apple and Microsoft
40(1)
2.4 Evaluation of Options on Several Assets
41(6)
2.4.1 Partial Differential Equation for Option Values
41(1)
2.4.2 Option Value as an Expectation
42(1)
2.4.2.1 Vanilla Options
43(1)
2.4.3 Exchange Option
43(1)
2.4.4 Quanto Options
44(3)
2.5 Greeks
47(3)
2.5.1 Error on the Option Value
47(1)
2.5.2 Extension of the Broadie-Glasserman Methodologies for Options on Several Assets
48(2)
2.6 Suggested Reading
50(1)
2.7 Exercises
51(2)
2.8 Assignment Questions
53(1)
2.A Auxiliary Result
54(1)
2.A.1 Evaluation of E {eaZ N(b + cZ)}
54(1)
2.B Proofs of the Results
54(9)
2.B.1 Proof of Proposition 2.1.1
54(1)
2.B.2 Proof of Proposition 2.2.1
55(1)
2.B.3 Proof of Proposition 2.3.1
56(1)
2.B.4 Proof of Proposition 2.3.2
56(1)
2.B.5 Proof of Proposition 2.4.1
57(2)
2.B.6 Proof of Proposition 2.4.2
59(1)
2.B.7 Proof of Proposition 2.5.1
59(1)
2.B.8 Proof of Proposition 2.5.3
59(2)
Bibliography
61(2)
3 Discussion of the Black-Scholes Model
63(40)
Summary
63(1)
3.1 Critiques of the Model
63(6)
3.1.1 Independence
63(3)
3.1.2 Distribution of Returns and Goodness-of-Fit Tests of Normality
66(2)
3.1.3 Volatility Smile
68(1)
3.1.4 Transaction Costs
68(1)
3.2 Some Extensions of the Black-Scholes Model
69(3)
3.2.1 Time-Dependent Coefficients
69(1)
3.2.1.1 Extended Black-Scholes Formula
70(1)
3.2.2 Diffusion Processes
70(2)
3.3 Discrete Time Hedging
72(2)
3.3.1 Discrete Delta Hedging
73(1)
3.4 Optimal Quadratic Mean Hedging
74(15)
3.4.1 Offline Computations
74(1)
3.4.2 Optimal Solution of the Hedging Problem
75(1)
3.4.3 Relationship with Martingales
76(1)
3.4.3.1 Market Price vs Theoretical Price
76(1)
3.4.4 Markovian Models
77(1)
3.4.5 Application to Geometric Random Walks
77(2)
3.4.5.1 Illustrations
79(4)
3.4.6 Incomplete Markovian Models
83(6)
3.4.7 Limiting Behavior
89(1)
3.5 Suggested Reading
89(1)
3.6 Exercises
90(2)
3.7 Assignment Questions
92(1)
3.A Tests of Serial Independence
93(1)
3.B Goodness-of-Fit Tests
94(2)
3.B.1 Cramer-von Mises Test
95(1)
3.B.1.1 Algorithms for Approximating the P-Value
95(1)
3.B.2 Lilliefors Test
96(1)
3.C Density Estimation
96(1)
3.C.1 Examples of Kernels
97(1)
3.D Limiting Behavior of the Delta Hedging Strategy
97(1)
3.E Optimal Hedging for the Binomial Tree
98(2)
3.F A Useful Result
100(3)
Bibliography
100(3)
4 Measures of Risk and Performance
103(44)
Summary
103(1)
4.1 Measures of Risk
103(5)
4.1.1 Portfolio Model
103(1)
4.1.2 VaR
104(1)
4.1.3 Expected Shortfall
104(1)
4.1.4 Coherent Measures of Risk
105(1)
4.1.4.1 Comments
106(1)
4.1.5 Coherent Measures of Risk with Respect to a Stochastic Order
107(1)
4.1.5.1 Simple Order
107(1)
4.1.5.2 Hazard Rate Order
107(1)
4.2 Estimation of Measures of Risk by Monte Carlo Methods
108(8)
4.2.1 Methodology
109(1)
4.2.2 Nonparametric Estimation of the Distribution Function
109(1)
4.2.2.1 Precision of the Estimation of the Distribution Function
109(2)
4.2.3 Nonparametric Estimation of the VaR
111(2)
4.2.3.1 Uniform Estimation of Quantiles
113(1)
4.2.4 Estimation of Expected Shortfall
114(2)
4.2.5 Advantages and Disadvantages of the Monte Carlo Methodology
116(1)
4.3 Measures of Risk and the Delta-Gamma Approximation
