Financial market risk: measurement and analysis
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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London ; New York
Routledge
2003
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| Series: | Routledge international studies in money and banking
24 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXXI, 460 S. graph. Darst. |
| ISBN: | 041527866X |
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MARC
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| 490 | 1 | |a Routledge international studies in money and banking |v 24 | |
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| adam_text | Contents
List of figures xiii
List of tables xix
Preface xxj
Introduction xxvii
PARTI
Financial risk processes 1
1 Risk - asset class, horizon and time 3
1.1 Introduction 3
1.2 Uncertainty 7
1.3 Nonparametric and parametric distributions 17
1.4 Random processes and time series 31
1.5 Software 41
1.6 Exercises 41
2 Competing financial market hypotheses 47
2.1 Introduction 47
2.2 EMH: martingale theory 47
2.3 FMH: fractal theory 53
2.4 Importance of identifying the degree of market efficiency 65
2.5 Software 67
2.6 Exercises 67
3 Stable scaling distributions in finance 71
3.1 Introduction 71
3.2 Affine traces of speculative prices 72
3.3 Invariant properties: stationarity versus scaling 76
3.4 Invariances of (Pareto-Levy) scaling distributions 77
3.5 Zolotarev parametrization of stable distributions 80
3.6 Examples of closed form stable distributions 92
x Contents
3.7 Stable parameter estimation and diagnostics 94
3.8 Software 96
3.9 Exercises 97
4 Persistence of financial risk 102
4.1 Introduction 102
4.2 Serial dependence 102
4.3 Global dependence 105
4.4 (G)ARCHprocesses 111
4.5 Fractional Brownian Motion 117
4.6 Range/Scale analysis 120
4.7 Critical color categorization of randomness 122
4.8 Software 128
4.9 Exercises 128
PART II
Financial risk measurement 133
5 Frequency analysis of financial risk 135
5.7 Introduction 135
5.2 Visualization of long-term financial risks 135
5.3 Correlation and time convolution 136
5.4 Fourier analysis of stationary price innovations 141
5.5 Software 152
5.6 Exercises 152
6 Fourier time-frequency analysis of risk 155
6.1 Introduction 155
6.2 FT for aperiodic variables 156
6.3 Hurst exponent identification from risk spectrum 169
6.4 Heisenberg Uncertainty Principle 171
6.5 Windowed FT for transient price innovations 173
6.6 Software 186
6.7 Exercises 186
7 Wavelet time-scale analysis of risk 190
7.1 Introduction 190
7.2 Wavelet analysis of transient pricing 192
7.3 Mallat sMRA 209
7.4 Wavelet Parseval Risk Decomposition Theorem 223
Contents xi
7.5 Software 224
7.6 Exercises 224
8 Multiresolution analysis of local risk 230
8.1 Introduction 230
8.2 Measurement of local financial market risk 237
8.3 Homogeneous Hurst exponents of monofractal price series 250
8.4 Multiresolution analysis of multifractal price series 265
8.5 Software 281
8.6 Exercises 281
PART III
Term structure dynamics 287
9 Chaos - nonunique equilibria processes 289
9.1 Introduction 289
9.2 Logistic parabola regimes 292
9.3 General nonlinear dynamic systems 317
9.4 Detecting attracting points and aperiodic orbits 327
9.5 Summary of aperiodic cyclical steady-state equilibria 328
9.6 Software 330
9.7 Exercises 331
10 Measuring term structure dynamics 337
10.1 Introduction 337
10.2 Dynamic investment cashflow theory 340
10.3 Nonlinear relationships in finance 347
10.4 Liquidity and financial turbulence 358
10.5 Software 372
10.6 Exercises 373
11 Simulation of financial turbulence 380
11.1 Introduction 380
11.2 Theories of physical and financial turbulence 381
11.3 Measurement and simulation of turbulence 388
11.4 Simulation of financial cashflow turbulence 396
11.5 Multiresolution analysis of financial turbulence 398
11.6 Wavelet solutions of financial diffusion equations 402
11.7 Software 414
11.8 Exercises 414
xii Contents
PART IV
Financial risk management 423
12 Managing VaR and extreme values 425
12.1 Introduction 425
12.2 Global dependence of financial returns 425
12.3 VaR for stable distributions 428
12.4 VaR for parametric distributions 431
12.5 Extreme value theory 437
12.6 VaR and fractal pricing processes 440
12.7 Software 445
12.8 Exercises 445
Appendix A: original scaling in financial economics 449
Appendix B: S P500 daily closing prices for 1988 450
Index 453
Figures
1 Risk = Danger (Wei) + Opportunity (Ji) xxx
1.1 Historical average annual returns and return volatility 4
1.2 Levels and returns of empirical financial time series: AMEX
stock and oil indices and DEM-USD exchange rate 9
1.3 Simple and relatively inexpensive radiation monitor 11
1.4 The nth-order moments and cumulants for n = 1,2, 3,4
of the Laplace, Gaussian and Uniform p.d.fs 21
1.5 The nth-order moments and cumulants for n = 1,2,3,4
of Exponential, Rayleigh and K-distribution p.d.fs 22
