Risk Neutral Pricing and Financial Mathematics

A Primer
Academic Press
  • 1. Auflage
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  • erschienen am 29. Juli 2015
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  • 348 Seiten
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978-0-12-801727-2 (ISBN)

Risk Neutral Pricing and Financial Mathematics: A Primer provides a foundation to financial mathematics for those whose undergraduate quantitative preparation does not extend beyond calculus, statistics, and linear math. It covers a broad range of foundation topics related to financial modeling, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, and term structure models, along with related valuation and hedging techniques. The joint effort of two authors with a combined 70 years of academic and practitioner experience, Risk Neutral Pricing and Financial Mathematics takes a reader from learning the basics of beginning probability, with a refresher on differential calculus, all the way to Doob-Meyer, Ito, Girsanov, and SDEs. It can also serve as a useful resource for actuaries preparing for Exams FM and MFE (Society of Actuaries) and Exams 2 and 3F (Casualty Actuarial Society).

  • Includes more subjects than other books, including probability, discrete and continuous time and space valuation, stochastic processes, equivalent martingales, option pricing, term structure models, valuation, and hedging techniques
  • Emphasizes introductory financial engineering, financial modeling, and financial mathematics
  • Suited for corporate training programs and professional association certification programs

Peter Knopf obtained his Ph.D. from Cornell University and subsequently taught at Texas A&M University and Rutgers University. He is currently Professor of Mathematics at Pace University. He has numerous research publications in both pure and applied mathematics. His recent research interests have been in the areas of difference equations and stochastic delay equation models for pricing securities.
  • Englisch
  • San Diego
  • |
  • USA
Elsevier Science
  • 5,67 MB
978-0-12-801727-2 (9780128017272)
0128017279 (0128017279)
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  • Front Cover
  • Risk Neutral Pricing and Financial Mathematics
  • Copyright Page
  • Dedication
  • Contents
  • About the Authors
  • Preface
  • 1 Preliminaries and Review
  • 1.1 Financial Models
  • 1.2 Financial Securities and Instruments
  • 1.3 Review of Matrices and Matrix Arithmetic
  • 1.3.1 Matrix Arithmetic
  • Matrix Arithmetic Properties
  • The Inverse Matrix
  • Illustration: The Gauss-Jordan Method
  • Illustration: Solving Systems of Equations
  • 1.3.2 Vector Spaces, Spanning, and Linear Dependence
  • Linear Dependence and Linear Independence
  • Illustrations: Linear Dependence and Independence
  • Spanning the Vector Space and the Basis
  • Illustration: Spanning the Vector Space and the Basis
  • 1.4 Review of Differential Calculus
  • 1.4.1 Essential Rules for Calculating Derivatives
  • The Power Rule
  • The Sum Rule
  • The Chain Rule
  • Product and Quotient Rules
  • Exponential and Log Function Rules
  • 1.4.2 The Differential
  • Illustration: The Differential and the Error
  • 1.4.3 Partial Derivatives
  • The Chain Rule for Two Independent Variables
  • 1.4.4 Taylor Polynomials and Expansions
  • 1.4.5 Optimization and the Method of Lagrange Multipliers
  • Illustration: Lagrange Optimization
  • 1.5 Review of Integral Calculus
  • 1.5.1 Antiderivatives
  • 1.5.2 Definite Integrals
  • Reimann Sums
  • 1.5.3 Change of Variables Technique to Evaluate Integrals
  • Illustration: Change of Variables Technique for the Indefinite Integral
  • Change of Variables Technique for the Definite Integral
  • 1.6 Exercises
  • Notes
  • 2 Probability and Risk
  • 2.1 Uncertainty in Finance
  • 2.2 Sets and Measures
  • 2.2.1 Sets
  • Illustration: Toss of Two Dice
  • Finite, Countable, and Uncountable Sets
  • 2.2.2 Measurable Spaces and Measures
  • 2.3 Probability Spaces
  • 2.3.1 Physical and Risk-Neutral Probabilities
