Preface
Main Abbreviations and Notations
Acknowledgments
Contents
1: Introduction: Economics and Organization of Financial Markets
1.1 The Role of Financial Markets
1.1.1 The Allocation of Cash Resources Over Time
1.1.2 Risk Allocation
1.1.3 The Market as a Supplier of Information
1.2 Securities as Sequences of Cash Flows
1.2.1 Definition of a Security (or Financial Asset)
1.2.2 Characterizing the Cash Flow Sequence
1.3 Equilibrium, Absence of Arbitrage Opportunity, Market Efficiency and Liquidity
1.3.1 Equilibrium and Price Setting
1.3.2 Absence of Arbitrage Opportunity (AAO) and the Notion of Redundant Assets
1.3.3 Efficiency
1.3.3.1 The Notion of Efficiency
1.3.3.2 Theoretical and Empirical Considerations
1.3.4 Liquidity
1.3.5 Perfect Markets
1.4 Organization, a Typology of Markets, and Listing
1.4.1 The Banking System and Financial Markets
1.4.2 A Simple Typology of Financial Markets
1.4.2.1 Primitive Spot Assets: Allocation of Cash
1.4.2.2 Derivative Product Markets: Risk Allocation
1.4.3 Market Organization
1.4.3.1 Over-the-Counter/Exchange Traded Markets
1.4.3.2 Intermediation
1.4.3.3 Centralized and Decentralized Markets
1.4.3.4 Quotation on an Exchange; Order Book, Fixing and Clearinghouse
Example of Order Book and Fixing
1.4.3.5 Primary Markets, Secondary Markets and Over-the-Counter (OTC)
1.5 Summary
Appendix: The World´s Principal Financial Markets
Stock markets, market indexes and interest rate instruments
Organized Derivative Markets (Futures and Options, Unless Otherwise Indicated)
Suggestion for Further Reading
Books
Articles
Part I: Basic Financial Instruments
2: Basic Finance: Interest Rates, Discounting, Investments, Loans
2.1 Cash Flow Sequences
Example 1
2.2 Transactions Involving Two Cash Flows
2.2.1 Transactions of Lending and Borrowing Giving Rise to Two Cash Flows over One Period
2.2.2 Transactions with Two Cash Flows over Several Periods
Example 2
Example 3
2.2.3 Comparison of Simple and Compound Interest
Example 4
Example 5
2.2.4 Two ``Complications´´ in Practice
Example 6
Example 7
Example 8 (Bank Discount)
Example 9
2.2.5 Continuous Rates
2.2.6 General Equivalence Formulas for Rates Differing in Convention and the Length of the Reference Period
Example 10
Example 11
2.3 Transactions Involving an Arbitrary Number of Cash Flows: Discounting and the Analysis of Investments
2.3.1 Discounting
Example 12
Example 13
2.3.2 Yield to Maturity (YTM), Discount Rate and Internal Rate of Return (IRR)
2.3.3 Application to Investment Selection: The Criteria of the NPV and the IRR
2.3.4 Interaction Between Investing and Financing, and Financial Leverage
Example 14
2.3.5 Some Guidelines for the Choice of an Appropriate Discount Rate
Example 15
2.3.6 Inflation, Real and Nominal Cash Flows and Rates
2.4 Analysis of Long-Term Loans
2.4.1 General Considerations and Definitions: YTM and Interest Rates
Example 16
Example 17
2.4.2 Amortization Schedule for a Loan
Example 18
Example 19
2.5 Summary
Appendix 1: Geometric Series and Discounting
Example 20
Example 21
Appendix 2: Using Financial Tables and Spreadsheets for Discount Computations
1. Financial Tables
Example 22
Suggested Reading
3: The Money Market and Its Interbank Segment
3.1 Interest Rate Practices and the Valuation of Securities
3.1.1 Interest Rate Practices on the Euro-Zone´s Money Market
Example 1 (Interests in arrears)
Example 2 (Interests in advance)
Example 3
Example 4
3.1.2 Alternative Practices and Conventions
3.2 Money Market Instruments and Operations
3.2.1 The Short-Term Securities of the Money Markets
3.2.2 Repos, Carry Trades, and Temporary Transfers of Claims
Example 5
3.2.3 Other Trades
3.3 Participants and Orders of Magnitude of Trades
3.3.1 The Participants
3.3.2 Orders of Magnitude
3.4 Role of the Interbank Market and Central Bank Intervention
3.4.1 Central Bank Money and the Interbank Market
3.4.1.1 Central Bank Money, Bank Reserves, and Interbank Settlement Payments
3.4.1.2 A Simple Analysis of the Macroeconomic Effects of Monetary Policy
3.4.1.3 The Role of the Interbank Market
3.4.2 Central Bank Interventions and Their Influence on Interest Rates
3.4.2.1 Open Market Refinancing
Example 6 (A stylized example of weekly refinancing operation with a variable rate)
3.4.2.2 Permanent Access to Central Bank Money (Standing Facilities)
3.5 The Main Monetary Indices
3.5.1 Indices Reflecting the Value of a Money-Market Rate on a Given Date
3.5.2 Indices Reflecting the Average Value of a Money-Market Rate During a Given Period
3.6 Summary
Suggestions for Further Reading
4: The Bond Markets
4.1 Fixed-Rate Bonds
4.1.1 Financial Characteristics and Yield to Maturity at the Date of Issue
Example 1
Example 2 (to be contrasted with Example 1)
4.1.2 The Market Bond Value at an Arbitrary Date; the Influence of Market Rates and of the Issuer´s Rating
4.1.2.1 Market Value and Interest Rate
Example 3
Example 4
4.1.2.2 Market Value, Credit Risk, and Rating
Example 5
4.1.3 The Quotation of Bonds
4.1.3.1 Definitions and Conventions
Example 6
4.1.3.2 Some Important Properties of Bond Quotes
Example 7
4.1.4 Bond Yield References and Bond Indices
4.2 Floating-Rate Bonds, Indexed Bonds, and Bonds with Covenants
4.2.1 Floating-Rate Bonds and Notes
4.2.2 Indexed Bonds
4.2.3 Bonds with Covenants (Optional Clauses)
4.2.3.1 Bonds Convertible to Shares
4.2.3.2 Bonds with a Detachable Warrant
4.3 Issuing and Trading Bonds
4.3.1 Primary and Secondary Markets
4.3.2 Treasury Bonds and Treasury Notes Issues: Reopening and STRIPS
4.3.2.1 Fungible Treasury Bonds and Reopening
Example 8
4.3.2.2 Stripping T-Bonds and Creating Zero-Coupon Bonds
4.4 International and Institutional Aspects; the Order of Magnitude of the Volume of Transactions
4.4.1 Brief Presentation of the International Bond Markets
4.4.1.1 General Considerations
4.4.1.2 The Euro-Bond Market
4.4.2 The Main National Markets
4.4.2.1 The American Market
4.4.2.2 European Bond Markets
4.4.2.3 The Japanese Market
4.4.2.4 Other Markets
4.5 Summary
Suggested Readings
5: Introduction to the Analysis of Interest Rate and Credit Risks
5.1 Interest Rate Risk
5.1.1 Introductory Examples: The Influence of the Maturity of a Security on Its Sensitivity to Interest Rates
Example 1
Example 2
5.1.2 Variation, Sensitivity and Duration of a Fixed-Income Security
5.1.2.1 Definitions
Example 3
5.1.2.2 Two Interesting Special Cases: Zero-Coupon Bonds and Perpetual Annuities
Example 4
5.1.2.3 Practical Computation of the Sensitivity and the Duration
Example 5
5.1.3 Alternative Expressions for the Variation, Sensitivity and Duration
5.1.3.1 Expressions for S and D as a Function of Proportional Rates
5.1.3.2 Expressions for S and D as Functions of Continuous Rates
5.1.3.3 A Simple Expression for the Sensitivity as a Function of Zero-Coupon Rates
5.1.4 Some Properties of Sensitivity and Duration
5.1.4.1 The Influence of Rates on Sensitivity and Duration
5.1.4.2 The Influence of the Passage of Time on S and D
5.1.5 The Sensitivity of a Portfolio of Assets and Liabilities or of a Balance Sheet: Sensitivity and Gaps
5.1.5.1 Interest Rate Risk of a Portfolio of Assets and Liabilities Evaluated at Market Value
5.1.5.2 Interest Rate Risk of a Balance Sheet Made Up of Assets and Liabilities Valued in Terms of the Principal Remaining Due
Example 6
Example 7
5.1.6 A More Accurate Estimate of Interest Rate Risk: Convexity
Example 8
Example 9
5.2 Introduction to Credit Risk
5.2.1 Analysis of the Determinants of the Credit Spread
Example 10
5.2.2 Simplified Modeling of the Credit Spread; the Credit Triangle
Example 11
Example 12
5.3 Summary
Appendix 1
Default Probability, Recovery Rate and Credit Spread
Suggested Reading
6: The Term Structure of Interest Rates
6.1 Spot Rates and Forward Rates
6.1.1 The Yield Curve
6.1.1.1 The Interest Rate as a Function of Its Maturity
6.1.1.2 Different Yield Curves for Different Markets and Definitions of Market Rates
6.1.2 Yields to Maturity and Zero-Coupon Rates
6.1.2.1 Bullet Bonds and YTM Curves
6.1.2.2 Zero-coupon Bonds and Rate Curves; Discount Factors
6.1.2.3 Estimating the Zero-coupon Rate Curve from a YTM Curve
Example 1
6.1.3 Forward Interest Rates Implicit in the Spot Rate Curve
6.1.3.1 Equations Involving the YTM