116(10)
4.3.1 Delta-Gamma Approximation
117(1)
4.3.2 Delta-Gamma-Normal Approximation
117(1)
4.3.3 Moment Generating Function and Characteristic Function of Q
118(1)
4.3.4 Partial Monte Carlo Method
119(1)
4.3.4.1 Advantages and Disadvantages of the Methodology
120(1)
4.3.5 Edgeworth and Cornish-Fisher Expansions
120(1)
4.3.5.1 Edgeworth Expansion for the Distribution Function
120(1)
4.3.5.2 Advantages and Disadvantages of the Edge-worth Expansion
121(1)
4.3.5.3 Cornish-Fisher Expansion
121(1)
4.3.5.4 Advantages and Disadvantages of the Cornish-Fisher Expansion
122(1)
4.3.6 Saddlepoint Approximation
122(1)
4.3.6.1 Approximation of the Density
123(1)
4.3.6.2 Approximation of the Distribution Function
124(1)
4.3.6.3 Advantages and Disadvantages of the Methodology
124(1)
4.3.7 Inversion of the Characteristic Function
125(1)
4.3.7.1 Davies Approximation
125(1)
4.3.7.2 Implementation
125(1)
4.4 Performance Measures
126(5)
4.4.1 Axiomatic Approach of Cherny-Madan
126(1)
4.4.2 The Sharpe Ratio
127(1)
4.4.3 The Sortino Ratio
127(1)
4.4.4 The Omega Ratio
128(1)
4.4.4.1 Relationship with Expectiles
128(1)
4.4.4.2 Gaussian Case and the Sharpe Ratio
129(1)
4.4.4.3 Relationship with Stochastic Dominance
130(1)
4.4.4.4 Estimation of Omega and G
130(1)
4.5 Suggested Reading
131(1)
4.6 Exercises
131(3)
4.7 Assignment Questions
134(1)
4.A Brownian Bridge
134(1)
4.B Quantiles
135(1)
4.C Mean Excess Function
135(1)
4.C.1 Estimation of the Mean Excess Function
136(1)
4.D Bootstrap Methodology
136(1)
4.D.1 Bootstrap Algorithm
136(1)
4.E Simulation of QF,a,b
137(1)
4.F Saddlepoint Approximation of a Continuous Distribution Function
137(1)
4.G Complex Numbers in MATLAB
138(1)
4.H Gil-Pelaez Formula
139(1)
4.1 Proofs of the Results
139(8)
4.I.1 Proof of Proposition 4.1.1
139(1)
4.I.2 Proof of Proposition 4.1.3
140(1)
4.I.3 Proof of Proposition 4.2.1
141(1)
4.I.4 Proof of Proposition 4.2.2
141(1)
4.I.5 Proof of Proposition 4.3.1
142(1)
4.I.6 Proof of Proposition 4.4.1
143(1)
4.I.7 Proof of Proposition 4.4.2
143(1)
4.I.8 Proof of Proposition 4.4.4
144(1)
Bibliography
144(3)
5 Modeling Interest Rates
147(36)
Summary
147(1)
5.1 Introduction
147(1)
5.1.1 Vasicek Result
147(1)
5.2 Vasicek Model
148(12)
5.2.1 Ornstein-Uhlenbeck Processes
149(1)
5.2.2 Change of Measurement and Time Scales
149(1)
5.2.3 Properties of Ornstein-Uhlenbeck Processes
150(1)
5.2.3.1 Moments of the Ornstein-Uhlenbeck Process
150(1)
5.2.3.2 Stationary Distribution of the Ornstein-Uhlenbeck Process
151(1)
5.2.4 Value of Zero-Coupon Bonds under a Vasicek Model
151(1)
5.2.4.1 Vasicek Formula for the Value of a Bond
152(1)
5.2.4.2 Annualized Bond Yields
152(1)
5.2.5 Estimation of the Parameters of the Vasicek Model Using Zero-Coupon Bonds
153(1)
5.2.5.1 Measurement and Time Scales
154(1)
5.2.5.2 Duan Approach for the Estimation of Non Observable Data
154(1)
5.2.5.3 Joint Conditional Density of the Implied Rates
155(1)
5.2.5.4 Change of Variables Formula
156(1)
5.2.5.5 Application of the Change of Variable Formula to the Vasicek Model
156(2)
5.2.5.6 Precision of the Estimation
158(2)
5.3 Cox-Ingersoll-Ross (CIR) Model
160(10)
5.3.1 Representation of the Feller Process
