Essentials of Stochastic Finance

The following sections are included:

• Contents

• Foreword

The following sections are included:

• Financial Structures and Instruments
  • Key Objects and Structures
  • Financial Markets
  • Market of Derivatives. Financial Instruments

• Financial Markets under Uncertainty. Classical Theories of the Dynamics of Financial Indexes, their Critics and Revision. Neoclassical Theories
  • Random Walk Conjecture and Concept of Efficient Market
  • Investment Portfolio. Markowitz's Diversification
  • CAPM: Capital Asset Pricing Model
  • APT: Arbitrage Pricing Theory
  • Analysis, Interpretation, and Revision of the Classical Concepts of Efficient Market. I
  • Analysis, Interpretation, and Revision of the Classical Concepts of Efficient Market. II

• Aims and Problems of Financial Theory, Engineering, and Actuarial Calculations
  • Role of Financial Theory and Financial Engineering. Financial Risks
  • Insurance: a Social Mechanism of Compensation for Financial Losses
  • A Classical Example of Actuarial Calculations: the Lundberg–Cramér Theorem

The following sections are included:

• Necessary Probabilistic Concepts and Several Models of the Dynamics of Market Prices
  • Uncertainty and Irregularity in the Behavior of Prices. Their Description and Representation in Probabilistic Terms
  • Doob Decomposition. Canonical Representations
  • Local Martingales. Martingale transformations. Generalized Martingales
  • Gaussian and Conditionally Gaussian Models
  • Binomial Model of Price Evolution
  • Models with Discrete Intervention of Chance

• Linear Stochastic Models
  • Moving Average Model MA(q)
  • Autoregressive Model AR(p)
  • Mixed Autoregressive Moving Average Model ARMA(p,q) and Integrated Model ARIMA(p,d,q)
  • Prediction in Linear Models

• Nonlinear Stochastic Conditionally Gaussian Models
  • ARCH and GARCH Models
  • EGARCH, TGARCH, HARCH, and Other Models
  • Stochastic Volatility Models

• Supplement: Dynamical Chaos Models
  • Nonlinear Chaotic Models
  • Distinguishing between 'Chaotic' and 'Stochastic' Sequences

The following sections are included:

• Non-Gaussian Models of Distributions and Processes
  • Stable and Infinitely Divisible Distributions
  • Lévy Processes
  • Stable Processes
  • Hyperbolic Distributions and Processes

• Models with Self-Similarity. Fractality
  • Hurst's Statistical Phenomenon of Self-Similarity
  • A Digression on Fractal Geometry
  • Statistical Self-Similarity. Fractional Brownian Motion
  • Fractional Gaussian Noise: a Process with Strong Aftereffect

• Models Based on a Brownian Motion
  • Brownian Motion and its Role of a Basic Process
  • Brownian Motion: a Compendium of Classical Results
  • Stochastic Integration with respect to a Brownian Motion
  • Itô Processes and Itô's Formula
  • Stochastic Differential Equations
  • Forward and Backward Kolmogorov's Equations. Probabilistic Representation of Solutions

• Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices
  • Stochastic Interest Rates
  • Standard Diffusion Model of Stock Prices (Geometric Brownian Motion) and its Generalizations
  • Diffusion Models of the Term Structure of Prices in a Family of Bonds

• Semimartingale Models
  • Semimartingales and Stochastic Integrals
  • Doob–Meyer Decomposition. Compensators. Quadratic Variation
  • Itô's Formula for Semimartingales. Generalizations

The following sections are included:

• Empirical Data. Probabilistic and Statistical Models of Their Description. Statistics of 'Ticks'
  • Structural Changes in Financial Data Gathering and Analysis
  • Geography-Related Features of the Statistical Data on Exchange Rates
  • Description of Financial Indexes as Stochastic Processes with Discrete Intervention of Chance
  • On the Statistics of 'Ticks'

• Statistics of One-Dimensional Distributions
  • Discretizing Statistical Data
  • One-Diniensional Distributions of the Logarithms of Relative Price Changes. Deviation from the Gaussian Property and Leptokurtosis of Empirical Densities
  • One-Dimensional Distributions of the Logarithms of Relative Price Changes. 'Heavy Tails' and Their Statistics
  • One-Dimensional Distributions of the Logarithms of Relative Price Changes. Structure of the 'Central' Parts of Distributions

• Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices
  • Volatility. Definition and examples
  • Periodicity and Fractal Structure of Volatility in Exchange Rates
  • Correlation Properties
  • 'Devolatization'. Operational Time
  • 'Cluster' Phenomenon and Aftereffect in Prices

• Statistical -Analysis
  • Sources and Methods of -Analysis
  • -Analysis of Some Financial Time Series

The following sections are included:

• Investment Portfolio on a (B, S)-Market
  • Strategies Satisfying Balance Conditions
  • Notion of 'Hedging'. Upper and Lower Prices. Complete and Incomplete Markets
  • Upper and Lower Prices in a Single-Step Model
  • CRR-Model: an Example of a Complete Market

• Arbitrage-Free Market
  • 'Arbitrage' and 'Absence of Arbitrage'
  • Martingale Criterion of the Absence of Arbitrage. First Fundamental Theorem
  • Martingale Criterion of the Absence of Arbitrage. Proof of Sufficiency
  • Martingale Criterion of the Absence of Arbitrage. Proof of Necessity (by Means of the Esscher Conditional Transformation)
  • Extended Version of the First Fundamental Theorem