1.6 Construction of the histogram of a time series by binning 24
1.7 Raw and transformed daily returns of the DAX. The histograms
on the right show the relative frequencies of the returns in the
same scale 25
1.8 Empirical histogram of minute-by-minute log-increments of the
JPY in June 1997 25
1.9 Volatility smile of foreign currrency options 26
1.10 Implied distribution and (log-) normal distribution of foreign
currency options 26
1.11 Skewed volatility smile of equities 27
1.12 Implied distribution and (log-) normal distribution 28
1.13 The empirical cumulative distributions for USD/DEM and USD
6 months cash interest rate, shown for different time horizons 30
1.14 Semi-annual cumulative distributions of THB-FX increments,
January-December 1997 36
1.15 Semi-annual cumulative distributions of DEM-FX increments,
January-December 1997 37
2.1 Annualized volatility of theoretical Random Walk model of
constant, normalized, asset return volatilities 55
2.2 Empirical annualized volatility of financial market returns 56
2.3 Mandelbrot s Julia set 58
2.4 Clear-air turbulence 59
2.5 Intraday return variance of the HSI 61
2.6 Intraday return variance of the HSIF 62
xiv List of figures
2.7 One-month Eurodollar yield and time-varying (a) turbulent
volatility and (b) daily data 63
2.8 Five-year CMT yield and time-varying (a) turbulent volatility
and (b) daily data 64
3.1 Stable probabilistic schemes 78
3.2 Stable density in the Zolotarev S(az, /?, y, 8; 0) =
S(az, 0.8, 1, 0; 0) parametrization 86
3.3 Stable density in the Zolotarev S(az, P, y, $; 1) =
S(az, 0.8, 1, 0; 1) parametrization 87
3.4 Comparison of the Af = 1 minute p.d.f. for high-frequency
S P500 price changes with the Gaussian p.d.f. and with a Levy
stable p.d.f. 90
3.5 Non-convergent moving variance of 253 daily rates of return
(in 100 percent) of the S P500 stock market index in 1998 90
3.6 Estimates of four parameters of the Zolotarev parametrization
of FX distributions 95
4.1 Successive increments of ARCH(l) simulations with the same
unconditional variance (a2 = 1) 114
4.2 Probability density function of the successive increments shown
in Figure 4.1. 115
4.3 Comparison of the scaling properties of the unconditional p.d.f.
of a GARCH( 1,1) stochastic process 117
4.4 Autocorrelograms of equally-weighted CRSP daily and
monthly stock return indices 119
4.5 Sample of power spectra of white, pink and brown noise 125
4.6 Relations between and constraints on d, H and az 127
5.1 Fourier series approximation of a square wave 144
5.2 Heat diffusion analysis 146
5.3 A sample signal constructed from sine functions representing
three pulsations 150
5.4 The FT of the sampled signal s(t) 151
5.5 Fourier series analysis of pure musical harmonics: dominant
frequencies of the clarinet, violin and bagpipe 151
6.1 Granger and Morgenstern s global risk spectrum of Standard
and Poor series, based on annual data, 1875-1952 166
6.2 Semi-log plot of the autocorrelation function y (r) for the
S P500 index, sampled at a 1-minute time scale 170
6.3 Spectral density of high-frequency data from the S P500 index 171
6.4 Gdbor s atom go,£ as a function of time for three frequencies:
(a) high |i, (b) middle £2 and (c) low £3 176
6.5 Heisenberg boxes of two windowed Fourier atoms gu^ and gv y 177
6.6 Time-frequency analysis by the Gabor Transform with a
adapted to the time coherence of frequencies a and 3 180
6.7 Signal, spectrograms and scalogram 181
List of figures xv
6.8 Spectrogram Ps (z, |) of time series with two superimposed
time-varying frequencies 182
6.9 Laughter data and their global histogram 183
6.10 Spectrogram of laughter data with three dominant harmonics 184
6.11 Changes in the daily level of the three-month Treasury yield 184
6.12 Comparison of the modulated spectrogram of empirical
DEM/USD increments with the flat spectogram of white noise 185
7.1 A sine wave and a Daubechies wavelet ^D20 193
7.2 Self-similarity of wavelets: translation (every fourth k) and
scaling of a wavelet f 194
7.3 Wavelet coefficients are correlation or resonance
coefficients. Here a wavelet is correlated with an irregular
signal. Different sections of the signal produce different
resonance coefficients 196
7.4 A scalogram: a plot of the magnitude of wavelet coefficients 197
7.5 A 3D scalogram: a plot of the magnitude of the wavelet
coefficients in three dimensions 197
7.6 Heisenberg boxes of two wavelets. Smaller scales decrease the
time dispersion, but increase the frequency support, which is
shifted towards higher frequencies 198
7.7 Time-frequency resolution and basis functions of the
Windowed FT and the Wavelet Transform 199
7.8 A scalogram with modulus | W(r, a)| using a Morlet wavelet
for 25 different scales 201
7.9 Normalized scalogram ^/r))Pw(r, a) 202
7.10 Time-scale tiling for a sinusoidal function with an isolated
singularity at to 203
7.11 Empirical 3D scalogram of Thai Baht increments in July 1997 204