160(2)
5.3.1.1 Properties of the Feller Process
162(1)
5.3.1.2 Measurement and Time Scales
163(1)
5.3.2 Value of Zero-Coupon Bonds under a CIR Model
163(1)
5.3.2.1 Formula for the Value of a Zero-Coupon Bond under the CIR Model
164(1)
5.3.2.2 Annualized Bond Yields
165(1)
5.3.2.3 Value of a Call Option on a Zero-Coupon Bond
165(1)
5.3.2.4 Put-Call Parity
166(1)
5.3.3 Parameters Estimation of the CIR Model Using Zero-Coupon Bonds
166(1)
5.3.3.1 Measurement and Time Scales
167(1)
5.3.3.2 Joint Conditional Density of the Implied Rates
167(1)
5.3.3.3 Application of the Change of Variable Formula for the CIR Model
168(1)
5.3.3.4 Precision of the Estimation
169(1)
5.4 Other Models for the Spot Rates
170(1)
5.4.1 Affine Models
171(1)
5.5 Suggested Reading
171(1)
5.6 Exercises
172(3)
5.7 Assignment Questions
175(1)
5.A Interpretation of the Stochastic Integral
175(1)
5.B Integral of a Gaussian Process
176(1)
5.C Estimation Error for a Ornstein-Uhlenbeck Process
176(2)
5.D Proofs of the Results
178(5)
5.D.1 Proof of Proposition 5.2.1
178(1)
5.D.2 Proof of Proposition 5.2.2
178(1)
5.D.3 Proof of Proposition 5.3.1
179(1)
5.D.4 Proof of Proposition 5.3.2
180(1)
5.D.5 Proof of Proposition 5.3.3
180(1)
Bibliography
180(3)
6 Levy Models
183(40)
Summary
183(1)
6.1 Complete Models
183(1)
6.2 Stochastic Processes with Jumps
184(4)
6.2.1 Simulation of a Poisson Process over a Fixed Time Interval
185(1)
6.2.2 Jump-Diffusion Models
185(1)
6.2.3 Merton Model
186(1)
6.2.4 Kou Jump-Diffusion Model
187(1)
6.2.5 Weighted-Symmetric Models for the Jumps
187(1)
6.3 Levy Processes
188(4)
6.3.1 Random Walk Representation
188(1)
6.3.2 Characteristics
189(1)
6.3.3 Infinitely Divisible Distributions
190(1)
6.3.4 Sample Path Properties
190(1)
6.3.4.1 Number of Jumps of a Levy Process
191(1)
6.3.4.2 Finite Variation
191(1)
6.4 Examples of Levy Processes
192(5)
6.4.1 Gamma Process
192(1)
6.4.2 Inverse Gaussian Process
193(1)
6.4.2.1 Simulation of Tα,β
193(1)
6.4.3 Generalized Inverse Gaussian Distribution
194(1)
6.4.4 Variance Gamma Process
194(1)
6.4.5 Levy Subordinators
195(2)
6.5 Change of Distribution
197(6)
6.5.1 Esscher Transforms
197(1)
6.5.2 Examples of Application
198(1)
6.5.2.1 Merton Model
198(1)
6.5.2.2 Kou Model
199(1)
6.5.2.3 Variance Gamma Process
199(1)
6.5.2.4 Normal Inverse Gaussian Process
199(1)
6.5.3 Application to Option Pricing
199(1)
6.5.4 General Change of Measure
200(1)
6.5.5 Incompleteness
201(2)
6.6 Model Implementation and Estimation of Parameters
203(12)
6.6.1 Distributional Properties
204(1)
6.6.1.1 Serial Independence
204(1)
6.6.1.2 Levy Process vs Brownian Motion
204(1)
6.6.2 Estimation Based on the Cumulants
205(1)
6.6.2.1 Estimation of the Cumulants
206(1)
6.6.2.2 Application
207(2)
6.6.2.3 Discussion
209(1)
6.6.3 Estimation Based on the Maximum Likelihood Method
209(6)
6.7 Suggested Reading
215(1)
6.8 Exercises
215(1)
6.9 Assignment Questions
216(1)
6.A Modified Bessel Functions of the Second Kind
217(1)
6.B Asymptotic Behavior of the Cumulants
218(1)
6.C Proofs of the Results
219(4)
6.C.1 Proof of Lemma 6.5.1
219(1)
6.C.2 Proof of Corollary 6.5.2
219(1)
6.C.3 Proof of Proposition 6.6.1