  • Illustration: Probability Space
  • 2.3.2 Random Variables
  • Illustration: Discrete Random Variables
  • 2.4 Statistics and Metrics
  • 2.4.1 Metrics in Discrete Spaces
  • Expected Value, Variance, and Standard Deviation
  • Illustration
  • Co-movement Statistics
  • 2.4.2 Metrics in Continuous Spaces
  • Illustration: Distributions in a Continuous Space
  • Expected Value and Variance
  • 2.5 Conditional Probability
  • Illustration: Drawing a Spade
  • 2.5.1 Bayes Theorem
  • Illustration: Detecting Illegal Insider Trading
  • 2.5.2 Independent Random Variables
  • Illustration
  • Multiple Random Variables
  • 2.6 Distributions and Probability Density Functions
  • 2.6.1 The Binomial Random Variable
  • Illustration: Coin Tossing
  • Illustration: DK Trades
  • 2.6.2 The Uniform Random Variable
  • Illustration: Uniform Random Variable
  • 2.6.3 The Normal Random Variable
  • Calculating Cumulative Normal Density
  • Linear Combinations of Independent Normal Random Variables
  • 2.6.4 The Lognormal Random Variable
  • The Expected Value of the Lognormal Distribution
  • Illustration: Risky Securities
  • 2.6.5 The Poisson Random Variable
  • Deriving the Poisson Distribution
  • Illustration: Stock Price Jumps
  • Illustration: Bank Auditor
  • 2.7 The Central Limit Theorem
  • Illustration of the Central Limit Theorem
  • 2.8 Joint Probability Distributions
  • 2.8.1 Transaction Execution Delay Illustration
  • 2.8.2 The Bivariate Normal Distribution
  • 2.8.3 Calculating Cumulative Bivariate Normal Densities
  • Illustration: Calculating Bivariate Normal Densities
  • 2.9 Portfolio Mathematics
  • 2.9.1 Portfolio Arithmetic
  • 2.9.2 Optimal Portfolio Selection
  • Portfolio Selection Illustration
  • 2.10 Exercises
  • Notes
  • References
  • 3 Discrete Time and State Models
  • 3.1 Time Value
  • 3.1.1 Annuities and Growing Annuities
  • Illustration of Using Cash Flows to Value a Bond
  • Geometric Series and Expansions
  • Annuities and Perpetuities
  • Illustration of an annuity
  • Growing Annuities and Perpetuities
  • 3.1.2 Coupon Bonds and Yield Curves
  • The Term Structure of Interest Rates
  • Illustration: Term Structure
  • Term Structure Estimation with Coupon Bonds
  • Illustration: Coupon Bonds and the Yield Curve
  • Illustration: Pricing Bonds
  • 3.2 Discrete Time Models
  • 3.2.1 Arbitrage and No Arbitrage
  • No-Arbitrage Bond Markets
  • Illustration: No-Arbitrage Bond Markets
  • Pricing Bonds
  • Illustration: Pricing Bonds
  • The Pricing Kernel
  • 3.2.2 Arbitrage with Riskless Bonds
  • Replicating the Future Cash Flow Structure of Bond D
  • Creating the Arbitrage Portfolio
  • Illustration: Obtaining the Pricing Kernel
  • 3.3 Discrete State Models
  • 3.3.1 Outcomes, Payoffs, and Pure Securities
  • Payoff Vectors and Pure Securities
  • Illustration: Payoff Vectors and Pure Securities
  • Spanning and Complete Markets
  • Illustration: Spanning and Complete Markets
  • 3.3.2 Arbitrage and No Arbitrage Revisited
  • The Pricing Kernel
  • Illustration: Obtaining the Pricing Kernel
  • 3.3.3 Synthetic Probabilities
  • Discount Factors
  • The Risk-Neutrality Argument
  • 3.3.4 Binomial Option Pricing: One Time Period
  • 3.3.5 Put-Call Parity: One Time Period
  • 3.3.6 Completing the State Space
  • 3.4 Discrete Time-Space Models
  • Illustration: Multiple Time Periods and States
  • 3.5 Exercises
  • Notes
  • 4 Continuous Time and State Models
  • 4.1 Single Payment Model
  • 4.1.1 Forward Contracts
  • Simple Illustration: Forward Contract
  • 4.1.2 Forward Market Complications
  • Dividends
  • Foreign Exchange
  • Carry Costs
  • 4.1.3 Pricing a Zero-Coupon Bond with Continuous Deterministic Interest Rates
  • Illustration: Pricing a Zero-Coupon Bond with a Deterministic Continuous Rate
  • 4.2 Continuous Time Multipayment Models
  • 4.2.1 Differential Equations in Financial Modeling: An Introduction
  • Separable Differential Equations
  • Growth Models
  • Security Returns in Continuous Time
  • Illustration: Doubling an Investment Amount
  • Illustration: Mean Reverting Interest Rates