7.12 Time signal observations on f(x) 211
7.13 Haar and triangle scaling functions and their respective MRA
equations 216
7.14 Haar and triangle wavelets and their respective MRA equations 220
8.1 USD/DEM exchange rate on a time scale of At = 20 minutes 232
8.2 Scaling law behavior of the USD/DEM exchange rate in the
period 1993-1994, for various subperiods 233
8.3 Trading transaction density based on daily and weekly averages
of tick-by-tick data 234
8.4 Daily and weekly averaged volatility per hour of the USD/DEM
exchange rate 235
8.5 The cone of influence of an abscissa singularity v 245
8.6 Singularity cones of influence of a Dirac pulse at t = t0 245
8.7 Wavelet decomposition of a time series with singularities 248
8.8 How to measure the degree of irregularity of local risk of a
series of price singularities x (r) 249
8.9 Complete wavelet tiling 251
xvi List of figures
8.10 Wavelet-based persistence analysis of heartbeat interarrival
times for a healthy patient with a Daubechies(5) wavelet 254
8.11 Wavelet-based persistence analysis of weekly Dow Jones
Industrial Index data with Daubechies(5) wavelet 256
8.12 Wavelet MRA by Morlet(6) wavelet of the various exchange
rate regimes of the Mexican Peso/USD and the various
Brazilian financial market crises in the 1990s 257
8.13 Wavelet MRA of the various exchange rate regimes of the first
(log) differences of the Mexican Peso/USD in the 1990s 260
8.14 Wavelet MRA, based on daily data, of Chilean stock index rate
of returns in the 1990s 261
8.15 The first four monthly moments of the distributions of the
minute-by-minute quotations of nine currency rates in
May-August 1997 (USD is the numeraire) 264
8.16 Development of Koch s snowflake with Hausdorff dimension
D= 1.2619 267
8.17 Schematic convex multifractal singularity spectrum D{ai),
with various Gibbs exponent regimes 271
8.18 Computation of singularity spectrum of the devil s staircase, its
partition function Z(q, a) scaling exponents r(q) and its
theoretical spectrum D{ai) 214
8.19 Wavelet MRA of Fractional Brownian Motion 275
8.20 Time-warped Geometric Brownian Motion 276
8.21 Multifractal spectrum analysis of time-warped GBM 277
8.22 Mandelbrot s early multifractal turbulence trace modeling in a
laboratory experiment 278
8.23 Multifractal spectrum of physical (windtunnel generated)
turbulence 279
8.24 Turbulent correlation between the S P500 spot and futures
market, 1982-1991 280
9.1 Nonunique dynamic equilibria of the logistic parabola 296
9.2 The Hurst exponent may not be the best measure of the global
dependence of intermittent and chaotic processes 297
9.3 The various stability regimes of the logistic process are
determined by the value of the scaling parameter k 298
9.4 When a dynamic process is chaotic, its later values are directly
dependent on the precision of its initial condition 299
9.5 After a fixed attraction point turns unstable, an orbit of period
length p = 2 emerges 301
9.6 Period-doubling appears first at a scaling parameter value just
above k = 3 302
9.7 The relationship between the parabolic map f(x) for an orbit of
period length p = 2 and the 1 x iterated map /(2)(;r) 303
9.8 Oscillation of the logistic process between two steady-state
equilibria at x* = 0.5 and x* = 0.809 304
List of figures xvii
9.9 The 1 x iterated map f{2)(x) for period length p = 4, with
2x2 stable steady-state equilibria and one unstable equilibrium 305
9.10 Oscillation of the logistic process between four steady-state
equilibria at x* = 0.5, 0.875, 0.383 and 0.827, respectively 306
9.11 The 2 x iterated parabolic map for the scaling parameter
k = l+v/8+10-3 306
9.12 A sample window of 100 observations of an undefined orbit, or
frequency, of infinite period length with scaling parameter
k = 3.6 309
9.13 Intermittency in time series is characterized by periods of
stability alternating with periods of chaos 310
9.14 The reappearance of a period of apparent stability 310
9.15 Another period of apparent stability with periodicity with six
steady-state equilibria 311
9.16 Another instance of intermittency in the time series of the
logistic process, after the birth of period length 3 311
9.17 Complete chaos is defined by the coexistence of an infinite
number of deterministic unstable equilibrium orbits 313
9.18 Complete logistic chaos consists of infinitely many coexisting
steady-state dynamic equilibria and is not white noise 315
9.19 Complete chaos exhibits infinitely many aperiodic oscillations
with each oscillation having its own amplitude 315
9.20 Complete chaos exhibits infinitely many aperiodic oscillations
with each oscillation having its own amplitude 316
9.21 Wavelet scalogram and scalegram of the completely chaotic
logistic parabola process with scaling parameter k = 4.0 317
9.22 The trajectory of a billiard ball depends on the shape of the
elastic boundary that constraints it 318
