The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Since that time, numerous researchers have contributed to the theory. The present paper begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new "down-and-out" option are derived. The effects of dividends and call provisions on the warrant price are examined. The possibilities for further extension of the theory to the pricing of corporate liabilities are discussed.

https://www.jstor.org/stable/pdf/3003143.pdf

Theory of rational option pricing

A theory of memory retrieval is developed and is shown to apply over a range of experimental paradigms. Access to memory traces is viewed in terms of a resonance metaphor. The probe item evokes the search set on the basis of probe-memory item relatedness, just as a ringing tuning fork evokes sympathetic vibrations in other tuning forks. Evidence is accumulated in parallel from each probe-memory item comparison, and each comparison is modeled by a continuous random walk process. In item recognition, the decision process is self-terminating on matching comparisons and exhaustive on nonmatching comparisons. The mathematical model produces predictions about accuracy, mean reaction time, error latency, and reaction time distributions that are in good accord with experimental data. The theory is applied to four item recognition paradigms (Sternberg, prememorized list, study-test, and continuous) and to speed-accuracy paradigms; results are found to provide a basis for comparison of these paradigms. It is noted that neural network models can be interfaced to the retrieval theory with little difficulty and that semantic memory models may benefit from such a retrieval scheme.

A Theory of Memory Retrieval.

1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

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Wavelet Methods for Time Series Analysis

Acknowledgements 1. Introduction 2. Electromagnetic wave propagation 3. The absorption of light 4. Specular reflection 5. Single particle scattering: perfect spheres 6. Single particle scattering: irregular particles 7. Propagation in a nonuniform medium: the equation of radiative transfer 8. The bidirectional reflectance of a semi-infinite medium 9. The opposition effect 10. A miscellany of bidirectional reflectances and related quantities 11. Integrated reflectances and planetary photometry 12. Photometric effects of large scale roughness 13. Polarization 14. Reflectance spectroscopy 15. Thermal emission and emittance spectroscopy 16. Simultaneous transport of energy by radiation and conduction Appendix A. A brief review of vector calculus Appendix B. Functions of a complex variable Appendix C. The wave equation in spherical coordinates Appendix D. Fraunhoffer diffraction by a circular hole Appendix E. Table of symbols Bibliography Index.

Theory of Reflectance and Emittance Spectroscopy

In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.

Functions of a Complex Variable

Complex Variables and Applications

1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Preface 1 Fourier Series Piecewise Continuous Functions Fourier Cosine Series Examples Fourier Sine Series Examples Fourier Series Examples Adaptations to Other Intervals 2 Convergence of Fourier Series One-Sided Derivatives A Property of Fourier Coefficients Two Lemmas A Fourier Theorem A Related Fourier Theorem Examples Convergence on Other Intervals A Lemma Absolute and Uniform Convergence of Fourier Series The Gibbs Phenomenon Differentiation of Fourier Series Integration of Fourier Series 3 Partial Differential Equations of Physics Linear Boundary Value Problems One-Dimensional Heat Equation Related Equations Laplacian in Cylindrical and Spherical Coordinates Derivations Boundary Conditions Duhamel's Principle A Vibrating String Vibrations of Bars and Membranes General Solution of the Wave Equation Types of Equations and Boundary Conditions 4 The Fourier Method Linear Operators Principle of Superposition Examples Eigenvalues and Eigenfunctions A Temperature Problem A Vibrating String Problem Historical Development 5 Boundary Value Problems A Slab with Faces at Prescribed Temperatures Related Temperature Problems Temperatures in a Sphere A Slab with Internally Generated Heat Steady Temperatures in Rectangular Coordinates Steady Temperatures in Cylindrical Coordinates A String with Prescribed Initial Conditions Resonance An Elastic Bar Double Fourier Series Periodic Boundary Conditions 6 Fourier Integrals and Applications The Fourier Integral Formula Dirichlet's Integral Two Lemmas A Fourier Integral Theorem The Cosine and Sine Integrals Some Eigenvalue Problems on Undounded Intervals More on Superposition of Solutions Steady Temperatures in a Semi-Infinite Strip Temperatures in a Semi-Infinite Solid Temperatures in an Unlimited Medium 7 Orthonormal Sets Inner Products and Orthonormal Sets Examples Generalized Fourier Series Examples Best Approximation in the Mean Bessel's Inequality and Parseval's Equation Applications to Fourier Series 8 Sturm-Liouville Problems and Applications Regular Sturm-Liouville Problems Modifications Orthogonality of Eigenfunctions adn Real Eigenvalues Real-Valued Eigenfunctions Nonnegative Eigenvalues Methods of Solution Examples of Eigenfunction Expansions A Temperature Problem in Rectangular Coordinates Steady Temperatures Other Coordinates A Modification of the Method Another Modification A Vertically Hung Elastic Bar 9 Bessel Functions and Applications The Gamma Function Bessel Functions Jn(x) Solutions When v = 0,1,2,... Recurrence Relations Bessel's Integral Form Some Consequences of the Integral Forms The Zeros of Jn(x) Zeros of Related Functions Orthogonal Sets of Bessel Functions Proof of the Theorems Two Lemmas Fourier-Bessel Series Examples Temperatures in a Long Cylinder A Temperature Problem in Shrunken Fittings Internally Generated Heat Temperatures in a Long Cylindrical Wedge Vibration of a Circular Membrane 10 Legendre Polynomials and Applications Solutions of Legendre's Equation Legendre Polynomials Rodrigues' Formula Laplace's Integral Form Some Consequences of the Integral Form Orthogonality of Legendre Polynomials Normalized Legendre Polynomials Legendre Series The Eigenfunctions Pn(cos theta) Dirichlet Problems in Spherical Regions Steady Temperatures in a Hemisphere 11 Verification of Solutions and Uniqueness Abel's Test for Uniform Convergence Verification of Solution of Temperature Problem Uniqueness of Solutions of the Heat Equation Verification of Solution of Vibrating String Problem Uniqueness of Solutions of the Wave Equation Appendixes Bibliography Some Fourier Series Expansions Solutions of Some Regular Sturm-Liouville Problems Some Fourier-Bessel Series Expansions Index

James Ward Brown

Papers

Complex Variables and Applications

Complex Variables and Applications

Fourier series and boundary value problems

Variable compleja y aplicaciones

Complex Variables and Applications Sixth Edition