Stochastic Volatility in Financial Markets: Crossing the Bridge to Continuous Time

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Springer Science & Business Media, May 31, 2000 - Business & Economics - 145 pages
Stochastic Volatility in Financial Markets presents advanced topics in financial econometrics and theoretical finance, and is divided into three main parts. The first part aims at documenting an empirical regularity of financial price changes: the occurrence of sudden and persistent changes of financial markets volatility. This phenomenon, technically termed `stochastic volatility', or `conditional heteroskedasticity', has been well known for at least 20 years; in this part, further, useful theoretical properties of conditionally heteroskedastic models are uncovered. The second part goes beyond the statistical aspects of stochastic volatility models: it constructs and uses new fully articulated, theoretically-sounded financial asset pricing models that allow for the presence of conditional heteroskedasticity. The third part shows how the inclusion of the statistical aspects of stochastic volatility in a rigorous economic scheme can be faced from an empirical standpoint.
 

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Contents

INTRODUCTION
1
12 Empirical models in discrete time
7
122 EXTENSIONS
8
RAMIFICATIONS
11
124 DIFFUSIONS AS ARCH APPROXIMATIONS
14
13 Theoretical issues
16
132 THE TERM STRUCTURE OF INTEREST RATES
20
14 Statistical inference
23
331 FEYNMANKAC REPRESENTATION OF THE SOLUTIONS
71
HEDGING COST PROCESSES AND DECOMPOSITION FORMULAE
74
34 On mean selffinancing strategies and the minimal martingale measure
75
relative entropy and the minimal martingale measure
80
MODELS OF THE TERM STRUCTURE WITH STOCHASTIC VOLATILITY
81
42 From the one factor model to the modeling of conditional heteroskedasticity
83
43 Searching for affinity
87
44 Early equilibriumbased models
90

15 Plan
29
CONTINUOUS TIME BEHAVIOR OF NON LINEAR ARCH MODELS
31
23 Interpretation of the moment conditions
36
24 Effectiveness of ARCH as diffusion approximations of theoretical models
37
25 Limiting behavior of the error process
40
26 Continuous time behavior of the volatility switching models
44
262 CONVERGENCE ISSUES
45
263 INVARIANT MEASURES
46
proofs on convergence issues
48
proofs on distributional issues
52
Appendix C
55
CONTINUOUS TIME STOCHASTIC VOLATILITY OPTION PRICING FOUNDATIONAL ISSUES
57
32 The reference model
59
322 INCOMPLETENESS ISSUES
63
323 MODELS COMPLETED BY NONPRODUCTIVE ASSETS
65
A BASIC EXAMPLE
66
33 Applications to stochastic volatility
70
45 A class of equilibrium models of the term structure with stochastic volatility
91
451 PRELIMINARY OPTIMALITY RESULTS
92
taking account of nonlinearities
96
FORMULATING SOLVING AND ESTIMATING MODELS OF THE TERM STRUCTURE USING ARCH MODELS AS DIFFUSION APPROXIMA...
99
52 Specification of the theoretical models
100
53 Econometric strategy
102
54 The pure numerical solution of the theoretical models
109
542 THE GENERAL CASE
112
543 LIMITING AND TRANSVERSALITY CONDITIONS
113
55 An illustrative example
115
a solution method based on the approach of iterated approximations
119
5A2 GENERALITIES
120
5A3 AN ITERATED APPROXIMATION RESULT
123
REFERENCES
129
Index
143
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