How to Understand and Use Mathematics for Derivatives: Vol. 2

  • ID: 5378
  • January 2001
  • 347 Pages
  • Euromoney Trading Ltd
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These two companion volumes offer a comprehensive guide to the ‘new maths’, analysing and explaining the behaviour of the markets and providing a practical guide to the key mathematical models underlying trading and risk management.

Volume II - Advanced Modelling Methods - offers a comprehensive explanation of new terms and techniques used in financial analysis. Ten chapters evaluate the latest developments in the field, formulae calculation and analysis of the different ways in which data can be interpreted and profited from including: computational finance; trading techniques; financial engineering; non-linear maths; mathematical analysis and risk management; statistical inference, skewness and kyrosis; high frequency data and why banks use it; fractals and chaos theory in financial markets.

Author information

Professor Dr. Dimitris N. Chorafas has, since 1961, advised financial institutions and industrial corporations on strategic planning, risk management, computer and communications systems, and internal controls. A graduate of the University of California, Los Angeles, the University of Paris, and the Technical University of Athens, Dr Chorafas READ MORE >

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Foreword 1
Preface 5

Chapter 1 11 The sense of analytical computational finance
1 Introduction
2 Derivative financial instruments and the need for analytics
3 Rocket scientists and non-traditional financial research
4 Analytical finance with high-frequency financial data
5 Thomas Edison: inventor and model for rocket scientists
6 Theory and practice in financial engineering
7 Sir Isaac Newton, Albert Einstein and their theories
8 Blind beliefs are the tombstone of physics and of finance
9 Notes and references

Chapter 2 35 Leading-edge techniques in prognosis and trading
1 Introduction
2 Why classical technical analysis is not enough
3 Predictive technology and evolutionary modelling solutions
4 Non-traditional financial research and differences in time horizons
5 Benefits to be provided through a cross-disciplinary approach
6 Is there a correlation between mathematics and reality?
7 The use of leading-edge techniques in prognostication by Chemical Banking
8 What should be done by a good predictive model?
9 Using models for training and trading
10 Pitfalls with prognostication: an example with gold and derivatives
11 Notes and references

Chapter 3 57 Fair value, marking-to-model, implied volatility and realtime simulation
1 Introduction
2 The difference between marking-to-market and marking-to-model
3 Developing and using models in connection to fair value
4 Discovering gems of beauty and merit in financial analysis
5 Gaining advantages from more advanced use of the Black-Scholes model
6 Mapping the market into the computer
7 Continuous time, stochastic and implied volatility
8 Using diffusion equations in estimating volatility
9 The role of realtime simulation in finance
10 Notes and references

Chapter 4 81 Applying chaos theory in financial transactions
1 Introduction
2 Challenges posed by chaotic systems
3 Sensitivity dependence on initial conditions and the edge of chaos
4 The art of interpreting complex, chaotic patterns
5 Prognosis, entropy and data encoding
6 What can be done about noisy patterns?
7 Forex markets, chaos theory and information asymmetry
8 Deterministic chaos and strange attractors
9 Why do we need complexity theory?
10 Notes and references


Chapter 5 107 Logarithmic distributions, black boxes, Laplace transforms and Fourier series
1 Introduction
2 Appreciating the geometric aspects of Napierian logarithms
3 Notions underpinning the logarithmic spiral
4 Logarithmic functions, complex numbers, lognormal distribution, statistical moments
5 Three metrics for change of logarithmic price at time t
6 Understanding the practical use of black boxes in interpreting time series
7 Volatility ratio and the direction of volatility changes
8 Laplace transformations and Fourier analysis
9 Fourier series and fast Fourier transform
10 Notes and references

Chapter 6 135 Statistical design, leptokyrtotic distributions, confidence intervals and test of hypothesis
1 Introduction
2 Studying skewness and kyrtosis of a distribution
3 Leptokyrtosis, allokyrtosis and prognostication
4 Types of errors, producer's risk a, and consumer's risk b
5 Sampling plans and operating characteristics curves
6 Boundary conditions, limits and confidence levels
7 Hypotheses, test of significance and t-test
8 An introduction to the variance ratio test
9 Maximum likelihood, square-root rule and boxplots
10 Statistical design, Weiner's process and transition probabilities
11 Using the Weiner process for risk-factor calculations
12 Applying the notion of confidence intervals to risk factors and tolerance limits
13 Notes and references