6.1.3.2 Alternative Relationships
6.1.3.3 Numerical Examples
Example 2
Example 3
Example 4
6.2 Factors Determining the Shape of the Curve
6.2.1 The Curve Shape
6.2.2 Expectations Hypothesis with Term Premiums
6.2.2.1 The Basic Hypothesis
6.2.2.2 Arguments for the Hypothesis
6.2.2.3 The Long-term Rate as the Geometric Mean of Anticipated Short-term Rates Augmented by Premiums
6.2.2.4 Implications for the Dynamics of the Yield Curve
6.2.2.5 The Effect of Expectations and Term Premiums
Example 5
6.2.3 Influence of the Credit Spread on Yield Curves
6.3 Analysis of Interest Rate Risk: Impact of Changes in the Slope and Shape of the Yield Curve
6.3.1 The Risk of a Change in the Slope of the Yield Curve
6.3.2 Multifactor Variation and Sensitivity and Models of Yield Curves
6.3.2.1 Analysis of Non-parallel Variations of the Zero-coupon Curve Using a Model
6.3.2.2 Examples of Yield Curve Models Applied to Interest Rate Risk Analysis
Example 6
6.3.2.3 Analysis of Non-parallel Variations in the YTM Curve
6.4 Summary
Suggested Readings
7: Vanilla Floating Rate Instruments and Swaps
7.1 Floating Rate Instruments
7.1.1 General Discussion and Notation
7.1.1.1 Definition and Basic Principles
7.1.1.2 Notation
7.1.1.3 Types of Floating-rate Assets and Main Reference Values
Example 1 Forward-Looking Rate
Example 2 Capitalized Euribor
Example 3 Backward-Looking (Post-Determined) Rate
Example 4 Backward-Looking Rate
Example 5
7.1.2 ``Replicable´´ Assets: Valuation and Interest Rate and Spread Risks
7.1.2.1 Valuation and Risk for Floating-rate Assets: Generalities
7.1.2.2 Analysis of Forward-Looking Rate Instruments Depending on a Money-Market Benchmark
Example 6 Value and Modified Duration of a Replicable FL Floater
7.1.2.3 Replicable Backward-Looking (Post-Determined) Money-Market Rates
7.1.2.4 Credit Risk, Spreads, Spread Risk
Example 7 Spread Risk
7.2 Vanilla Swaps
7.2.1 Definitions and Generalities About Swaps
7.2.1.1 Definition of a Standard Interest Rate Swap
Example 8 Overnight Indexed Swap (OIS)
7.2.1.2 Managing Interest Rate Risk with Swaps
Example 9 Transforming a Floating rate into a Fixed Rate
7.2.1.3 Benefiting from a Comparative Advantage: The Quality Spread Differential
Example 10 Benefiting from a Quality spread Differential
7.2.2 Replication and Valuation of an Interest Rate Swap
7.2.2.1 Replication of a Swap
7.2.2.2 Valuing a Replicable Swap
7.2.2.3 Examples of Valuing Swaps
Example 11 (Vanilla Forward-Looking Rate)
Example 12 (Vanilla Forward-Looking Floating Leg)
Example 13 Overnight-Indexed-Swap
7.2.3 Interest Rate, Counterparty and Credit Risks for an Interest Rate Swap
7.2.3.1 Interest Rate Risk: Modified Durations for a Fixed-for-Floating Interest Rate Swap
Example 14 Valuing and Assessing Interest Rate Risk for a Vanilla Interest Rate Swap
7.2.3.2 Intermediation and Counterparty Risk on Interest Rate Swaps
7.2.3.3 Credit Spread Risk on the Reference Rate and the LIBOR-OIS Spread
7.2.4 Summary of the Various Types of Swaps
7.2.4.1 Fixed-for-Floating Interest Rate Swaps
7.2.4.2 Currency Swaps
Example 15 Currency Swap (Currency-Interest Swap)
7.2.4.3 Basis Swaps (Floating-for-Floating)
7.2.4.4 Nonstandard Swaps
7.3 Summary
Appendix
Proof of the Equivalence Between Eq. (7.2´) and Proposition 1
Suggested Reading
Books
Articles
8: Stocks, Stock Markets, and Stock Indices
8.1 Stocks
8.1.1 Basic Notions: Equity, Stock Market Capitalization, and Share Issuing
8.1.1.1 General Considerations
Example 1
8.1.1.2 Some Definitions About Equity and Market Capitalization (Total and Floating)
8.1.1.3 Different Forms of Issue: Partnership Shares and Stocks
8.1.1.4 Listing and Initial Public Offering (IPO)
8.1.1.5 Reduction of Equity Capital and Share Repurchase
8.1.2 Analysis of Stock Issues, Dilution, and Subscription Rights
8.1.2.1 Impact of the Issue on Share Value and Market Capitalization
Example 2
8.1.2.2 Protection of Former Shareholders and Subscription Rights
Example 3
8.1.3 Market Performance of a Share and Adjusted Share Price
Example 4
8.1.4 Introduction to the Valuation of Firms and Shares; Interpretation and Use of the PER
8.1.4.1 Valuation Using Static or Asset-Based Methods
8.1.4.2 Dynamic Methods
8.1.4.3 The PER Method
Example 5
8.1.4.4 Mixed Methods
8.1.4.5 The Choice of the Discount Rate
8.2 Return Probability Distributions and the Evolution of Stock Market Prices
8.2.1 Stock Price on a Future Date, Stock Return, and Its Probability Distribution: Static Analysis
8.2.1.1 A Refresher on Return and Log-Return Calculations
8.2.1.2 Probability Distributions of Future Stock Prices and Returns
8.2.2 Modeling a Stock Price Evolution with a Stochastic Process: Dynamic Analysis
8.2.2.1 Representation of Price Evolution Using a Geometric Brownian Motion
8.2.2.2 Mean Return: Interest Rate and Risk Premium
Example 6
8.2.2.3 Volatility
8.3 Placing and Executing Orders and the Functioning of Stock Markets
8.3.1 Types of Orders
8.3.1.1 Limit Orders
8.3.1.2 Market Orders
8.3.1.3 Stop-Loss Orders
8.3.1.4 Futures
8.3.2 The Clearing and Settlement System
8.3.2.1 Transfer of Securities
8.3.2.2 Transfer of Cash: The Payment System
8.3.3 Investment Management
8.3.3.1 General Principles
8.3.3.2 Discretionary Management and the Investment Mandate
8.3.3.3 Collective Management and the Workings of Funds
8.3.4 The Main Stock Markets
8.4 Stock Market Indices
8.4.1 Composition and Calculation
8.4.1.1 The Composition of an Index
8.4.1.2 Weighting by Market Capitalizations (Total or Floating)
8.4.1.3 Other Weightings and Ways of Calculating the Index
8.4.2 The Main Indices
8.4.2.1 North American Indices
8.4.2.2 European Indices
8.4.2.3 Main Asian Indices
8.4.2.4 Main Worldwide Global Indices
8.5 Summary
Appendix 1
Skewness and Kurtosis of Log-Returns
Appendix 2
Modeling Volatility with ARCH and GARCH
Suggestions for Further Reading
Book Chapters
Articles
For an Online Comparative Description of Investment Funds from Different Countries
For an Online Description and Analysis of the Asset Management Industry
Part II: Futures and Options
9: Futures and Forwards
9.1 General Analysis of Forward and Futures Contracts
9.1.1 Definition of a Forward Contract: Terminology and Notation
9.1.1.1 General Definitions
9.1.1.2 Notation
9.1.1.3 Gains for the Buyer and the Seller
9.1.2 Futures Contracts: Comparison of Futures and Forward Contracts
9.1.2.1 Forwards and Futures
9.1.2.2 Comparison of (Pure) Forward and Futures Contracts
9.1.3 Unwinding a Position Before Expiration
9.1.4 The Value of Forward and Futures Contracts
9.2 Cash-and-carry and the Relation Between Spot and Forward Prices
9.2.1 Arbitrage, Cash-and-Carry, and Spot-Forward Parity
9.2.1.1 Cash-and-carry Arbitrage and Spot-Forward Parity: Fundamental Formulation
9.2.1.2 Alternative Formulations of the Spot-Forward Parity
9.2.2 Forward Prices, Expected Spot Prices, and Risk Premiums
9.3 Maximum and Optimal Hedging with Forward and Futures Contracts
9.3.1 Perfect or Maximum Hedging
9.3.1.1 A Model of Maximum Hedging
Example 1
9.3.1.2 Basis and Correlation Risks
Example 2
9.3.1.3 Hedging by Rolling Over Forward Contracts
9.3.2 Optimal Hedging and Speculation
9.4 The Main Forward and Futures Contracts
9.4.1 Contracts on Commodities
9.4.1.1 Brief Summary of Contracts and Markets
Example 3
9.4.1.2 Relation Between Forward and Spot Prices, Warehousing Cost, and Convenience Yield
Example 4
Example 5
9.4.2 Contracts on Currencies (Foreign Exchanges)
9.4.2.1 Brief Summary of Contracts and Markets
9.4.2.2 Analysis and Valuation
Example 6
9.4.3 Forward and Futures Contracts on Financial Securities (Stocks, Bonds, Negotiable Debt Securities), FRA, and Contracts on...
9.4.3.1 Brief Summary of Contracts and Markets
9.4.3.2 Analysis and Valuation of a Contract on Fixed-Income Instruments or Stocks
Example 7
9.4.3.3 Analysis of Forward Contracts on Fixed-Income Securities
9.4.3.4 Forward Rate Agreement (FRA)
Example 8
9.4.3.5 Forward Contracts on a Market Index
Example 9
9.5 Summary
Appendix
The Relationship Between Forward and Futures Prices
Suggestions for Further Reading
Books
Articles
10: Options (I): General Description, Parity Relations, Basic Concepts, and Valuation Using the Binomial Model