9.23 The state space trajectory of a chaotic system shows aperiodic
cyclicity with non-overlapping orbits 319
9.24 First 10 observations of the state space trajectory of the chaotic
logistic process x(t) for k = 4.0 320
9.25 First 10 observations of the steady-state equilibrium points
where the trajectory touches the parabolic constraint of the
chaotic logistic process x(t) for k = 4.0 320
9.26 First 20 observations of the state space trajectory of the chaotic
logistic process x(t) for k — 4.0 321
9.27 First 20 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for k = 4.0 321
9.28 First 50 observations of the state space trajectory of the chaotic
logistic process x{t) for k = 4.0 322
9.29 First 50 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for *¦ = 4.0 322
9.30 First 90 observations of the state space trajectory of the chaotic
logistic process x(t) for k = 4.0 323
xviii List of figures
9.31 First 90 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for k = 4.0 323
9.32 The physical or institutional resource constraint of the chaotic
process determines its global, long-term predictability 324
9.33 A close return or recurrence plot of the Belousov-Zhabotinsky
chemical reaction 327
9.34 Close return histograms of (a) a chaotic time series with
aperiodic cyclicity and (b) Geometric Brownian Motion 328
10.1 Nelson and Siegel curve fitted to UK government bond (gilt)
rates derived from nine bonds for to = September 4, 1996 358
10.2 Vertical ocean shear at various depths 359
10.3 A 6-scale wavelet coeficient sequence decomposition of ocean
shear series 360
10.4 More than 9,300 minute-by-minute quotations on the Philippine
pesos collected in real time for the month of July 1997 361
10.5 Three-scale wavelet resonance coefficient series of the
minute-by-minute quotations on the Philippine pesos 362
10.6 The term structure gradient, TSG(?) 364
11.1 An indictment of global (average) statistical analysis by two
time series with the same global risk spectrum P(a)) 382
11.2 Simulated evolution, / = 10, 20,40, of a 2D vortex spiral,
based on a pseudo-wavelet computation of 2D Navier-Stokes
equations 384
11.3 Theoretical and empirical representations of a shock wave 386
11.4 Data in the time domain from nine different turbulent flows 391
11.5 Gibbs phenomenon 405
11.6 Approximation of the Heaviside function f(x) in panel of
Figure 11.5, using a wavelet basis 406
12.1 Typical time dependence of financial price volatility, log a2 426
12.2 Empirical distribution of daily revenues of JP Morgan in 1994 431
12.3 Potential large-scale catastrophic flow risk: the Yangtze River 434
12.4 Drawing of the completed Three Gorges Dam: the spillway to
release water and control flooding is in the center 435
12.5 Emergence of large-scale dynamic catastrophic flow risk
management 436
A. 1 Mandelbrot s original evidence for scaling in economic pricing
processes 449
B. 1 S P500 daily closing prices taken from table 2.7 in Sherry
(1992), pp. 29-32 451
Tables
1.1 Ellsberg Paradox payoffs: Options 1 and 2 16
1.2 Ellsberg Paradox payoffs: Options 3 and 4 16
1.3 First four moments of FX returns: USD/DEM and USD/JPY 28
1.4 First four moments of FX returns by time interval 30
2.1 Volatility matrix of European option prices for various strike
prices and expiration dates 60
3.1 Comparison of tail P(X c) probabilities 93
4.1 ARCH( 1) limit kurtosis 114
4.2 ACFs of long- and short-memory series 118
4.3 Equivalence of various critical irregularity exponents 124
6.1 Risk spectrum of FBM increments 168
8.1 Degree of Lipschitz irregularity of Daubechies wavelets 242
8.2 Heterogeneous Hurst exponents of subsequent exchange rate
regimes in Mexico in the 1990s 259
8.3 Measured homogeneous Hurst exponents of Latin American
stock and foreign exchange markets 262
8.4 Values of homogeneous Hurst exponents for nine currencies 265
9.1 Levels of short- and long-term predictability 290
9.2 Steady-state equilibrium regimes of the logistic process 329
|
| adam_txt |
Contents
List of figures xiii
List of tables xix
Preface xxj
Introduction xxvii
PARTI
Financial risk processes 1
1 Risk - asset class, horizon and time 3
1.1 Introduction 3
1.2 Uncertainty 7
1.3 Nonparametric and parametric distributions 17
1.4 Random processes and time series 31
1.5 Software 41
1.6 Exercises 41
2 Competing financial market hypotheses 47
2.1 Introduction 47
2.2 EMH: martingale theory 47
2.3 FMH: fractal theory 53
2.4 Importance of identifying the degree of market efficiency 65
2.5 Software 67
2.6 Exercises 67
3 Stable scaling distributions in finance 71
3.1 Introduction 71
3.2 Affine traces of speculative prices 72
3.3 Invariant properties: stationarity versus scaling 76
3.4 Invariances of (Pareto-Levy) scaling distributions 77
3.5 Zolotarev parametrization of stable distributions 80
3.6 Examples of closed form stable distributions 92
x Contents
3.7 Stable parameter estimation and diagnostics 94