220(1)
6.C.4 Proof of Proposition 6.4.1
220(1)
Bibliography
221(2)
7 Stochastic Volatility Models
223(34)
Summary
223(1)
7.1 Garch Models
223(5)
7.1.1 Garch(1,1)
224(2)
7.1.2 Garch(p,q)
226(1)
7.1.3 Egarch
226(1)
7.1.4 Ngarch
227(1)
7.1.5 Gjr-Garch
227(1)
7.1.6 Augmented Garch
227(1)
7.2 Estimation of Parameters
228(7)
7.2.1 Application for GARCH(p,q) Models
229(1)
7.2.2 Tests
230(1)
7.2.3 Goodness-of-Fit and Pseudo-Observations
230(2)
7.2.4 Estimation and Goodness-of-Fit When the Innovations Are Not Gaussian
232(3)
7.3 Duan Methodology of Option Pricing
235(4)
7.3.1 Lrnvr Criterion
235(2)
7.3.2 Continuous Time Limit
237(1)
7.3.2.1 A New Parametrization
238(1)
7.4 Stochastic Volatility Model of Hull-White
239(7)
7.4.1 Market Price of Volatility Risk
239(1)
7.4.2 Expectations vs Partial Differential Equations
240(1)
7.4.3 Option Price as an Expectation
240(2)
7.4.4 Approximation of Expectations
242(1)
7.4.4.1 Monte Carlo Methods
242(1)
7.4.4.2 Taylor Series Expansion
242(1)
7.4.4.3 Edgeworth and Gram-Charlier Expansions
243(2)
7.4.4.4 Approximate Distribution
245(1)
7.5 Stochastic Volatility Model of Heston
246(1)
7.6 Suggested Reading
247(1)
7.7 Exercises
247(2)
7.8 Assignment Questions
249(1)
7.A Khmaladze Transform
250(1)
7.A.1 Implementation Issues
250(1)
7.B Proofs of the Results
251(6)
7.B.1 Proof of Proposition 7.1.1
251(2)
7.B.2 Proof of Proposition 7.4.1
253(1)
7.B.3 Proof of Proposition 7.4.2
254(1)
Bibliography
254(3)
8 Copulas and Applications
257(88)
Summary
257(1)
8.1 Weak Replication of Hedge Funds
257(2)
8.1.1 Computation of g
258(1)
8.2 Default Risk
259(7)
8.2.1 n-th to Default Swap
259(1)
8.2.2 Simple Model for Default Time
260(1)
8.2.3 Joint Dynamics of Xi and Yi
261(1)
8.2.4 Simultaneous Evolution of Several Markov Chains
262(1)
8.2.4.1 Credit Metrics
262(2)
8.2.5 Continuous Time Model
264(2)
8.2.5.1 Modeling the Default Time of a Firm
266(1)
8.2.6 Modeling Dependence Between Several Default Times
266(1)
8.3 Modeling Dependence
266(5)
8.3.1 An Image is Worth a Thousand Words
267(2)
8.3.2 Joint Distribution, Margins and Copulas
269(1)
8.3.3 Visualizing Dependence
269(2)
8.4 Bivariate Copulas
271(5)
8.4.1 Examples of Copulas
271(1)
8.4.2 Sklar Theorem in the Bivariate Case
272(2)
8.4.3 Applications for Simulation
274(1)
8.4.4 Simulation of (U1, U2) ~ C
274(1)
8.4.5 Modeling Dependence with Copulas
275(1)
8.4.6 Positive Quadrant Dependence (PQD) Order
276(1)
8.5 Measures of Dependence
276(17)
8.5.1 Estimation of a Bivariate Copula
278(1)
8.5.1.1 Precision of the Estimation of the Empirical Copula
278(1)
8.5.1.2 Tests of Independence Based on the Empirical Copula
278(2)
8.5.2 Kendall Function
280(1)
8.5.2.1 Estimation of Kendall Function
281(1)
8.5.2.2 Precision of the Estimation of the Kendall Function
282(1)
8.5.2.3 Tests of Independence Based on the Empirical Kendall Function
282(4)
8.5.3 Kendall Tau
286(1)
8.5.3.1 Estimation of Kendall Tau
286(1)
8.5.3.2 Precision of the Estimation of Kendall Tau
287(1)
8.5.4 Spearman Rho
287(1)
8.5.4.1 Estimation of Spearman Rho
288(1)
8.5.4.2 Precision of the Estimation of Spearman Rho
288(1)
8.5.5 van der Waerden Rho
289(1)