10.1 Basic Concepts, Call-Put Parity, and Other Restrictions from No Arbitrage
10.1.1 Definitions, Value at Maturity, Intrinsic Value, and Time Value
10.1.2 The Standard Call-Put Parity
10.1.3 Other Parity Relations
10.1.3.1 Call-Put Parity for European Options Written on an Underlying Spot Asset Paying Discrete Dividends
10.1.3.2 Call-Put Parity for European Options on Forward Contracts
10.1.4 Other Arbitrage Restrictions
10.1.4.1 Some Simple Relations Satisfied by European and American Options
10.1.4.2 Irrelevance of the Early Exercise Right for an American Call Written on a Spot, Non-dividend Paying Asset
Comments and Clarifications
10.1.4.3 Convexity of Option Prices with Respect to the Strike
10.2 A Pricing Model for One Period and Two States of the World
10.2.1 Two Markets, Two States
10.2.2 Hedging Strategy and Option Value in the Absence of Arbitrage
Example 1
10.2.3 The ``Risk-Neutral´´ Probability
10.2.4 The Risk Premium and the Market Price of Risk
10.3 The Multi-period Binomial Model
10.3.1 The Model Framework and the Dynamics of the Underlying´s Price
10.3.2 Risk-Neutral Probability and Martingale Processes
10.3.3 Valuation of an Option Using the Cox-Ross-Rubinstein Binomial Model
10.3.3.1 Recursive Backward Application of the One-Period Model
Example 2
10.3.3.2 A Closed-Form Solution for the Premiums of Calls and Puts
10.4 Calibration of the Binomial Model and Convergence to the Black-Scholes Formula
10.4.1 An Interpretation of Premiums in Terms of Probabilities of Exercise
10.4.2 Calibration and Convergence
10.4.2.1 Calibration of the Binomial Model
10.4.2.2 Convergence of the Binomial Model Results to Those of Black and Scholes
10.5 Summary
Appendix 1
Calibration of the Binomial Model
*Appendix 2
Suggestions for Further Reading
Books
Articles
11: Options (II): Continuous-Time Models, Black-Scholes and Extensions
11.1 The Standard Black-Scholes Model
11.1.1 The Analytical Framework and BS Model´s Assumptions
11.1.2 Self-Financing Dynamic Strategies
11.1.3 Pricing Using a Partial Differential Equation and the Black-Scholes Formula
11.1.3.1 The Fundamental Idea
11.1.3.2 The Partial Differential Equation for Pricing
11.1.3.3 The Black-Scholes Pricing Formula (1973)
Example 1
11.1.4 Probabilistic Interpretation
11.1.4.1 The Fundamental Idea
11.1.4.2 Price Dynamics in the Risk-Neutral Universe and the Value of an Option as an Expectation
11.1.4.3 Proof of Proposition 2 (Black-Scholes Formula) by Integration
11.2 Extensions of the Black-Scholes Formula
11.2.1 Underlying Assets That Pay Out (Dividends, Coupons, etc.)
11.2.1.1 Model with Continuous Dividends
Example 2
11.2.1.2 Model with a Discrete Dividend
Example 3
11.2.2 Options on Commodities
11.2.3 Options on Exchange Rates
11.2.4 Options on Futures and Forwards
Example 4
11.2.5 Variable But Deterministic Volatility
11.2.6 Stochastic Interest Rates: The Black-Scholes-Merton (BSM) Model
11.2.7 Exchange Options (Margrabe)
11.2.8 Stochastic Volatility (*)
11.2.8.1 Justification for the Model
11.2.8.2 The Heston Model (1993)
11.2.8.3 An Alternative Model
11.3 Summary
Appendix 1
Historical and Risk-Neutral Probabilities and Changes in Probability
Appendix 2
Changing the Probability Measure and the Numeraire
Definition and Examples
Existence of a Martingale Measure for Each Numeraire
Application to the Numeraire S
Appendix 3
Alternative Interpretations of the Black-Scholes Formula
Suggested Reading
Books
Articles
12: Option Portfolio Strategies: Tools and Methods
12.1 Basic Static Strategies
12.1.1 The General P&L Profile at Maturity
12.1.2 The Main Static Strategies
12.1.3 Replication of an Arbitrary Payoff by a Static Option Portfolio (*)
12.2 Historical and Implied Volatilities, Smile, Skew and Term Structure
12.2.1 Historical Volatility
12.2.2 The Implied Volatility
12.2.3 Smile, Skew, Term Structure, and Volatility Surface
12.2.3.1 Definitions and Use of the Volatility Surface and the Smile or Skew
12.2.3.2 Explanations for the Existence of the Volatility Term Structure and the Smile; The Method´s Coherence
12.3 Option Sensitivities (Greek Parameters)
12.3.1 The Delta (δ)
Example 1
12.3.2 The Gamma (Γ)
Example 2
12.3.3 The Vega (υ)
Example 3
12.3.4 The Theta (θ)
Example 4
12.3.5 The Rho (ρ)
Example 5
12.3.6 Sensitivity to the Dividend Rate
12.3.7 Elasticity and Risk-Expected Return Tradeoff
12.4 Dynamic Management of an Option Portfolio Using Greek Parameters
12.4.1 Variation in the Value of a Position in the Short Term and General Considerations
12.4.2 Delta-Neutral Management
12.4.2.1 Preliminaries
12.4.2.2 Impact of the Underlying´s Price Variation on a Delta-Neutral Position According to the Sign of Gamma
Example 6
12.4.2.3 Variation in the Value of a Delta-Neutral Position According to the Signs of Γ and θ
12.4.2.4 Taking into Account Variations in Volatility
12.4.2.5 Dynamic Pseudo-Arbitrages
12.4.2.6 Obtaining Greek Parameters of Any Sign
Example 7
12.4.3 A Tool for Risk Management: The P&L Matrix
Example 8 (Simplified)
12.5 Summary
Appendix 1
Computing Partial Derivatives (Greeks)
The Black-Scholes Model
Other Models
Appendix 2
Option Prices and the Underlying Price Probability Distribution
Appendix 3
Replication of an Arbitrary Payoff with a Static Option Portfolio
Suggestions for Further Reading
Books
Articles
13: American Options and Numerical Methods
13.1 Early Exercise and Call-Put Parity for American Options
13.1.1 Early Exercise of American Options
13.1.1.1 A Refresher on the Early Exercise of a Call Written on a Dividend Paying Asset
13.1.1.2 Early Exercise of an American Call Written on a Spot Underlying Paying a Single Discrete Dividend
Comments and Interpretations
13.1.1.3 Early Exercise of an American Call on a Spot Asset Paying a Continuous Dividend
13.1.1.4 Early Exercise of American Puts Written on a Spot Asset Paying a Continuous Dividend
13.1.1.5 American Put on a One-Dividend Paying Asset
13.1.1.6 Early Exercise of American Options Written on a Forward Contract
13.1.2 Call-Put ``Parity´´ for American Options
13.2 Pricing American Options: Analytical Approaches
13.2.1 Pricing an American Call on a Spot Asset Paying a Single Discrete Dividend or Coupon
13.2.1.1 Black´s Approximation