• Construction of Martingale Measures by Means of an Absolutely Continuous Change of Measure
  • Main Definitions. Density Process
  • Discrete Version of Girsanov's Theorem. Conditionally Gaussian Case
  • Martingale Property of the Prices in the Case of a Conditionally Gaussian and Logarithmically Conditionally Gaussian Distributions
  • Discrete Version of Girsanov's Theorem. General Case
  • Integer-Valued Random Measures and Their Compensators. Transformation of Compensators under Absolutely Continuous Changes of Measures. 'Stochastic Integrals'
  • 'Predictable' Criteria of Arbitrage-Free (B, S)-Markets

• Complete and Perfect Arbitrage-Free Markets
  • Martingale Criterion of a Complete Market. Statement of the Second Fundamental Theorem. Proof of Necessity
  • Representability of Local Martingales. 'S-Representability'
  • Representability of Local Martingales ('μ-Representability' and '(μ–ν)-Representability')
  • 'S-Representability' in the Binomial CRR-Model
  • Martingale Criterion of a Complete Market. Proof of Necessity for d = 1
  • Extended Version of the Second Fundamental Theorem

The following sections are included:

• European Hedge Pricing on Arbitrage-Free Markets
  • Risks and Their Reduction
  • Main Hedge Pricing Formula. Complete Markets
  • Main Hedge Pricing Formula. Incomplete Markets
  • Hedge Pricing on the Basis of the Mean Square Criterion
  • Forward Contracts and Futures Contracts

• American Hedge Pricing on Arbitrage-Free Markets
  • Optimal Stopping Problems. Supermartingale Characterization
  • Complete and Incomplete Markets. Supermartingale Characterization of Hedging Prices
  • Complete and Incomplete Markets. Main Formulas for Hedging Prices
  • Optional Decomposition

• Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage
  • One Model of 'Large' Financial Markets
  • Criteria of the Absence of Asymptotic Arbitrage
  • Asymptotic Arbitrage and Contiguity
  • Some Issues of Approximation and Convergence in the Scheme of Series of Arbitrage-Free Markets

• European Options on a Binomial (B ,S)-Market
  • Problems of Option Pricing
  • Rational Pricing and Hedging Strategies. Pay-Off Functions
  • Rational Pricing and Hedging Strategies. Markovian Pay-Off Functions
  • Standard Call and Put Options
  • Option-Based Strategies (Combinations and Spreads)

• American Options on a Binomial (B ,S)-Market
  • American Option Pricing
  • Standard Call Option Pricing
  • Standard Put Option Pricing
  • Options with Aftereffect. 'Russian Option' Pricing

The following sections are included:

• Investment Portfolio in Semimartingale Models
  • Admissible Strategies. Self-Financing. Stochastic Vector Integral
  • Discounting Processes
  • Admissible Strategies. Some Special Classes

• Semimartingale Models without Opportunities for Arbitrage. Completeness
  • Concept of Absence of Arbitrage and its Modifications
  • Martingale Criteria of the Absence of Arbitrage. Sufficient Conditions
  • Martingale Criteria of the Absence of Arbitrage. Necessary and Sufficient Conditions (a List of Results)
  • Completeness in Semimartingale Models

• Semimartingale and Martingale Measures
  • Canonical Representation of Semimartingales. Random Measures. Triplets of Predictable Characteristics
  • Construction of Marginal Measures in Diffusion Models. Girsanov's Theorem
  • Construction of Martingale Measures for Lévy Processes. Esscher Transformation
  • Predictable Criteria of the Martingale Property of Prices. I
  • Predictable Criteria of the Martingale Property of Prices. II
  • Representability of Local Martingales ('(H^c, μ–ν)-Representability')
  • Girsanov's Theorem for Semimartingales. Structure of the Densities of Probabilistic Measures

• Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock
  • Arbitrage and Conditions of its Absence. Completeness
  • Price of Hedging in Complete Markets
  • Fundamental Partial Differential Equation of Hedge Pricing

• Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Bonds
  • Models without Opportunities for Arbitrage
  • Completeness
  • Fundamental Partial Differentai Equation of the Term Structure of Bonds

The following sections are included:

• European Options in Diffusion (B, S)-Stockmarkets
  • Bachelier's formula
  • Black–Scholes Formula. Martingale Inference
  • Black–Scholes Formula. Inference Based on the Solution of the Fundamental Equation
  • Black–Scholes Formula. Case with Dividends

• American Options in Diffusion (B, S)-Stockmarkets. Case of an Infinite Time Horizon
  • Standard Call Option
  • Standard Put Option
  • Combinations of Put and Call Options
  • Russian Option

• American Options in Diffusion (B, S)-Stockmarkets. Finite Time Horizons
  • Special Features of Calculations on Finite Time Intervals
  • Optimal Stopping Problems and Stephan Problems
  • Stephan Problem for Standard Call and Put Options
  • Relations Between the Prices of European and American Options

• European and American Options in a Diffusion (B, )-Bondmarket
  • Option Pricing in a Bondmarket
  • European Option Pricing in Single-Factor Gaussian Models
  • American Option Pricing in Single-Factor Gaussian Models

The following sections are included:

• Bibliography

• Index

• Index of Symbols