  • 4.2.2 Annuities and Growing Annuities
  • Annuities and Perpetuities
  • Perpetuities
  • Illustration: Continuous Annuity
  • Illustration: Continuous Dividend Streams
  • Growing Annuities
  • 4.2.3 Duration and Convexity in Continuous Time
  • Zero-Coupon Instruments
  • Coupon Instruments
  • Illustration: Duration and Convexity
  • 4.2.4 The Yield Curve in Continuous Time
  • 4.2.5 Term Structure Theories
  • 4.3 Continuous State Models
  • 4.3.1 Option Pricing: The Elements
  • Expected Values of European Options
  • Illustration: Call Options and Uniformly Distributed Stock Prices
  • 4.4 Exercises
  • Notes
  • References
  • 5 An Introduction to Stochastic Processes and Applications
  • 5.1 Random Walks and Martingales
  • 5.1.1 Stochastic Processes: A Brief Introduction
  • Illustration: Filtrations in a Two-Time Period Random Walk
  • 5.1.2 Random Walks and Martingales
  • Illustration: The Random Walk
  • Markov Processes and Independent Increments
  • Illustration: Families of Markov Processes
  • Martingales
  • Submartingales
  • 5.1.3 Equivalent Probabilities and Equivalent Martingale Measures
  • Numeraires
  • Equivalent Probability Measures
  • Equivalent Martingale Measures
  • Pricing with Submartingales
  • 5.2 Binomial Processes: Characteristics and Modeling
  • 5.2.1 Binomial Processes
  • Binomial Returns Process
  • Illustration: Binomial Outcome and Event Spaces
  • Pure Security Prices
  • Bond Prices
  • Physical Probabilities
  • The Equivalent Martingale Measure
  • Change of Numeraire and Martingales
  • 5.2.2 Binomial Pricing, Change of Numeraire, and Martingales
  • Pricing the Stock and Bond from Time 0 to Time 1
  • Verifying the Martingale Property
  • 5.2.3 Binomial Option Pricing
  • One-Time Period Case
  • Multitime Period Case
  • The Dynamic Hedge
  • Illustration: Binomial Option Pricing
  • One-Time Period Case
  • Extending the Binomial Model to Two Periods
  • 5.3 Brownian Motion and Itô Processes
  • 5.3.1 Brownian Motion Processes
  • Illustration: Brownian Motion
  • 5.3.2 Stopping Times
  • Illustration: Stock Price Hitting Times with Arithmetic Brownian Motion
  • 5.3.3 The Optional Stopping Theorem
  • High and Low Hitting Times
  • Illustration: Stock Price Hitting Times with Arithmetic Brownian Motion
  • Expected Stopping Time
  • Illustration: Expected Minimum Hitting Time
  • 5.3.4 Brownian Motion Processes with Drift
  • 5.3.5 Itô Processes
  • 5.4 Option Pricing: A Heuristic Derivation of Black-Scholes
  • 5.4.1 Estimating Exercise Probability in a Black-Scholes Environment
  • 5.4.2 The Expected Expiry Date Call Value
  • 5.4.3 Observations Concerning N(d1), N(d2), and c0
  • 5.5 The Tower Property
  • 5.6 Exercises
  • Notes
  • References
  • 6 Fundamentals of Stochastic Calculus
  • 6.1 Stochastic Calculus: Introduction
  • 6.1.1 Differentials of Stochastic Processes
  • An Example of a Real-Valued Differential from Ordinary Calculus
  • An Example of a Stochastic Differential
  • 6.1.2 Stochastic Integration
  • Contrasting Integration of Real-Valued Functions with Integration of Stochastic Processes
  • 6.1.3 Elementary Properties of Stochastic Integrals
  • Integral of a Stochastic Differential
  • Linearity
  • 6.1.4 More on Defining Stochastic Integrals
  • 6.1.5 Significant Results Based on Stochastic Integration
  • The Martingale Theorem
  • Itô Isometry
  • Characteristics of Integrals of Real-Valued Functions with Respect to Brownian Motion
  • 6.2 Change of Probability and the Radon-Nikodym Derivative
  • 6.2.1 The Radon-Nikodym Process
  • 6.2.2 The Radon-Nikodym Derivative
  • Illustration 1: The Radon-Nikodym Derivative and Two Coin Tosses
  • Illustration 2: Calculating Risk-Neutral Probabilities and the Radon-Nikodym Derivative
  • 6.2.3 The Radon-Nikodym Derivative and Binomial Pricing
  • Radon-Nikodym Derivatives in a Multiple Period Binomial Environment
  • 6.2.4 Radon-Nikodym Derivatives and Multiple Time Periods
  • Illustration: The Binomial Pricing Model
  • 6.2.5 Change of Normal Density
  • Illustration: Change of Normal Density
  • 6.2.6 More General Shifts