8.5.5.1 Estimation of van der Waerden Rho
290(1)
8.5.5.2 Precision of the Estimation of van der Waerden Rho
290(1)
8.5.6 Other Measures of Dependence
291(1)
8.5.6.1 Estimation of ρ(J)
291(1)
8.5.6.2 Precision of the Estimation of ρ(J)
292(1)
8.5.7 Serial Dependence
292(1)
8.6 Multivariate Copulas
293(4)
8.6.1 Kendall Function
294(1)
8.6.2 Conditional Distributions
294(1)
8.6.2.1 Applications of Theorem 8.6.2
294(1)
8.6.3 Stochastic Orders for Dependence
295(1)
8.6.3.1 Frechet-Hoeffding Bounds
295(1)
8.6.3.2 Application
296(1)
8.6.3.3 Supermodular Order
296(1)
8.7 Families of Copulas
297(14)
8.7.1 Independence Copula
297(1)
8.7.2 Elliptical Copulas
297(1)
8.7.2.1 Estimation of ρ
298(1)
8.7.3 Gaussian Copula
298(1)
8.7.3.1 Simulation of Observations from a Gaussian Copula
299(1)
8.7.4 Student Copula
299(1)
8.7.4.1 Simulation of Observations from a Student Copula
300(1)
8.7.5 Other Elliptical Copulas
300(1)
8.7.6 Archimedean Copulas
301(1)
8.7.6.1 Financial Modeling
301(1)
8.7.6.2 Recursive Formulas
301(2)
8.7.6.3 Conjecture
303(1)
8.7.6.4 Kendall Tau for Archimedean Copulas
303(1)
8.7.6.5 Simulation of Observations from an Archimedean Copula
304(1)
8.7.7 Clayton Family
304(1)
8.7.7.1 Simulation of Observations from a Clayton Copula
305(1)
8.7.8 Gumbel Family
305(1)
8.7.8.1 Simulation of Observations from a Gumbel Copula
306(1)
8.7.9 Frank Family
306(1)
8.7.9.1 Simulation of Observations from a Frank Copula
307(1)
8.7.10 Ali-Mikhail-Haq Family
308(1)
8.7.10.1 Simulation of Observations from an Ali-Mikhail-Haq Copula
308(1)
8.7.11 PQD Order for Archimedean Copula Families
309(1)
8.7.12 Farlie-Gumbel-Morgenstern Family
309(1)
8.7.13 Plackett Family
310(1)
8.7.14 Other Copula Families
310(1)
8.8 Estimation of the Parameters of Copula Models
311(4)
8.8.1 Considering Serial Dependence
311(1)
8.8.2 Estimation of Parameters: The Parametric Approach
312(1)
8.8.2.1 Advantages and Disadvantages
312(1)
8.8.3 Estimation of Parameters: The Semiparametric Approach
312(1)
8.8.3.1 Advantages and Disadvantages
313(1)
8.8.4 Estimation of ρ for the Gaussian Copula
313(1)
8.8.5 Estimation of ρ and ν for the Student Copula
313(1)
8.8.6 Estimation for an Archimedean Copula Family
314(1)
8.8.7 Nonparametric Estimation of a Copula
314(1)
8.8.8 Nonparametric Estimation of Kendall Function
315(1)
8.9 Tests of Independence
315(1)
8.9.1 Test of Independence Based on the Copula
316(1)
8.10 Tests of Goodness-of-Fit
316(3)
8.10.1 Computation of P-Values
317(1)
8.10.2 Using the Rosenblatt Transform for Goodness-of-Fit Tests
318(1)
8.10.2.1 Computation of P-Values
318(1)
8.11 Example of Implementation of a Copula Model
319(6)
8.11.1 Change Point Tests
320(1)
8.11.2 Serial Independence
320(1)
8.11.3 Modeling Serial Dependence
320(1)
8.11.3.1 Change Point Tests for the Residuals
320(1)
8.11.3.2 Goodness-of-Fit for the Distribution of Innovations
320(1)
8.11.4 Modeling Dependence Between Innovations
321(1)
8.11.4.1 Test of Independence for the Innovations
321(2)
8.11.4.2 Goodness-of-Fit for the Copula of the Innovations
323(2)
8.12 Suggested Reading
325(1)
8.13 Exercises
326(4)
8.14 Assignment Questions
330(1)
8.A Continuous Time Markov Chains