Chapter 7 165 Parameter-driven statistical design, parametric statistics and non-parametric statistics
1 Introduction
2 A parametric and a non-parametric methodology
3 The connection between para metric statistics and parameter-driven modelling
4 Benefits to be obtained from a parameter-driven design
5 A parameter-driven model in currency exchange
6 Sample size and the use of non-parametric statistics
7 Non-parametric runs and sign tests
8 Understanding how to do sign tests and runs tests
9 A brief applications example with the sign test
10 Mean square successive differences and von Neumann's algorithm
11 The importance of distribution-free methods and Chebyshev's inequality
12 Notes and references


Chapter 8 191 Understanding and applying non-linear models
1 Introduction
2 What distinguishes linear from non-linear systems?
3 Characteristic solution, particular solution, steady state and instability
4 Oscillatory outputs and unintentional non-linearities
5 Challenges presented by slow and fast non-linearities
6 The method of describing function analysis
7 Determining the variation of the describing function and its rationale
8 Studying non-linearities through plane-phase analysis
9 Exploiting the potential of a phase portrait
10 Notes and references

Chapter 9 215 Feedback control systems, root locus, the Nyquist criterion and auto-oscillations
1 Introduction
2 Open-loop and closed-loop systems
3 Statistical design theory, mean square error and discontinuities
4 Transient response and locus of a system
5 Stability criteria
6 The Nyquist solution
7 The system transfer function
8 Reconsidering the definition of stability
9 Forced oscillations of non-linear systems
10 An introduction to harmonic functions and auto-oscillations
11 The special case of solitons
12 Notes and references

Chapter 10 239 Brownian motion, Mandelbrot's fractals and currency exchange applications
1 Introduction
2 The concept of an underlying Brownian motion
3 Brownian motion and the drift parameter
4 Fractal geometry and fractal trajectories
5 Capitalizing on fractal prices and the market inefficiency they represent
6 Intraday fractal structures and the drift exponent
7 What can we learn from the study of bid/ask prices?
8 Studies based on the absolute value of price changes
9 Market stress, seasonal and microseasonal results
10 Psychological time, activity variables and directional change
11 Notes and references


Chapter 11 265 Weibull distribution and the risk-management algorithm
1 Introduction
2 Why high-frequency data is important to risk management
3 Reliability, dynamic boundary conditions and safety margins
4 The risk from big derivatives contracts with counterparties
5 Poisson distribution, expected failures and the survival equation
6 Implementing the Weibull distribution in reliability and risk management
7 Rules which apply to the management of reliability and of financial exposure
8 Exponential Weibull distribution and the Chorafas algorithm
9 Rethinking and restructuring the concept of risk management
10 Capitalizing on the use of the addition theorem of the Poisson distribution
11 Placing emphasis on analytics, experimentation and levels of confidence
12 Notes and references

Chapter 12 295 Intraday information, micro-seasonality and high- frequency financial data
1 Introduction
2 Intraday financial transactions and expected analytical results
3 The financial markets are a 24-hour realtime system
4 Exploring time series in ways never before available
5 When financial instruments are traded around the clock, realtime is the only solution
6 Derivatives, high-frequency data and risk management
7 Exploiting information flows in the securities and in foreign exchange
8 Derivative financial instruments and the characteristics of trades
9 A non-traditional viewpoint regarding non-events
10 Capitalizing on the concept of ergodicity
11 Understanding microseasonality and its role in world-wide transaction volume
12 Notes and references
Chapter 13 319 Heteroschedasticity, autocorrelation, autoregression
and the Arch family of models
1 Introduction
2 High-frequency financial data and heteroschedasticity
3 Sampling space, the effects of sampling procedures on heteroschedasticity and devolatilization
4 Independent, identically distributed random variables and stochastic volatility
5 Serial dependence, autocorrelation and dynamic strategies
6 Implementing the mean square error algorithm and measuring the autocorrelation
7 Autocorrelation as qualitative measure of the regularity of the function, and convolution integral
8 Capitalizing on correlation, covariance and lagged correlation
9 What is needed for a careful study of autoregression?
10 Autoregressive conditional heteroschedasticity: the Arch and Garch models
11 Taking another look at conditional autoregression, Arch, Garch and Harch
12 Autoregressive conditional duration and the Weibull distribution
13 Notes and references

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