  • 6.2.7 Change of Brownian Motion
  • 6.3 The Cameron-Martin-Girsanov Theorem and the Martingale Representation Theorem
  • 6.3.1 Multiple Time Periods: Discrete Case to Continuous Case
  • 6.3.2 The Cameron-Martin-Girsanov Theorem
  • Illustration: Applying the Cameron-Martin-Girsanov Theorem
  • 6.3.3 The Martingale Representation Theorem
  • 6.4 Itô's Lemma
  • 6.4.1 A Discussion on Taylor Series Expansions
  • Taylor Series and Two Independent Variables
  • 6.4.2 The Itô Process
  • Demonstration that ?t Can Replace (?Zt)2 in the Differential ?y
  • Itô's Formula
  • 6.4.3 Itô's Lemma
  • 6.4.4 Applying Itô's Lemma
  • Illustration: Applying Itô's Lemma
  • 6.4.5 Application: Geometric Brownian Motion
  • Returns and Price Relatives
  • Itô's Formula: Numerical Illustration
  • 6.4.6 Application: Forward Contracts
  • 6.5 Exercises
  • Notes
  • References
  • 7 Derivatives Pricing and Applications of Stochastic Calculus
  • 7.1 Option Pricing Introduction
  • 7.2 Self-Financing Portfolios and Derivatives Pricing
  • 7.2.1 Introducing the Self-Financing Replicating Portfolio
  • 7.2.2 The Martingale Approach to Valuing Derivatives
  • Implementing the Change of Measure
  • Demonstrating the Self-Financing Characteristic
  • 7.2.3 Pricing a European Call Option and the Black-Scholes Formula
  • 7.3 The Black-Scholes Model
  • 7.3.1 Black-Scholes Assumptions
  • 7.3.2 Deriving Black-Scholes
  • 7.3.3 The Black-Scholes Model: A Simple Numerical Illustration
  • 7.4 Implied Volatility
  • 7.4.1 The Method of Bisection
  • 7.4.2 The Newton-Raphson Method
  • 7.4.3 Smiles, Smirks, and Aggregating Procedures
  • 7.5 The Greeks
  • Illustration: Greeks Calculations for Calls
  • Illustration: Greeks Calculations for Puts
  • 7.6 Compound Options
  • 7.6.1 Estimating Exercise Probabilities
  • 7.6.2 Valuing the Compound Call
  • Illustration: Valuing the Compound Call
  • 7.6.3 Put-Call Parity for Compound Options
  • 7.7 The Black-Scholes Model and Dividend Adjustments
  • 7.7.1 Lumpy Dividend Adjustments
  • The European Known Dividend Model
  • Modeling American Calls
  • Black's Pseudo-American Call Model
  • The Roll-Geske-Whaley Compound Option Formula
  • Illustration: Calculating the Value of an American Call on Dividend-Paying Stock
  • 7.7.2 Merton's Continuous Leakage Formula
  • Illustration: Continuous Dividend Leakage
  • 7.8 Beyond Plain Vanilla Options on Stock
  • 7.8.1 Exchange Options
  • Changing the Numeraire for Pricing the Exchange Call
  • Exchange Option Illustration
  • 7.8.2 Currency Options
  • Currency Option Illustration
  • 7.9 Exercises
  • Notes
  • References
  • 8 Mean-Reverting Processes and Term Structure Modeling
  • 8.1 Short- and Long-Term Rates
  • 8.1.1 Rates and Arithmetic Brownian Motion
  • 8.1.2 Rates and Geometric Brownian Motion
  • 8.2 Ornstein-Uhlenbeck Processes
  • 8.2.1 The Ornstein-Uhlenbeck Path
  • 8.2.2 Solving the Ornstein-Uhlenbeck Stochastic Differential Equation
  • Illustration: Conditional Expected Short Rates and the Vasicek Model
  • Expected Short Rates and Variances
  • 8.3 Single Risk Factor Interest Rate Models
  • 8.3.1 Pricing a Zero-Coupon Bond
  • The Interest Rate Process and Itô's Lemma
  • Setting the Self-Financing Portfolio Combinations
  • The Market Price of Risk and the Bond Pricing Differential Equation
  • The Martingale Approach
  • The Risk-Neutral Probability Space and the Vasicek Model
  • The Bond Pricing Formula
  • 8.3.2 The Yield Curve
  • Illustration: Mean-Reverting Interest Rates and Bond Pricing
  • 8.3.3 The Problem with Vasicek Models
  • 8.4 Alternative Interest Rate Processes
  • 8.4.1 The Merton Model
  • 8.4.2 The Cox, Ingersoll, and Ross Process
  • 8.4.3 Bond Pricing and the Yield Curve with CIR
  • The Yield Curve
  • Numerical Illustration
  • The Yield Curve
  • 8.4.4 Yield Curve Models: Summary and Further Development
  • 8.5 Where Do We Go from Here?
  • 8.6 Exercises
  • Notes
  • References
  • Appendix A: The z-table
  • Appendix B: Exercise Solutions
  • Appendix C: Glossary of Symbols
  • Glossary of Terms
  • Index
Chapter 1