331(1)
8.B Tests of Independence
332(1)
8.C Polynomials Related to the Gumbel Copula
333(1)
8.D Polynomials Related to the Frank Copula
334(1)
8.E Change Point Tests
334(2)
8.E.1 Change Point Test for the Copula
335(1)
8.F Auxiliary Results
336(1)
8.G Proofs of the Results
336(9)
8.G.1 Proof of Proposition 8.4.1
336(1)
8.G.2 Proof of Proposition 8.4.2
337(1)
8.G.3 Proof of Proposition 8.5.1
338(1)
8.G.4 Proof of Theorem 8.7.1
338(1)
Bibliography
339(6)
9 Filtering
345(30)
Summary
345(1)
9.1 Description of the Filtering Problem
345(1)
9.2 Kalman Filter
346(8)
9.2.1 Model
346(1)
9.2.2 Filter Initialization
347(1)
9.2.3 Estimation of Parameters
348(1)
9.2.4 Implementation of the Kalman Filter
348(1)
9.2.4.1 Solution
348(5)
9.2.5 The Kalman Filter for General Linear Models
353(1)
9.3 IMM Filter
354(2)
9.3.1 IMM Algorithm
354(2)
9.3.2 Implementation of the IMM Filter
356(1)
9.4 General Filtering Problem
356(2)
9.4.1 Kallianpur-Striebel Formula
356(1)
9.4.2 Recursivity
357(1)
9.4.3 Implementing the Recursive Zakai Equation
358(1)
9.4.4 Solving the Filtering Problem
358(1)
9.5 Computation of the Conditional Densities
358(2)
9.5.1 Convolution Method
359(1)
9.5.2 Kolmogorov Equation
360(1)
9.6 Particle Filters
360(6)
9.6.1 Implementation of a Particle Filter
360(1)
9.6.2 Implementation of an Auxiliary Sampling/Importance Resampling (ASIR) Particle Filter
361(2)
9.6.2.1 ASIR0
363(1)
9.6.2.2 ASIR1
363(1)
9.6.2.3 ASIR2
364(1)
9.6.3 Estimation of Parameters
365(1)
9.6.3.1 Smoothed Likelihood
365(1)
9.7 Suggested Reading
366(1)
9.8 Exercises
367(1)
9.9 Assignment Questions
368(1)
9.A Schwartz Model
369(1)
9.B Auxiliary Results
370(1)
9.C Fourier Transform
371(1)
9.D Proofs of the Results
371(4)
9.D.1 Proof of Proposition 9.2.1
371(1)
Bibliography
372(3)
10 Applications of Filtering
375(32)
Summary
375(1)
10.1 Estimation of ARMA Models
375(5)
10.1.1 AR(p) Processes
375(1)
10.1.1.1 MA(q) Processes
376(1)
10.1.2 MA Representation
376(1)
10.1.3 ARMA Processes and Filtering
377(1)
10.1.3.1 Implementation of the Kalman Filter in the Gaussian Case
378(1)
10.1.4 Estimation of Parameters of ARMA Models
379(1)
10.2 Regime-Switching Markov Models
380(9)
10.2.1 Serial Dependence
380(1)
10.2.2 Prediction of the Regimes
381(1)
10.2.3 Conditional Densities and Predictions
382(1)
10.2.4 Estimation of the Parameters
383(1)
10.2.4.1 Implementation of the E-step
383(1)
10.2.5 M-step in the Gaussian Case
384(1)
10.2.6 Tests of Goodness-of-Fit
385(3)
10.2.7 Continuous Time Regime-Switching Markov Processes
388(1)
10.3 Replication of Hedge Funds
389(6)
10.3.0.1 Measurement of Errors
390(1)
10.3.1 Replication by Regression
391(1)
10.3.2 Replication by Kalman Filter
391(1)
10.3.3 Example of Application
391(4)
10.4 Suggested Reading
395(1)
10.5 Exercises
396(1)
10.6 Assignment Questions
397(1)
10.A EM Algorithm
398(3)
10.B Sampling Moments vs Theoretical Moments
401(1)
10.C Rosenblatt Transform for the Regime-Switching Model
401(2)
10.D Proofs of the Results
403(4)
10.D.1 Proof of Proposition 10.1.1
403(1)
10.D.2 Proof of Proposition 10.1.2
404(1)
Bibliography
404(3)
A Probability Distributions
407(28)
Summary
407(1)