13.2.1.2 Pricing the Call on a Spot Asset Detaching A Single Dividend with a Compound Option
13.2.2 Pricing an American Option (Call and Put) on a Spot Asset Paying a Continuous Dividend or Coupon
13.2.2.1 The PDE Approach: The Free Boundary and the Linear Complementarity Formulations
13.2.2.2 Stopping Time Formulation
13.2.2.3 An Approximate Analytical Solution (Barone-Adesi and Whaley)
13.2.3 Prices of American and European Options: Orders of Magnitude
13.3 Pricing American Options with the Binomial Model
13.3.1 Binomial Dynamics of Price S: The Case of a Discrete Dividend
13.3.2 Binomial Dynamics of Price S: The Continuous Dividend Case
13.3.3 Pricing an American Option Using the Binomial Model
13.3.4 Improving the Procedure with a Control Variate
13.4 Numerical Methods: Finite Differences, Trinomial and Three-Dimensional Trees
13.4.1 Finite Difference Methods (*)
13.4.1.1 The Standard Implicit Method
13.4.1.2 The Implicit Method with a Free Boundary
13.4.2 Trinomial Trees
13.4.3 Three-Dimensional Trees Representing Two Correlated Processes
13.4.3.1 Construction of the Tree with Independent S1(t) and S2(t)
13.4.3.2 Construction of the Tree with S1(t) and S2(t) Correlated
13.5 Summary
Appendix 1
Proof of the Smooth Pasting (Tangency) Condition (13.5b)
Appendix 2
Orthogonalization of the Processes ln S1 and ln S2 and Construction of a Three-Dimensional Tree
Suggestion for Further Reading
Books
Articles
14: *Exotic Options
14.1 Path-Independent Options
14.1.1 The Forward Start Option (with Deferred Start)
14.1.2 Digital and Double Digital Options
Example 1
14.1.3 Multi-underlying (Rainbow) Options (*)
14.1.3.1 Exchange Options
14.1.3.2 Best of or Worst of Options
14.1.3.3 Options on the Minimum or on the Maximum
14.1.4 Options on Options or ``Compounds´´
14.1.5 Quantos and Compos
Example 2
14.1.5.1 The Quanto Call
Example 3
14.1.5.2 A Compo Call
Example 4
14.2 Path-Dependent Options
14.2.1 Barrier Options
14.2.1.1 Valuing Barrier Options
14.2.1.2 Value of Rebates
14.2.1.3 Other Barriers
14.2.2 Digital Barriers
14.2.3 Lookback Options (*)
14.2.4 Options on Averages (Asians)
14.2.4.1 Options on a Geometric Average Price
14.2.4.2 Options with a Geometric Average Strike
14.2.4.3 Options on Arithmetic Means
14.2.5 Chooser Options (*)
14.3 Summary
Appendix 1
**Value of a Compo Call
Appendix 2
**Lemmas on Hitting Probabilities for a Drifted Brownian Motion
Appendix 3
**Proof of the ``Inverses´´ Relation for Barrier Options
Appendix 4
**Valuing a Call Up-and-Out with L (Barrier) > K (Strike)
Appendix 5
**Valuing Rebates
Appendix 6
**Proof of the Price of a Lookback Call
Appendix 7
**Options on an Average Price
Appendix 8
**Options with an Average Strike
Suggestions for Further Reading
Books
Articles
15: Futures Markets (2): Contracts on Interest Rates
15.1 Notional Contracts
15.1.1 Basket of Deliverable Securities (DS) and Notional Security
15.1.2 The Euro-Bund Contract
15.1.2.1 Contract Description
15.1.2.2 Example (1) of Transactions for a Euro-Bund Contract Wound Up Before Its Expiry
15.1.3 Settlement and Conversion Factors
Example 2
15.1.4 Cheapest to Deliver and Quoting Futures at Expiration
15.1.4.1 Seller´s Choice and Quotation at Expiry
Example 3
15.1.4.2 Detailed Examination of the Cheapest to Deliver
Example 4
15.1.5 Arbitrage and Cash-Futures Relationship
15.1.5.1 Cash and Carry
15.1.5.2 Reverse Cash and Carry and Spot-Futures Parity
15.1.6 Interest Rate Sensitivity of Futures Prices
15.1.6.1 Parallel Shift of the Yield Curve
15.1.6.2 Multifactor Deformations of the Rate Curve
15.1.7 Hedging Interest Rate Risk Using Notional Bond Contracts
15.1.7.1 Hedging a Current Position
Example 5
15.1.7.2 Hedging an Expected, but Known, Position
Example 6
15.1.8 The Main Notional Contracts
15.1.8.1 Brief Description of the Main Medium- and Long-Term Notional Contracts
15.1.8.2 Contracts on Swap Notes
15.2 Short-Term Interest Rate Contracts (STIR) (3-Month Forward-Looking Rates and Backward-Looking Overnight Averages)
15.2.1 STIR 3-Month Contracts (LIBOR Type, Forward-Looking)
15.2.1.1 Quotation, General Description and Margin Calls for the 3-Months STIR Contracts
Example 7: Eurodollar Futures Transactions
15.2.1.2 Alternative Formulation and Definition of the Underlying Security
15.2.1.3 Forward-Looking STIR and FRA
15.2.1.4 The Main STIR Futures on 3-Month Forward-Looking Rates
15.2.2 Futures Contracts on an Average Overnight Rate
15.2.2.1 General Description of 3-Month Contracts on a Compound Average of Overnight Rates
15.2.2.2 Arbitrage and Prices of Overnight Rate Futures
Example 8: 3-Month SOFR Contracts (with Negative Interest Rates)
15.2.2.3 The Case of a Reference Period of Duration K Different from 0.25
15.2.2.4 The Main Futures Contracts on Overnight Rates Averages
15.2.3 Hedging Interest Rate Risk with STIR Contracts
15.2.3.1 Simple and Extended Durations
15.2.3.2 Hedge Ratios
Example 9: Hedging Future Borrowing
15.3 Summary
Appendices
1 Valuation of the Delivery Option
2 Relationship Between Forward and Futures Prices
Suggestions for Further Reading
Books
Articles
Internet Sites
16: Interest Rate Instruments: Valuation with the BSM Model, Hybrids, and Structured Products
16.1 Valuation of Interest Rate Instruments Using Standard Models
16.1.1 Principles of Valuation and the Black-Scholes-Merton Model Generalized to Stochastic Interest Rates
16.1.1.1 Valuation Principles
16.1.1.2 Revisiting the Generalized BSM or Gaussian Model
16.1.2 Valuation of a Bond Option Using the BSM-Price Model
Example 1
16.1.3 Valuation of the Right to a Cash Flow Expressed as a Function of a Rate and the BSM-Rate Model
16.1.3.1 Analysis of a Vanilla Cash Flow: Forward Rate and FN Expectation of a Spot Rate
16.1.3.2 Valuation of a Caplet or a Floorlet: the BSM-Rate Model
Example 2
16.1.3.3 Digital Option on a Rate
16.1.4 Convexity Adjustments for Non-vanilla Cash Flows (*)
16.1.4.1 Adjustment for Convexity
16.1.4.2 Application: Accounting for Time Lags
16.2 Nonstandard Swaps and Swaptions
16.2.1 Review of Swaps and Notation
16.2.2 Some Nonstandard Swaps
16.2.2.1 Forward Swaps (Forward Start)
Example 3
16.2.2.2 Step-Down (Amortization) Swaps
Example 4
16.2.2.3 In Arrears Swaps
Example 5
16.2.2.4 Constant Maturity Swaps
16.2.3 Swap Options (or Swaptions)