3.8 Software 96
3.9 Exercises 97
4 Persistence of financial risk 102
4.1 Introduction 102
4.2 Serial dependence 102
4.3 Global dependence 105
4.4 (G)ARCHprocesses 111
4.5 Fractional Brownian Motion 117
4.6 Range/Scale analysis 120
4.7 Critical color categorization of randomness 122
4.8 Software 128
4.9 Exercises 128
PART II
Financial risk measurement 133
5 Frequency analysis of financial risk 135
5.7 Introduction 135
5.2 Visualization of long-term financial risks 135
5.3 Correlation and time convolution 136
5.4 Fourier analysis of stationary price innovations 141
5.5 Software 152
5.6 Exercises 152
6 Fourier time-frequency analysis of risk 155
6.1 Introduction 155
6.2 FT for aperiodic variables 156
6.3 Hurst exponent identification from risk spectrum 169
6.4 Heisenberg Uncertainty Principle 171
6.5 Windowed FT for transient price innovations 173
6.6 Software 186
6.7 Exercises 186
7 Wavelet time-scale analysis of risk 190
7.1 Introduction 190
7.2 Wavelet analysis of transient pricing 192
7.3 Mallat'sMRA 209
7.4 Wavelet Parseval Risk Decomposition Theorem 223
Contents xi
7.5 Software 224
7.6 Exercises 224
8 Multiresolution analysis of local risk 230
8.1 Introduction 230
8.2 Measurement of local financial market risk 237
8.3 Homogeneous Hurst exponents of monofractal price series 250
8.4 Multiresolution analysis of multifractal price series 265
8.5 Software 281
8.6 Exercises 281
PART III
Term structure dynamics 287
9 Chaos - nonunique equilibria processes 289
9.1 Introduction 289
9.2 Logistic parabola regimes 292
9.3 General nonlinear dynamic systems 317
9.4 Detecting attracting points and aperiodic orbits 327
9.5 Summary of aperiodic cyclical steady-state equilibria 328
9.6 Software 330
9.7 Exercises 331
10 Measuring term structure dynamics 337
10.1 Introduction 337
10.2 Dynamic investment cashflow theory 340
10.3 Nonlinear relationships in finance 347
10.4 Liquidity and financial turbulence 358
10.5 Software 372
10.6 Exercises 373
11 Simulation of financial turbulence 380
11.1 Introduction 380
11.2 Theories of physical and financial turbulence 381
11.3 Measurement and simulation of turbulence 388
11.4 Simulation of financial cashflow turbulence 396
11.5 Multiresolution analysis of financial turbulence 398
11.6 Wavelet solutions of financial diffusion equations 402
11.7 Software 414
11.8 Exercises 414
xii Contents
PART IV
Financial risk management 423
12 Managing VaR and extreme values 425
12.1 Introduction 425
12.2 Global dependence of financial returns 425
12.3 VaR for stable distributions 428
12.4 VaR for parametric distributions 431
12.5 Extreme value theory 437
12.6 VaR and fractal pricing processes 440
12.7 Software 445
12.8 Exercises 445
Appendix A: original scaling in financial economics 449
Appendix B: S P500 daily closing prices for 1988 450
Index 453
Figures
1 Risk = Danger (Wei) + Opportunity (Ji) xxx
1.1 Historical average annual returns and return volatility 4
1.2 Levels and returns of empirical financial time series: AMEX
stock and oil indices and DEM-USD exchange rate 9
1.3 Simple and relatively inexpensive radiation monitor 11
1.4 The nth-order moments and cumulants for n = 1,2, 3,4
of the Laplace, Gaussian and Uniform p.d.fs 21
1.5 The nth-order moments and cumulants for n = 1,2,3,4
of Exponential, Rayleigh and K-distribution p.d.fs 22
1.6 Construction of the histogram of a time series by binning 24
1.7 Raw and transformed daily returns of the DAX. The histograms
on the right show the relative frequencies of the returns in the
same scale 25
1.8 Empirical histogram of minute-by-minute log-increments of the
JPY in June 1997 25
1.9 Volatility smile of foreign currrency options 26
1.10 Implied distribution and (log-) normal distribution of foreign
currency options 26
1.11 Skewed volatility smile of equities 27
1.12 Implied distribution and (log-) normal distribution 28
1.13 The empirical cumulative distributions for USD/DEM and USD
6 months cash interest rate, shown for different time horizons 30
1.14 Semi-annual cumulative distributions of THB-FX increments,
January-December 1997 36
1.15 Semi-annual cumulative distributions of DEM-FX increments,
January-December 1997 37
2.1 Annualized volatility of theoretical Random Walk model of
constant, normalized, asset return volatilities 55
2.2 Empirical annualized volatility of financial market returns 56
2.3 Mandelbrot's Julia set 58
2.4 Clear-air turbulence 59
2.5 Intraday return variance of the HSI 61
2.6 Intraday return variance of the HSIF 62
xiv List of figures
2.7 One-month Eurodollar yield and time-varying (a) turbulent
volatility and (b) daily data 63
2.8 Five-year CMT yield and time-varying (a) turbulent volatility
and (b) daily data 64
3.1 Stable probabilistic schemes 78
3.2 Stable density in the Zolotarev S(az, /?, y, 8; 0) =
S(az, 0.8, 1, 0; 0) parametrization 86
3.3 Stable density in the Zolotarev S(az, P, y, \$; 1) =
S(az, 0.8, 1, 0; 1) parametrization 87
