A Benchmark Approach to Quantitative Finance (Google eBook)

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Springer Science & Business Media, Oct 28, 2006 - Business & Economics - 716 pages
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The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk-neutral pricing theory. It permits a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. The existence of an equivalent risk-neutral pricing measure is not required. Instead, it leads to pricing formulae with respect to the real-world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models. The first part of the book describes the necessary tools from probability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part is devoted to financial modeling by the benchmark approach. Various quantitative methods for the real-world pricing and hedging of derivatives are explained. The general framework is used to provide an understanding of the nature of stochastic volatility. The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance. It aims to be a self-contained, accessible but mathematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quantitative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability.

  

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worst book I've ever seen, misleading, easily getting distracted with too many useless notations, this is not a student book for sure.

Contents

Preliminaries from Probability Theory
1
12 Continuous Random Variables and Distributions
11
13 Moments of Random Variables
22
14 Joint Distributions and Random Vectors
39
15 Copulas
50
16 Exercises for Chapter 1
53
Statistical Methods
55
22 Confidence Intervals
63
85 European Put Option
295
86 Hedge Simulation
298
87 Squared Bessel Processes
304
88 Exercises for Chapter 8
317
Various Approaches to Asset Pricing
319
92 Actuarial Pricing
329
93 Capital Asset Pricing Model
332
94 Risk Neutral Pricing
337

23 Estimation Methods
70
24 Maximum Likelihood Estimation
78
25 Normal Variance Mixture Models
81
26 Distribution of Index LogReturns Estimation of LogReturns
84
27 Convergence of Random Sequences
92
28 Exercises for Chapter 2
98
Modeling via Stochastic Processes
99
32 Certain Classes of Stochastic Processes
106
33 Discrete Time Markov Chains
110
34 Continuous Time Markov Chains
113
35 Poisson Processes
120
36 Levy Processes
126
37 Insurance Risk Modeling
128
38 Exercises for Chapter 3
131
Diffusion Processes
133
42 Examples for Continuous Markov Processes
136
43 Diffusion Processes
141
44 Kolmogorov Equations
145
45 Diffusions with Stationary Densities
154
46 MultiDimensional Diffusion Processes
159
47 Exercises for Chapter 4
161
Martingales and Stochastic Integrals
163
52 Quadratic Variation and Covariation
174
53 Gains from Trade as Stochastic Integral
187
54 Ito Integral for Wiener Processes
193
55 Stochastic Integrals for Semimartingales
197
56 Exercises for Chapter 5
203
The Ito Formula
204
62 Multivariate Ito Formula
209
63 Some Applications of the ItoFormula
213
64 Extensions of the Ito Formula
222
65 Levys Theorem
227
66 A Proof of the Ito Formula
230
67 Exercises for Chapter 6
234
Stochastic Differential Equations
237
72 Linear SDE with Additive Noise
241
73 Linear SDE with Multiplicative Noise
243
74 Vector Stochastic Differential Equations
246
75 Constructing Explicit Solutions of SDEs
248
76 Jump Diffusions
254
77 Existence and Uniqueness
261
78 Markovian Solutions of SDEs
272
79 Exercises for Chapter 7
275
Introduction to Option Pricing
276
82 Options under the BlackScholes Model
281
83 The BlackScholes Formula
288
84 Sensitivities for European Call Option
290
95 Girsanov Transformation and Bayes Rule
345
96 Change of Numeraire
350
97 FeynmanKac Formula
356
98 Exercises for Chapter 9
364
Continuous Financial Markets
367
102 Growth Optimal Portfolio
372
103 Supermartingale Property
375
104 Real World Pricing
378
105 GOP as Best Performing Portfolio
386
106 Diversified Portfolios in CFMs
389
107 Exercises for Chapter 10
402
Portfolio Optimization
403
111 Locally Optimal Portfolios
404
112 Market Portfolio and GOP
415
113 Expected Utility Maximization
419
114 Pricing Nonreplicable Payoffs
427
115 Hedging
430
116 Exercises for Chapter 11
437
Modeling Stochastic Volatility
438
122 Modified CEV Model
444
123 Local Volatility Models
461
124 Stochastic Volatility Models
472
125 Exercises for Chapter 12
481
Minimal Market Model
483
132 Stylized Minimal Market Model
488
133 Derivatives under the MMM
496
134 MMM with Random Scaling
502
135 Exercises for Chapter 13
511
Markets with Event Risk
512
142 Diversified Portfolios
523
143 MeanVariance Portfolio Optimization
532
144 Real World Pricing for Two Market Models
536
145 Exercises for Chapter 14
549
Numerical Methods
551
152 Scenario Simulation
558
153 Classical Monte Carlo Method
570
154 Monte Carlo Simulation for SDEs
578
155 Variance Reduction of Functionals of SDEs
587
156 Tree Methods
591
157 Finite Difference Methods
600
158 Exercises for Chapter 15
611
Solutions for Exercises
614
References
667
Author Index
682
Index
689
Copyright

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About the author (2006)

Professor Eckhard Platen is a joint appointment between the School of Finance and Economics and the Department of Mathematical Sciences to the 1997 created Chair in Quantitative Finance at the University of Technology Sydney. Prior to this appointment he was Founding Head of the Centre for Financial Mathematics at the Institute of Advanced Studies at the Australian National University in Canberra. He completed a PhD in Mathematics at the Technical University in Dresden in 1975 and obtained in 1985 his Dr. sc. from the Academy of Sciences in Berlin, where he headed at the Weierstrass Institute the Sector of Stochastics.
He is co-author of two successful books on Numerical Methods for Stochastic Differential Equations, published by Springer Verlag, and has authored more than 100 research papers in quantitative finance and mathematics.


Dr David Heath works as a Senior Research Fellow in Quantitative Finance at the University of Technology, Sydney. During the early 1990s he became interested in various aspects of quantitative finance. He completed his PhD in financial mathematics at the Australian National University at the Centre for Financial Mathematics in 1995. Since this time his main research interests have focussed on the application of advanced numerical methods for the pricing and hedging of index, equity, FX and interest rate derivatives. These numerical methods include PDE, Monte Carlo and Markov chain methods. He has developed a range of new quantitative methods that are specifically designed for the benchmark approach. Dr Heath has authored more than thirteen publications in financial mathematics.

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