16.3 Caps and Floors
16.3.1 Vanilla Caps
16.3.1.1 Definition and Description
Example 6
Example 7
16.3.1.2 Valuation of a Vanilla Cap
16.3.2 A Vanilla Floor
16.3.2.1 Definition and Description
16.3.2.2 Valuation of a Floor
16.4 Static Replications and Combinations; Structured Contracts
16.4.1 Basic Instruments: Notation and General Remarks
16.4.1.1 Fundamental Instruments: Definitions and Notation
16.4.1.2 Redundancy Between a Swap, a Fixed-Rate Asset, and a Floating-Rate Asset
16.4.1.3 Redundancy Between a Cap, a Floor, and a Swap
16.4.2 Replication of a Capped or Floored Floating-Rate Instrument Using a Standard Asset Associated with a Cap or a Floor
16.4.2.1 Replication of a Floored Floating-Rate Instrument
Example 8
16.4.2.2 Replication of a Capped Floating-Rate Instrument
16.4.3 Collars
16.4.3.1 The Collar
Example 9
16.4.3.2 The Reverse Collar
16.4.4 Non-standard Caps and Floors
16.4.4.1 Cap Spread and Floor Spread
16.4.4.2 Caps and Floors with Steps
16.4.4.3 Caps and Floors with Barriers
16.4.4.4 Cap and Floor with a Contingent Premium
16.4.4.5 Other Non-standard Caps and Floors
16.4.5 Other Static Combinations; Structured Products; Contracts on Interest Rates with Profit-Sharing
16.4.5.1 Generalities
16.4.5.2 Structured Products on Interest Rates
16.4.5.3 Example of an Interest Rate Contract with Profit-Sharing
Example 10
16.5 Bonds with Optional Features and Hybrid Products
16.5.1 Convertible Bonds
16.5.1.1 General Description and Qualitative Analysis
16.5.1.2 Quantitative Analysis and Valuation
16.5.2 Other Bonds with Optional Features
16.5.2.1 Subscription Warrants for Shares and Warrants
16.5.2.2 Bonds with Share Subscription Warrants
Example 11
16.5.2.3 Bonds with Optional Features Disconnected from the Stock´s Performance
16.5.2.4 Other Types of Convertible Bonds
16.6 Summary
Appendix
The Qa-Martingale Measure
Suggestions for Further Reading
Books
Articles
17: Modeling Interest Rates and Options on Interest Rates
17.1 Models Based on the Dynamics of Spot Rates
17.1.1 One-Factor Models (Vasicek, and Cox, Ingersoll and Ross)
17.1.1.1 General Presentation and Analysis of One-Factor Models
17.1.1.2 The Vasicek Model (1977)
17.1.1.3 The Cox-Ingersoll-Ross Model (1985)
17.1.2 Fitting the Initial Yield Curve; the Hull and White Model
17.1.2.1 Fitting the Initial Yield Curve
17.1.2.2 The Hull and White Model (1990)
17.1.3 Multifactor Structures
17.2 Models Grounded on the Dynamics of Forward Rates
17.2.1 The Heath-Jarrow-Morton Model (1992)
17.2.1.1 Representation of the Yield Curve
17.2.1.2 General Dynamics of Forward Rates and ZC Bond Prices
17.2.1.3 Application 1: Valuation of Options on Bonds and on Bond Forwards
17.2.1.4 Application 2: Forward-Futures Relationship and Options on Bond Futures Contracts
17.2.2 The Libor (LMM) and Swap (SMM) Market Models
17.2.2.1 The Libor Market Model (LMM)
Example
17.2.2.2 The Swap Market Model (SMM)
17.2.2.3 Numerical Estimates and Extension of the Basic Models
17.3 Summary
Appendix 1
*The Vasicek Model
Appendix 2
*The LMM and SMM Models
1 Proof of Eq. (17.28), the Dynamics of L(t,Ti) Under the Final Forward Measure Qn
2 Valuation of Swaptions in the LMM Framework
*3 Three Probability Measures for the SMM Model
Suggestions for Further Reading
Books
Articles
18: Elements of Stochastic Calculus
18.1 Definitions, Notation, and General Considerations About Stochastic Processes
18.1.1 Notation
18.1.2 Stochastic Processes: Definitions, Notation, and General Framework
18.1.2.1 Probability Framework (Simplified)
18.1.2.2 Processes Without Memory: Markov Processes
18.1.2.3 Processes with Continuous Paths
18.2 Brownian Motion
18.2.1 The One-Dimensional Brownian Motion
18.2.1.1 Introduction: Discrete Time
18.2.1.2 Continuous Time
18.2.2 Calculus Rules Relative to Brownian Motions
18.2.3 Multi-dimensional Arithmetic Brownian Motions
18.3 More General Processes Derived from the Brownian Motion; One-Dimensional Itô and Diffusion Processes
18.3.1 One-Dimensional Itô Processes
18.3.2 One-Dimensional Diffusion Processes
Example 1. The Geometric Brownian Motion (GBM)
Example 2. The Ornstein-Uhlenbeck Process
18.3.3 Stochastic Integrals (*)
18.3.3.1 The Itô Process Case
18.3.3.2 The Case of Diffusion Processes
18.4 Differentiation of a Function of an Itô Process: Itô´s Lemma
18.4.1 Itô´s Lemma
18.4.2 Examples of Application
18.4.2.1 Geometric Brownian Motion
18.4.2.2 The Ornstein-Uhlenbeck Process
18.5 Multi-dimensional Itô and Diffusion Processes (*)
18.5.1 Multivariate Itô and Diffusion Processes
18.5.2 Itô´s Lemma (Differentiation of a Function of an n-Dimensional Itô Process)
18.5.2.1 Itô´s Lemma for a Multivariate Process X
18.5.2.2 The Dynkin Operator
18.6 Jump Processes
18.6.1 Description of Jump Processes
18.6.2 Modeling Jump Processes
18.7 Summary
Suggestions for Further Reading
Books
19: *The Mathematical Framework of Financial Markets Theory
19.1 General Framework and Basic Concepts
19.1.1 The Probabilistic Framework
19.1.2 The Market, Securities, and Portfolio Strategies
19.1.2.1 Primitive Securities
19.1.3 Portfolio Strategies
19.1.4 Contingent Claims, AAO, and Complete Markets
19.1.5 Price Systems
19.1.5.1 Viable Price Systems
19.1.5.2 Existence and Uniqueness of a Viable Price System
19.1.5.3 Generalization to Non-self-Financing Strategies and Contingent Securities
19.2 Price Dynamics as Itô Processes, Arbitrage Pricing Theory and the Market Price of Risk
19.2.1 Price Dynamics as Itô Processes
19.2.2 Arbitrage Pricing Theory in Continuous Time
19.2.3 Redundant Securities and Characterizing the Base of Primitive Securities
19.2.3.1 Redundant Securities
19.2.3.2 More on Primitive assets and Conditions for Pricing by Arbitrage
19.3 The Risk-Neutral Universe and Transforming Prices into Martingales
19.3.1 Martingales, Driftless Processes, and Exponential Martingales
19.3.1.1 Definition and an Example
19.3.1.2 Representing a martingale as a Driftless Itô Process
19.3.1.3 Return Dynamics and Exponential Martingales
19.3.2 Price and Return Dynamics in the Risk-Neutral Universe, Transforming Prices into martingales and Pricing Contingent Cla...
Example. The Standard Black-Scholes (BS) Model
19.3.3 Characterizing a Complete market and Market Prices of Risk