3.4 Comparison of the Af = 1 minute p.d.f. for high-frequency
S P500 price changes with the Gaussian p.d.f. and with a Levy
stable p.d.f. 90
3.5 Non-convergent moving variance of 253 daily rates of return
(in 100 percent) of the S P500 stock market index in 1998 90
3.6 Estimates of four parameters of the Zolotarev parametrization
of FX distributions 95
4.1 Successive increments of ARCH(l) simulations with the same
unconditional variance (a2 = 1) 114
4.2 Probability density function of the successive increments shown
in Figure 4.1. 115
4.3 Comparison of the scaling properties of the unconditional p.d.f.
of a GARCH( 1,1) stochastic process 117
4.4 Autocorrelograms of equally-weighted CRSP daily and
monthly stock return indices 119
4.5 Sample of power spectra of white, pink and brown noise 125
4.6 Relations between and constraints on d, H and az 127
5.1 Fourier series approximation of a square wave 144
5.2 Heat diffusion analysis 146
5.3 A sample signal constructed from sine functions representing
three pulsations 150
5.4 The FT of the sampled signal s(t) 151
5.5 Fourier series analysis of pure musical harmonics: dominant
frequencies of the clarinet, violin and bagpipe 151
6.1 Granger and Morgenstern's global risk spectrum of Standard
and Poor series, based on annual data, 1875-1952 166
6.2 Semi-log plot of the autocorrelation function y (r) for the
S P500 index, sampled at a 1-minute time scale 170
6.3 Spectral density of high-frequency data from the S P500 index 171
6.4 Gdbor's atom go,£ as a function of time for three frequencies:
(a) high |i, (b) middle £2 and (c) low £3 176
6.5 Heisenberg boxes of two windowed Fourier atoms gu^ and gv y 177
6.6 Time-frequency analysis by the Gabor Transform with a
adapted to the time coherence of frequencies a \ and 3 180
6.7 Signal, spectrograms and scalogram 181
List of figures xv
6.8 Spectrogram Ps (z, |) of time series with two superimposed
time-varying frequencies 182
6.9 Laughter data and their global histogram 183
6.10 Spectrogram of laughter data with three dominant harmonics 184
6.11 Changes in the daily level of the three-month Treasury yield 184
6.12 Comparison of the modulated spectrogram of empirical
DEM/USD increments with the flat spectogram of white noise 185
7.1 A sine wave and a Daubechies' wavelet ^D20 193
7.2 Self-similarity of wavelets: translation (every fourth k) and
scaling of a wavelet f 194
7.3 Wavelet coefficients are "correlation" or "resonance"
coefficients. Here a wavelet is correlated with an irregular
signal. Different sections of the signal produce different
resonance coefficients 196
7.4 A scalogram: a plot of the magnitude of wavelet coefficients 197
7.5 A 3D scalogram: a plot of the magnitude of the wavelet
coefficients in three dimensions 197
7.6 Heisenberg boxes of two wavelets. Smaller scales decrease the
time dispersion, but increase the frequency support, which is
shifted towards higher frequencies 198
7.7 Time-frequency resolution and basis functions of the
Windowed FT and the Wavelet Transform 199
7.8 A scalogram with modulus | W(r, a)| using a Morlet wavelet
for 25 different scales 201
7.9 Normalized scalogram ^/r))Pw(r, a) 202
7.10 Time-scale tiling for a sinusoidal function with an isolated
singularity at to 203
7.11 Empirical 3D scalogram of Thai Baht increments in July 1997 204
7.12 Time signal observations on f(x) 211
7.13 Haar and triangle scaling functions and their respective MRA
equations 216
7.14 Haar and triangle wavelets and their respective MRA equations 220
8.1 USD/DEM exchange rate on a time scale of At = 20 minutes 232
8.2 Scaling law behavior of the USD/DEM exchange rate in the
period 1993-1994, for various subperiods 233
8.3 Trading transaction density based on daily and weekly averages
of tick-by-tick data 234
8.4 Daily and weekly averaged volatility per hour of the USD/DEM
exchange rate 235
8.5 The cone of influence of an abscissa singularity v 245
8.6 Singularity cones of influence of a Dirac pulse at t = t0 245
8.7 Wavelet decomposition of a time series with singularities 248
8.8 How to measure the degree of irregularity of local risk of a
series of price singularities x (r) 249
8.9 Complete wavelet tiling 251
xvi List of figures
8.10 Wavelet-based persistence analysis of heartbeat interarrival
times for a healthy patient with a Daubechies(5) wavelet 254
8.11 Wavelet-based persistence analysis of weekly Dow Jones
Industrial Index data with Daubechies(5) wavelet 256
8.12 Wavelet MRA by Morlet(6) wavelet of the various exchange
rate regimes of the Mexican Peso/USD and the various
Brazilian financial market crises in the 1990s 257
8.13 Wavelet MRA of the various exchange rate regimes of the first
(log) differences of the Mexican Peso/USD in the 1990s 260
8.14 Wavelet MRA, based on daily data, of Chilean stock index rate
of returns in the 1990s 261
8.15 The first four monthly moments of the distributions of the