A.1 Introduction
407(1)
A.2 Discrete Distributions and Densities
408(2)
A.2.1 Expected Value and Moments of Discrete Distributions
408(2)
A.3 Absolutely Continuous Distributions and Densities
410(2)
A.3.1 Expected Value and Moments of Absolutely Continuous Distributions
410(2)
A.4 Characteristic Functions
412(1)
A.4.1 Inversion Formula
413(1)
A.5 Moments Generating Functions and Laplace Transform
413(2)
A.5.1 Cumulants
414(1)
A.5.1.1 Extension
415(1)
A.6 Families of Distributions
415(14)
A.6.1 Bernoulli Distribution
415(1)
A.6.2 Binomial Distribution
416(1)
A.6.3 Poisson Distribution
416(1)
A.6.4 Geometric Distribution
417(1)
A.6.5 Negative Binomial Distribution
417(1)
A.6.6 Uniform Distribution
417(1)
A.6.7 Gaussian Distribution
418(1)
A.6.8 Log-Normal Distribution
418(1)
A.6.9 Exponential Distribution
419(1)
A.6.10 Gamma Distribution
420(1)
A.6.10.1 Properties of the Gamma Function
420(1)
A.6.11 Chi-Square Distribution
421(1)
A.6.12 Non-Central Chi-Square Distribution
421(1)
A.6.12.1 Simulation of Non-Central Chi-Square Variables
421(1)
A.6.13 Student Distribution
422(1)
A.6.14 Johnson SU Type Distributions
423(1)
A.6.15 Beta Distribution
423(1)
A.6.16 Cauchy Distribution
424(1)
A.6.17 Generalized Error Distribution
424(1)
A.6.18 Multivariate Gaussian Distribution
425(1)
A.6.18.1 Representation of a Random Gaussian Vector
425(1)
A.6.19 Multivariate Student Distribution
426(1)
A.6.20 Elliptical Distributions
426(3)
A.6.21 Simulation of an Elliptic Distribution
429(1)
A.7 Conditional Densities and Joint Distributions
429(1)
A.7.1 Multiplication Formula
429(1)
A.7.2 Conditional Distribution in the Markovian Case
430(1)
A.7.3 Rosenblatt Transform
430(1)
A.8 Functions of Random Vectors
430(3)
A.9 Exercises
433(2)
Bibliography
434(1)
B Estimation of Parameters
435(20)
Summary
435(1)
B.1 Maximum Likelihood Principle
435(2)
B.2 Precision of Estimators
437(1)
B.2.1 Confidence Intervals and Confidence Regions
437(1)
B.2.2 Nonparametric Prediction Interval
437(1)
B.3 Properties of Estimators
438(3)
B.3.1 Almost Sure Convergence
438(1)
B.3.2 Convergence in Probability
438(1)
B.3.3 Convergence in Mean Square
438(1)
B.3.4 Convergence in Law
439(1)
B.3.4.1 Delta Method
440(1)
B.3.5 Bias and Consistency
441(1)
B.4 Central Limit Theorem for Independent Observations
441(5)
B.4.1 Consistency of the Empirical Mean
442(1)
B.4.2 Consistency of the Empirical Coefficients of Skewness and Kurtosis
442(3)
B.4.3 Confidence Intervals I
445(1)
B.4.4 Confidence Ellipsoids
445(1)
B.4.5 Confidence Intervals II
445(1)
B.5 Precision of Maximum Likelihood Estimator for Serially Independent Observations
446(2)
B.5.1 Estimation of Fisher Information Matrix
446(2)
B.6 Convergence in Probability and the Central Limit Theorem for Serially Dependent Observations
448(1)
B.7 Precision of Maximum Likelihood Estimator for Serially Dependent Observations
448(2)
B.8 Method of Moments
450(2)
B.9 Combining the Maximum Likelihood Method and the Method of Moments
452(1)
B.10 M-estimators
453(1)
B.11 Suggested Reading
454(1)
B.12 Exercises
454(1)
Bibliography
454(1)
Index 455
Bruno Remillard
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