19.4 Change of Probability Measure, Radon-Nikodym derivative and Girsanov´s Theorem
19.4.1 Changing Probabilities and the Radon-Nikodym Derivative
19.4.2 Changing Probabilities and Brownian Motions: Girsanov´s Theorem
19.4.3 Formal Definition of RN Probabilities
19.4.4 Relations between Viable Price Systems, RN Probabilities, and MPR
19.4.4.1 Relationship between Π and
19.4.4.2 Relationship between Π, ΛP, and when Asset Prices Obey Itô Processes
19.4.4.3 The Case of Non-self-Financing Securities and Portfolios
19.5 Changing the Numeraire
19.5.1 Numeraires
19.5.1.1 Definition of a Numeraire
19.5.1.2 Examples of Numeraires
19.5.1.3 Properties of Numeraires
19.5.2 Numeraires and Probabilities that yield martingale Prices
19.5.2.1 Correspondence and Characterization of the Probabilities that Make Prices Denominated in numeraire N martingales (...
19.5.2.2 The Mapping and the Characterization of Numeraires
19.5.2.3 Volatility of numeraires and Market Prices of Risk in Complete Markets
19.6 The P-Numeraire (Optimal Growth or Logarithmic Portfolio)
19.6.1 Definition of the Portfolio (h, H) as the P-Numeraire
19.6.2 Characterization and Composition of the P-Numeraire Portfolio (h, H)
19.6.2.1 The Portfolio (h, H) Maximizes the Expectation of Logarithmic Utility
19.6.2.2 Other Properties of the P-Numeraire Portfolio
19.6.2.3 Composition, Volatility, and Dynamics of the Logarithmic Portfolio
19.6.2.4 P-Numeraire Portfolio and Radon-Nikodym Derivatives
Example
19.7 ** Incomplete Markets
19.7.1 MPR and the Kernel of the Diffusion Matrix (t)
19.7.1.1 Several Useful Results from Linear Algebra
19.7.1.2 Characterization of the Set ΛP of MPRs Compatible with AAO
19.7.1.3 Radon-Nikodym Derivatives
19.7.1.4 Decomposition of Random Variables; Replicable and Non-replicable Orthogonal Elements
19.7.1.5 P-Numeraires
19.7.2 Deflators
19.7.2.1 Deflators and the Pricing Kernel
19.7.2.2 Deflators, MPR, Radon-Nikodym Derivatives, and the P-Numeraire
19.8 Summary
Appendix
Construction of a One-to-one Correspondence between and Π
Suggestions for further reading
Books
Articles
20: The State Variables Model and the Valuation Partial Differential Equation
20.1 Analytical Framework and Notation
20.1.1 Dynamics of State Variables
20.1.2 The Asset Pricing Problem
20.2 Factor Decomposition of Returns
20.2.1 Expressing the Return dR as a Function of the dXj
20.2.2 Expressing the Return dR as a Function of the dWk
20.3 Expected Asset Returns and Arbitrage Pricing Theory (APT) in Continuous Time
20.3.1 First Formula for Expected Returns
20.3.2 Continuous Time APT in a State variables Model
20.4 The General valuation PDE
20.4.1 Derivation of the General valuation PDE
20.4.2 Market Prices of Risk and Risk Premia
20.4.3 The Relation between MPR and Excess Returns on Primitive Securities and the Condition for Market Completeness
Example of the Black-Scholes Model
20.5 Applications to the Term Structure of Interest Rates
20.5.1 Models with One State Variable
20.5.1.1 The Vasicek Model
20.5.1.2 The One-Factor Cox-Ingersoll-Ross Model
20.5.2 Multi-Factor models and valuation of Fixed-Income Securities
20.5.2.1 Models with Two State Variables; the Brennan and Schwartz Model (1979, 1982)
20.5.2.2 Multi-Factor Models; the APT Approach
20.5.2.3 Langetieg´s Multi-factor Model (1980)
20.6 Pricing in the Risk-Neutral Universe
20.6.1 Dynamics of Returns, of Brownian Motions and of State Variables in the Risk-Neutral Universe
20.6.2 The Valuation PDE
Examples
20.7 Discounting under Uncertainty and the Feynman-Kac Theorem
20.7.1 The Cauchy-Dirichlet PDE and the Feynman-Kac Theorem
20.7.2 Financial Interpretation of the Feynman-Kac Theorem and Discounting under Uncertainty
20.8 Summary
Appendix
Suggestions for Further Reading
Books
Part III: Portfolio Theory and Portfolio Management
21: Choice Under Uncertainty and Portfolio Optimization in a Static Framework: The Markowitz Model
21.1 Rational Choices Under Uncertainty: The Criteria of the Expected Utility and Mean-Variance
21.1.1 The Expected Utility Criterion
21.1.2 Some Features of Utility Functions
21.1.3 Risk Aversion and Concavity of the Utility Function
21.1.3.1 The Form of the Utility Function
21.1.3.2 Local Measure of the Degree of Risk Aversion
21.1.4 Some Standard Utility Functions
21.1.5 The Mean-Variance Criterion
21.1.5.1 Presentation of the Criterion
21.1.5.2 Mean-Variance Criterion and Expected Utility
21.2 Intuitive and Graphic Presentation of the Main Concepts of Portfolio Theory
21.2.1 Assumptions, General Framework and Efficient Portfolios
21.2.1.1 General Framework and Representation of Long and Short Positions
21.2.1.2 Efficient Portfolios
21.2.2 Two-Asset Portfolios
21.2.2.1 Notations and Analytic Forms of a Portfolio Return, Its Expected Value and Its Variance
Example 1
21.2.2.2 Geometric Representation of the Combinations of Two Assets
21.2.3 Portfolios with N Securities
21.2.3.1 First Case: All Assets Are Risky
21.2.3.2 Second Case: Existence of a Risk-Free Asset
Example 2 (risk-return trade-off)
21.2.4 Portfolio Diversification
21.2.4.1 General Considerations
21.2.4.2 Diversification in the Context of the Market Model (also Called Diagonal Model or Sharpe Model)
Example 3
21.3 Mathematical Analysis of Efficient Portfolio Choices
21.3.1 General Framework and Notations
21.3.1.1 Assets
21.3.1.2 Portfolios
21.3.1.3 Properties of the Variance-Covariance Matrix and Concept of Asset Redundancy
21.3.1.4 Definition of Efficient Portfolios
21.3.2 Efficient Portfolios and Portfolio Choice in the Absence of a Risk-Free Asset and of Portfolio Constraints
21.3.2.1 First Order Conditions and General Form of the Solution to (P)
21.3.2.2 Efficient Portfolios and Quadratic Investors
21.3.2.3 The Two-Fund Separation
21.3.3 Efficient Portfolios in the Presence of a Risk-Free Asset, with Allowed Short Positions; Tobin´s Two-Fund Separation
21.4 Some Extensions of the Standard Model and Alternatives
21.4.1 Problems Implementing the Markowitz Model; The Black-Litterman Procedure
21.4.2 Ban on Short Positions
21.4.2.1 Absence of a Risk-Free Asset
21.4.2.2 Presence of a Risk-Free Asset
21.4.3 Separation Results When Investors Maximize Expected Utility But Do Not Follow the Mean-Variance Criterion (Cass and Sti...