Preliminaries and Review

Chapter 1 provides a concise review of essential prerequisite material for financial mathematics along with brief discussions of financial securities and markets. The securities and markets sections introduce equity, fixed income, and derivatives instruments along with the markets in which they trade along with essential financial practices such as arbitrage and short selling. Mathematics reviews include matrix arithmetic, vector spaces and span, differential and integral calculus, all illustrated with relevant examples drawn from finance. Matrices are applied to solving systems of equations. Taylor series and LaGrange optimization are discussed along with differentials and Riemann sums. All of these topics are reviewed with the goal of applying their principles to understanding discrete and continuous valuation, basic stochastic calculus, and their applications to valuing and managing equity, fixed income, and derivative instruments.


Financial models; market efficiency; arbitrage; derivative securities; matrix; vector; Gauss-Jordan method; linear independence; spanning set of vectors; derivative and differential; antiderivative; definite integral; Riemann sum

1.1 Financial Models

A model can be characterized as an artificial structure describing the relationships among variables or factors. Practically all of the methodology in this book is geared toward the development and implementation of financial models to solve financial problems. For example, valuation models provide a foundation for investment decision-making and models describing stochastic processes provide an important tool to account for risk in decision-making.

The use of models is important in finance because "real world" conditions that underlie financial decisions are frequently extraordinarily complicated. Financial decision-makers frequently use existing models or construct new ones that relate to the types of decisions they wish to make. Models proposing decisions that ought to be made are called normative models.1

The purpose of models is to simulate or behave like real financial situations. When constructing financial models, analysts exclude the "real world" conditions that seem to have little or no effect on the outcomes of their decisions, concentrating on those factors that are most relevant to their situations. In some instances, analysts may have to make unrealistic assumptions in order to simplify their models and make them easier to analyze. After simple models have been constructed with what may be unrealistic assumptions, they can be modified to match more closely "real world" situations. A good financial model is one that accounts for the major factors that will affect the financial decision (a good model is complete and accurate), is simple enough for its use to be practical (inexpensive to construct and easy to understand), and can be used to predict actual outcomes. A model is not likely to be useful if it is not able to project an outcome with an acceptable degree of accuracy. Completeness and simplicity may directly conflict with one another. The financial analyst must determine the appropriate trade-off between completeness and simplicity in the model he wishes to construct.