minute-by-minute quotations of nine currency rates in
May-August 1997 (USD is the numeraire) 264
8.16 Development of Koch's snowflake with Hausdorff dimension
D= 1.2619 267
8.17 Schematic convex multifractal singularity spectrum D{ai),
with various Gibbs exponent regimes 271
8.18 Computation of singularity spectrum of the devil's staircase, its
partition function Z(q, a) scaling exponents r(q) and its
theoretical spectrum D{ai) 214
8.19 Wavelet MRA of Fractional Brownian Motion 275
8.20 Time-warped Geometric Brownian Motion 276
8.21 Multifractal spectrum analysis of time-warped GBM 277
8.22 Mandelbrot's early multifractal turbulence trace modeling in a
laboratory experiment 278
8.23 Multifractal spectrum of physical (windtunnel generated)
turbulence 279
8.24 Turbulent correlation between the S P500 spot and futures
market, 1982-1991 280
9.1 Nonunique dynamic equilibria of the logistic parabola 296
9.2 The Hurst exponent may not be the best measure of the global
dependence of intermittent and chaotic processes 297
9.3 The various stability regimes of the logistic process are
determined by the value of the scaling parameter k 298
9.4 When a dynamic process is chaotic, its later values are directly
dependent on the precision of its initial condition 299
9.5 After a fixed attraction point turns unstable, an orbit of period
length p = 2 emerges 301
9.6 Period-doubling appears first at a scaling parameter value just
above k = 3 302
9.7 The relationship between the parabolic map f(x) for an orbit of
period length p = 2 and the 1 x iterated map /(2)(;r) 303
9.8 Oscillation of the logistic process between two steady-state
equilibria at x* = 0.5 and x* = 0.809 304
List of figures xvii
9.9 The 1 x iterated map f{2)(x) for period length p = 4, with
2x2 stable steady-state equilibria and one unstable equilibrium 305
9.10 Oscillation of the logistic process between four steady-state
equilibria at x* = 0.5, 0.875, 0.383 and 0.827, respectively 306
9.11 The 2 x iterated parabolic map for the scaling parameter
k = l+v/8+10-3 306
9.12 A sample window of 100 observations of an undefined orbit, or
frequency, of infinite period length with scaling parameter
k = 3.6 309
9.13 Intermittency in time series is characterized by periods of
stability alternating with periods of chaos 310
9.14 The reappearance of a period of apparent stability 310
9.15 Another period of apparent stability with periodicity with six
steady-state equilibria 311
9.16 Another instance of intermittency in the time series of the
logistic process, after the birth of period length 3 311
9.17 Complete chaos is defined by the coexistence of an infinite
number of deterministic unstable equilibrium orbits 313
9.18 Complete logistic chaos consists of infinitely many coexisting
steady-state dynamic equilibria and is not white noise 315
9.19 Complete chaos exhibits infinitely many aperiodic oscillations
with each oscillation having its own amplitude 315
9.20 Complete chaos exhibits infinitely many aperiodic oscillations
with each oscillation having its own amplitude 316
9.21 Wavelet scalogram and scalegram of the completely chaotic
logistic parabola process with scaling parameter k = 4.0 317
9.22 The trajectory of a billiard ball depends on the shape of the
elastic boundary that constraints it 318
9.23 The state space trajectory of a chaotic system shows aperiodic
cyclicity with non-overlapping orbits 319
9.24 First 10 observations of the state space trajectory of the chaotic
logistic process x(t) for k = 4.0 320
9.25 First 10 observations of the steady-state equilibrium points
where the trajectory "touches" the parabolic constraint of the
chaotic logistic process x(t) for k = 4.0 320
9.26 First 20 observations of the state space trajectory of the chaotic
logistic process x(t) for k — 4.0 321
9.27 First 20 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for k = 4.0 321
9.28 First 50 observations of the state space trajectory of the chaotic
logistic process x{t) for k = 4.0 322
9.29 First 50 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for *¦ = 4.0 322
9.30 First 90 observations of the state space trajectory of the chaotic
logistic process x(t) for k = 4.0 323
xviii List of figures
9.31 First 90 steady-state equilibria points on the attractor set of the
chaotic logistic process x(t) for k = 4.0 323
9.32 The physical or institutional resource constraint of the chaotic
process determines its global, long-term predictability 324
9.33 A close return or recurrence plot of the Belousov-Zhabotinsky
chemical reaction 327
9.34 Close return histograms of (a) a chaotic time series with
aperiodic cyclicity and (b) Geometric Brownian Motion 328
10.1 Nelson and Siegel curve fitted to UK government bond (gilt)
rates derived from nine bonds for to = September 4, 1996 358
10.2 Vertical ocean shear at various depths 359
10.3 A 6-scale wavelet coeficient sequence decomposition of ocean