21.4.4 Loss Aversion and Introduction to Behavioral Finance
21.4.4.1 Loss Aversion
21.4.4.2 Elements of Behavioral Finance
21.5 Summary
Appendix 1: The Axiomatic of Von Neuman and Morgenstern and Expected Utility
A1.1 The Objects of Choice
A1.2 The Axioms Concerning Preferences
A1.3 The Expected Utility Criterion
A1.4 Notes and Complements
Appendix 2: A Reminder of Quadratic Forms and the Calculation of Gradients
Appendix 3: Expectations, Variances and Covariances-Definitions and Calculation Rules
A3.1 Definitions and Reminder
A3.2 Calculation Rules
Appendix 4: Reminder on Optimization Methods Under Constraints
A4.1 Optimization When the Constraints Take the Form of Equalities
A4.2 Optimization Under Inequality Constraints
Suggestions for Further Reading
Books
Articles
22: The Capital Asset Pricing Model
22.1 Derivation of the CAPM
22.1.1 Hypotheses
22.1.2 Intermediate Results in the Presence of a Risk-Free Asset
22.1.2.1 Tobin´s Separation Theorem
22.1.2.2 The Capital Market Line
22.1.3 The CAPM
22.1.3.1 Statement of the General CAPM
22.1.3.2 Black and Sharpe-Lintner-Treynor-Mossin CAPMs
22.1.3.3 Intuitive Justification of the Standard CAPM
Example 1
22.1.3.4 Interpretation of the CAPM
Example 2
22.1.3.5 The Equilibrium Price of Financial Assets
Example 3
22.2 Applications of the CAPM
22.2.1 Use of the CAPM for Financial Investment Purposes
Example 4
Example 5
22.2.2 Physical Investments by Firms
Example 6
22.2.3 Standard Performance Measures
22.2.3.1 The Sharpe Ratio
22.2.3.2 Jensen´s Alpha
Example 7
22.3 Extensions of the CAPM
22.3.1 Merton´s Intertemporal CAPM
22.3.2 International CAPM
22.4 Limits of the CAPM
22.4.1 Efficiency of the Market Portfolio and Roll´s Criticism
22.4.2 Stability of Betas
22.5 Tests of the CAPM
22.6 Summary
Suggestions for Further Reading
Books
23: Arbitrage Pricing Theory and Multi-factor Models
23.1 Multi-factor Models
23.1.1 Presentation of Models
23.1.2 Portfolio Management Models in Practice
23.2 Arbitrage Pricing Theory
23.2.1 Assumptions and Notations
23.2.2 The APT
23.2.2.1 Simplified Approach
Example 1
23.2.2.2 A More Rigorous Justification for APT
Example 2
23.2.3 Relationship with the CAPM
23.3 APT Applications and the Fama-French Model
23.3.1 Implementation of Multi-factor Models and APT
23.3.1.1 The Endogenous Method
23.3.1.2 The Exogenous Method
23.3.2 Portfolio Selection
23.3.3 The Three-Factor Model of Fama and French
23.4 Econometric Tests and Comparison of Models
23.4.1 Tests of the APT
23.4.2 Empirical and Practical CAPM-APT Comparison
23.4.3 Comparison of Factor Models
23.5 Summary
Appendix 1: Orthogonalization of Common Factors
Appendix 2: Compatibility of CAPM and APT
Example 3
Suggestions for Further Reading
Books
Articles
24: Strategic Portfolio Allocation
24.1 Strategic Asset Allocation Based on Common Sense Rules
24.1.1 Common Sense Rules
24.1.1.1 Consensual Rules Based on Common Sense and Reactions to Market Evolutions
24.1.1.2 Attempts to Rationalize Common Sense Rules, Puzzles, and Errors in Reasoning
24.1.2 Reactions to the Evolution of Market Conditions and of the Portfolio: Convex and Concave Strategies
24.2 Portfolio Insurance
24.2.1 The Stop Loss Method
24.2.2 Option-Based Portfolio Insurance
24.2.2.1 Portfolio Insurance with Long Puts or Replicated Puts
Example 1
24.2.2.2 Portfolio Insurance with Calls
24.2.2.3 A Special Case: Guaranteed Capital Fund
Example 2
24.2.3 CPPI Method
24.2.3.1 Presentation of the Method
Example 3
Example 4
24.2.3.2 Properties of the CPPI Strategy
24.2.3.3 Extensions of the CPPI Method
24.2.4 Variants and Extensions of the Basic Methods
24.2.5 Portfolio Insurance, Financial Markets Volatility and Stability
24.3 Dynamic Portfolio Optimization Models
24.3.1 Dynamic Strategies: General Presentation and Optimization Models
24.3.1.1 Presentation of the Problem and Notations
24.3.1.2 Dynamic Programming: Notations, Problems, and Principle
24.3.2 The Case of a Logarithmic Utility Function and the Optimal Growth Portfolio
24.3.3 The Merton Model
24.3.3.1 General Presentation of the Model and the General Form of the Solution
24.3.3.2 Principle of Separation into m + 2 Funds and Interpretation
24.3.3.3 Special Cases
24.3.4 The Model of Cox-Huang and Karatzas-Lehoczky-Shreve
24.3.4.1 The Notion of a Dynamically Complete Market
24.3.4.2 The Model
Example 7
24.4 Summary
Suggestions for Further Reading
Books
Articles
25: Benchmarking and Tactical Asset Allocation
25.1 Benchmarking
25.1.1 Definitions and Classification According to the Tracking Error
25.1.2 Pure Index Funds and Trackers
25.1.3 Replication Methods
25.1.4 Trackers or ETFs
25.2 Active Tactical Asset Allocation
25.2.1 Modeling and Solution to the Problem of an Active Manager Competing with a Benchmark
25.2.2 Analysis of the Performance of Active Portfolio Management: Empirical Information Ratio, Market Timing, and Security Pi...