In finance, mathematical models are usually the easiest to develop, manipulate, and modify. These models are usually adaptable to computers and electronic spreadsheets. Mathematical models are obviously most useful for those comfortable with math; the primary purpose of this book is to provide a foundation for improving the quantitative preparation of the less mathematically oriented analyst. Other models used in finance include those based on graphs and those involving simulations. However, these models are often based on or closely related to mathematical models.

The concepts of market efficiency and arbitrage are essential to the development of many financial models. Market efficiency is the condition in which security prices fully reflect all available information. Such efficiency is more likely to exist when wealth-maximizing market participants can instantaneously and costlessly execute transactions as information is revealed. Transactions costs, irrationality, and poor execution systems reduce efficiency. Arbitrage, in its simplest scenario, is the simultaneous purchase and sale of the same asset, or more generally, the nearly simultaneous purchase and sale of assets generating nearly identical cash flow structures. In either case, the arbitrageur seeks to produce a profit by purchasing at a price that is less than the selling price. Proceeds of the sales are used to finance purchases such that the portfolio of transactions is self-financing, and that over time, no additional capital is devoted to or lost from the portfolio. Thus, the portfolio is assured a non-negative profit at each time period. The arbitrage process is riskless if purchase and sale prices are known at the times they are initiated. Arbitrageurs frequently seek to profit from market inefficiencies. The existence of arbitrage profits is inconsistent with market efficiency.

1.2 Financial Securities and Instruments

A security is a tradable claim on assets. Real assets contribute to the productive capacity of the economy; securities are financial assets that represent claims on real assets or other securities. Most securities are marketable to the general public, meaning that they can be sold or assigned to other investors in the open marketplace. Some of the more common types of securities and tradable instruments are briefly introduced in the following:

1. Debt securities: Denote creditorship of an individual, firm or other institution. They typically involve payments of a fixed series of interest (often known as coupon payments) or amounts towards principal along with principal repayment (often known as face value). Examples include:

 Bonds: Long-term debt securities issued by corporations, governments, or other institutions. Bonds are normally of the coupon variety (they make periodic interest payments on the principal) or pure discount (they are zero coupon instruments that are sold at a discount from face value, the bond's final maturity value).

 Treasury securities: Debt securities issued by the Treasury of the United States federal government. They are often considered to be practically free of default risk.

2. Equity securities (stock): Denote ownership in a business or corporation. They typically permit for dividend payments if the firm's debt obligations have been satisfied.

3. Derivative securities: Have payoff functions derived from the values of other securities, rates, or indices. Some of the more common derivative securities are:

 Options: Securities that grant their owners rights to buy (call) or sell (put) an underlying asset or security at a specific price (exercise price) on or before its expiration date.

 Forward and futures contracts: Instruments that oblige their participants to either purchase or sell a given asset or security at a specified price (settlement price) on the future settlement date of that contract. A long position obligates the investor to purchase the given asset on the settlement date of the contract and a short position obligates the investor to sell the given asset on the settlement date of the contract.

 Swaps: Provide for the exchange of cash flows associated with one asset, rate, or index for the cash flows associated with another asset, rate, or index.

4. Commodities: Contracts, including futures and options on physical commodities such as oil, metals, corn, etc. Commodities are traded in spot markets, where the exchange of assets and money occurs at the time of the transaction or in forward and futures markets.

5. Currencies: Exchange rates denote the number of units of one currency that must be given up for one unit of a second currency. Exchange transactions can occur in either spot or forward markets. As with commodities, in the spot market, the exchange of one currency for another occurs when the agreement is made. In a forward market transaction, the actual exchange of one currency for another actually occurs at a date later than that of the agreement. Spot and forward contract participants take one position in each of two currencies:

 Long: An investor has a "long" position in that currency that he will accept at the later date.

 Short: An investor has a "short" position in that currency that he must deliver in the transaction.

6. Indices: Contracts pegged to measures of market performance such as the Dow Jones Industrials Average or the S&P 500 Index. These are frequently futures contracts on portfolios structured to perform exactly as the indices for which they are named. Index traders also trade options on these futures contracts.

This list of security types is far from comprehensive; it only reflects some of those instruments that will be emphasized in this book. In addition, most of the instrument types will have many different variations.

1.3 Review of Matrices and Matrix Arithmetic

A matrix is simply an ordered rectangular array of numbers. A matrix is an entity that enables one to...

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