shear series 360
10.4 More than 9,300 minute-by-minute quotations on the Philippine
pesos collected in real time for the month of July 1997 361
10.5 Three-scale wavelet resonance coefficient series of the
minute-by-minute quotations on the Philippine pesos 362
10.6 The term structure gradient, TSG(?) 364
11.1 An indictment of global (average) statistical analysis by two
time series with the same global risk spectrum P(a)) 382
11.2 Simulated evolution, / = 10, 20,40, of a 2D vortex spiral,
based on a pseudo-wavelet computation of 2D Navier-Stokes
equations 384
11.3 Theoretical and empirical representations of a shock wave 386
11.4 Data in the time domain from nine different turbulent flows 391
11.5 Gibbs phenomenon 405
11.6 Approximation of the Heaviside function f(x) in panel of
Figure 11.5, using a wavelet basis 406
12.1 Typical time dependence of financial price volatility, log a2 426
12.2 Empirical distribution of daily revenues of JP Morgan in 1994 431
12.3 Potential large-scale catastrophic flow risk: the Yangtze River 434
12.4 Drawing of the completed Three Gorges Dam: the spillway to
release water and control flooding is in the center 435
12.5 Emergence of large-scale dynamic catastrophic flow risk
management 436
A. 1 Mandelbrot's original evidence for scaling in economic pricing
processes 449
B. 1 S P500 daily closing prices taken from table 2.7 in Sherry
(1992), pp. 29-32 451
Tables
1.1 Ellsberg Paradox payoffs: Options 1 and 2 16
1.2 Ellsberg Paradox payoffs: Options 3 and 4 16
1.3 First four moments of FX returns: USD/DEM and USD/JPY 28
1.4 First four moments of FX returns by time interval 30
2.1 Volatility matrix of European option prices for various strike
prices and expiration dates 60
3.1 Comparison of tail P(X c) probabilities 93
4.1 ARCH( 1) limit kurtosis 114
4.2 ACFs of long- and short-memory series 118
4.3 Equivalence of various critical irregularity exponents 124
6.1 Risk spectrum of FBM increments 168
8.1 Degree of Lipschitz irregularity of Daubechies wavelets 242
8.2 Heterogeneous Hurst exponents of subsequent exchange rate
regimes in Mexico in the 1990s 259
8.3 Measured homogeneous Hurst exponents of Latin American
stock and foreign exchange markets 262
8.4 Values of homogeneous Hurst exponents for nine currencies 265
9.1 Levels of short- and long-term predictability 290
9.2 Steady-state equilibrium regimes of the logistic process 329 |
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| author | Los, Cornelis Albertus |
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| discipline | Wirtschaftswissenschaften |
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| id | DE-604.BV023566124 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:38:50Z |
| indexdate | 2024-07-09T21:24:38Z |
| institution | BVB |
| isbn | 041527866X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016882211 |
| oclc_num | 249009388 |
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| physical | XXXI, 460 S. graph. Darst. |
| publishDate | 2003 |
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| publisher | Routledge |
| record_format | marc |
| series | Routledge international studies in money and banking |
| series2 | Routledge international studies in money and banking |
| spelling | Los, Cornelis Albertus Verfasser aut Financial market risk measurement and analysis Cornelis A. Los London ; New York Routledge 2003 XXXI, 460 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Routledge international studies in money and banking 24 Hedging (Finance) Risk management Routledge international studies in money and banking 24 (DE-604)BV010787858 24 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016882211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Los, Cornelis Albertus Financial market risk measurement and analysis Routledge international studies in money and banking Hedging (Finance) Risk management |
| title | Financial market risk measurement and analysis |
| title_auth | Financial market risk measurement and analysis |
| title_exact_search | Financial market risk measurement and analysis |
| title_exact_search_txtP | Financial market risk measurement and analysis |
| title_full | Financial market risk measurement and analysis Cornelis A. Los |
| title_fullStr | Financial market risk measurement and analysis Cornelis A. Los |
| title_full_unstemmed | Financial market risk measurement and analysis Cornelis A. Los |
| title_short | Financial market risk |
| title_sort | financial market risk measurement and analysis |
| title_sub | measurement and analysis |
| topic | Hedging (Finance) Risk management |
| topic_facet | Hedging (Finance) Risk management |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016882211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010787858 |
| work_keys_str_mv | AT loscornelisalbertus financialmarketriskmeasurementandanalysis |