FORECASTING RISK AND RETURN

9.1 Introduction to Forecasting 248

9.2 Forecasting Volatility for Value-at-Risk 250

9.2.1 Three Simple Volatility Forecasting Models 250

9.2.2 Choosing the Best Volatility Forecasting Model 254

9.3 Forecasting Returns over Multiple Time Horizons 255

9.3.1 Intrinsic Time 255

9.3.2 Model Structure 256

9.3.3 A Linear Combination of Nonlinear Indicators 256

9.3.4 Moving Averages, Momenta, and Indicators 257

9.3.5 Continuous Coefficient Update 259

9.4 Measuring Forecast Quality 261

9.4.1 Appropriate Measures of Forecast Accuracy 262

9.4.2 Empirical Results for the Multi-Horizon Model 263

9.4.3 Forecast Effectiveness in Intraday Horizons 264

10

CORRELATION AND MULTIVARIATE RISK

10.1 Introduction 268

10.2 Estimating the Dependence of Financial Time Series 269

10.3 Covolatility Weighting 270

10.3.1 Formulation of an Adjusted Correlation Measure 272

10.3.2 Monte Carlo and Empirical Tests 274

10.4 Stability of Return Correlations 277

10.4.1 Correlation Variations over Time 278

10.4.2 The Exponential Memory of Return Correlations 282

10.5 Correlation Behavior at High Data Frequencies 287

10.6 Conclusions 293

TRADING MODELS

11.1 Introduction 295

11.2 Real-Time Trading Strategies 297

11.2.1 The Trading Model and Its Data-Processing Environment 299

11.2.2 Simulated Trader 303

11.3 Risk Sensitive Performance Measures 304

11.3.1 Xeff: A Symmetric Effective Returns Measure 305

11.3.2 Reff: An Asymmetric Effective Returns Measure 307

11.4 Trading Model Algorithms 309

11.4.1 An Example of a Trading Model 310

11.4.2 Model Design with Genetic Programming 311

11.5 Optimization and Testing Procedures 317

11.5.1 Robust Optimization with Genetic Algorithms 317

11.5.2 Testing Procedures 321

11.6 Statistical Study of a Trading Model 323

11.6.1 Heterogeneous Real-Time Trading Strategies 323

11.6.2 Price-Generation Processes and Trading Models 328

11.7 Trading Model Portfolios 338

11.8 Currency Risk Hedging 340

11.8.1 The Hedging Ratio and the "Neutral Point" 343

11.8.2 Risk/Return of an Overlay with Static and Dynamic

Positions 344

11.8.3 Dynamic Hedging with Exposure Constraints 345

11.8.4 Concluding Remarks 346

TOWARD A THEORY OF HETEROGENEOUS MARKETS

12.1 Definition of Efficient Markets 349

12.2 Dynamic Markets and Relativistic Effects 350

12.3 Impact of the New Technology 352

12.4 Zero-Sum Game or Perpetuum Mobile? 353

12.5 Discussion of the Conventional Definition 354

12.6 An Improved Definition of "Efficient Markets" 354

BIBLIOGRAPHY 356 INDEX 376

LIST OF FIGURES

I. I Size and data frequency of different samples 4

1.2 Models versus time scale

1.3 Volatility with daily versus high-frequency data

3.1 Types of time series operators

3.2 Interpolation methods

3.3 Overlapping time intervals

3.4 One week of USD-CHF prices

3.5 Moving average (MA) kernel

3.6 MA kernel on a logarithmic scale

3.7 Schematic differential kernel

3.8 Kernel of a differential operator

3.9 Decay of a differential kernel

3.10 Differential and return

3.11 Distribution of derivative operator

3.12 Annualized volatility as a moving norm 7

3.13 Moving moments of returns 7

3.14 Kernel of a windowed Fourier operator 7

3.15 Normed windowed Fourier transform 7

3.16 Microscopic volatility 7

3.17 Tick activity 8 4.1 Flowchart of a data-cleaning filter 8