25.2.3 Beta Coefficient Equal to 1
Example 1
25.2.4 Beta Coefficient Different from 1
25.2.5 Information Ratios, Sharpe Ratio, and Active Portfolio Management Theory
25.2.6 The Construction of a Maximum IR Portfolio from a Limited Number of Securities
25.2.7 The Construction of a Portfolio That Dominates the Benchmark (Higher Sharpe Ratio)
25.2.8 Synthesis, Interpretation and Application to Portfolio Management
25.3 Alternative Investment Management and Hedge Funds
25.3.1 General Description of Hedge Funds and Alternative Investment
25.3.2 Definition of the Main Alternative Investment Styles
25.3.3 The Interest of Alternative Investment
25.3.4 The Particular Difficulties of Measuring Performance in Alternative Investment
25.4 Summary
Appendix
Breakdown of the Tracking Error and Performance Attribution
Example 2
Suggestion for Reading
Books
Articles
Part IV: Risk Management, Credit Risk, and Credit Derivatives
26: Monte Carlo Simulations
26.1 Generation of a Sample from a Given Distribution Law
26.1.1 Sample Generation from a Given Probability Distribution
26.1.2 Construction of a Sample Taken from a Normal Distribution
26.2 Monte Carlo Simulations for a Single Risk Factor
26.2.1 Dynamic Paths Simulation of Y(t) and V(t, Y(t)) in the Interval (0, T)
Example 1
26.2.2 Simulations of Y(T) and V(T, Y(T)) at Time T (Static Simulations)
Example 2
26.2.3 Applications
26.2.3.1 Application 1: Calculation of VaR and ES (See Chap. 27)
26.2.3.2 Application 2: Evaluation of a European Option
Example 3
26.2.3.3 Application 3: Evaluation of a Path-Dependent Option
26.2.3.4 Application 4: Evaluation of the Greek Parameters of an Option
26.3 Monte Carlo Simulations for Several Risk Factors: Choleski Decomposition and Copulas
26.3.1 Simulation of a Multi-variate Normal Variable: Choleski Decomposition
26.3.2 Representation and Simulation of a Non-Gaussian Vector with Correlated Components Through the Use of a Copula
Example 4
26.3.3 General Definition of a Copula, and Student Copulas (*)
26.3.4 Simulation of Trajectories
Example 5. Simulations in a Three-Factor Model (Stochastic Price, Interest Rate, and Volatility)
26.4 Accuracy, Computation Time, and Some Variance Reduction Techniques
26.4.1 Antithetic Variables
26.4.2 Control Variate
Application 5 and Example 6
26.4.3 Importance Sampling
26.4.4 Stratified Sampling
26.5 Monte Carlo and American Options
26.5.1 General Description of the Problem and Methodology
26.5.2 Estimation of the Continuation Value by Regression (Carrière, Longstaff and Schwartz)
26.5.3 Overview of the Carrière Approach
26.5.4 Introduction to Longstaff and Schwartz Approach
Example 7
26.6 Summary
Suggestion for Further Reading
Books
Articles
27: Value at Risk, Expected Shortfall, and Other Risk Measures
27.1 Analytic Study of Value at Risk
27.1.1 The Problem of a Synthetic Risk Measure and Introduction to VaR
27.1.1.1 The Variance (or Standard Deviation) of Lh Is a First Measure of Risk
Example 1
27.1.1.2 A Quantile of the Probability Distribution of the Loss Lh as a Second Risk Measure; VaR Defined by Such a Quantile
Example 2
27.1.2 Definition of the VaR, Interpretations, and Calculation Rules
27.1.2.1 General Definition and Interpretations
27.1.2.2 Rules for Calculating with Quantiles of a Distribution
27.1.2.3 Alternative Expressions for the VaR
27.1.3 Analytic Expressions for the VaR in the Gaussian Case
27.1.3.1 Calculation of the VaR for a Gaussian Loss
Example 3
27.1.3.2 VaR Calculation When Vh Is Assumed Log-Normal
Example 4
27.1.3.3 Contribution of One Component to the VaR of a Portfolio
27.1.4 The Influence of Horizon h on the VaR of a Portfolio in the Absence or Presence of Serial Autocorrelation
27.1.4.1 In the Absence of AutoCorrelation
Example 5
Example 6
27.1.4.2 Serial Autocorrelation
27.2 Estimating the VaR
27.2.1 Preliminary Analysis and Modeling of a Complex Position
27.2.1.1 Standard Analysis
27.2.1.2 Representation of a Portfolio as a Combination of Elementary Standard Securities
27.2.1.3 Determining Risk Factors on Which the Value of the Portfolio Depends
27.2.1.4 Full Valuation and Partial Valuation
27.2.2 Estimating the VaR Through Simulations Based on Historical Data
27.2.2.1 Calculating the VaR of an Individual Asset
Example 7
27.2.2.2 The Case of a Portfolio of M Securities
Example 8
27.2.2.3 VaR of a Portfolio Whose Value Depends on Different Risk Factors
Example 9
Example 10
27.2.2.4 Reliability and Precision of the Empirical VaR
Example 11
27.2.3 Partial Valuation: Linear and Quadratic Approximations (the Delta-Normal and Delta-Gamma Methods)
27.2.3.1 General Sketch of the Linear Model (Delta-Normal Method)
27.2.3.2 Illustration of the Delta-Normal Method: RiskMetrics
Example 12
27.2.3.3 The Quadratic or Delta-Gamma Model
Example 13
27.2.4 Calculating the VaR Using Monte Carlo Simulations
Example 14
27.2.5 Comparison Between the Different Methods
27.3 Limitations and Drawbacks of the VaR, Expected Shortfall, Coherent Measures of Risk, and Portfolio Risks
27.3.1 The Drawbacks of VaR Measures
27.3.1.1 Technical Issues
27.3.1.2 Conceptual Difficulties
Example 15
27.3.2 An Improvement on the VaR: Expected Shortfall (or Tail-VaR, or C-VaR)
Example 16
Example 17
27.3.3 Coherent Risk Measures
27.3.3.1 Conditions for the Coherence of a Risk Measure
27.3.3.2 Construction of Coherent Risk Measures
27.3.4 Portfolio Risk Measures: Global, Marginal, and Incremental Risk
27.3.4.1 Portfolio Risk Measures
27.3.4.2 Risk Induced by a Component of a Portfolio: Marginal Risk, Contribution to Risk and Incremental Risk
27.4 Consequences of Non-normality and Analysis of Extreme Conditions
27.4.1 Non-normal Distributions with Fat Tails and Correlation at the Extremes
27.4.1.1 Skewness, Kurtosis, and the Cornish-Fisher Method of Computing a Quantile
Example 18
27.4.1.2 Correlation of Financial Variables Over the Extreme Ranges of Their Variation
27.4.1.3 Use of Copulas to Represent Non-Gaussian Multivariate Laws
27.4.2 Distributions of Extreme Values
27.4.2.1 Generalized Pareto Distributions
27.4.2.2 The Asymptotic Approximation of Distribution Tails
27.4.2.3 Estimation of the Parameters β and ξ
27.4.2.4 The Right-Hand Tail of the Loss Distribution L
27.4.2.5 Calculating the VaR and the Expected Shortfall (ES) from Extreme Distributions
Example 19
27.4.3 Stress Tests and Scenario Analysis
27.4.3.1 Developing Hypotheses and Scenarios
27.4.3.2 Analysis of the Consequences of Scenarios
27.5 Summary
Suggestions for Further Reading
Books
Articles
28: Modeling Credit Risk (1): Credit Risk Assessment and Empirical Analysis
28.1 Empirical Tools for Credit Risk Analysis
28.1.1 Reminder of Basic Concepts, Empirical Observations, and Notations
28.1.1.1 Basic Concepts and Notations
28.1.1.2 Empirical Observations on Yield Curves
28.1.2 Historical (Empirical) Default Probabilities and Transition Matrix
28.1.2.1 Historical Probabilities of Default
28.1.2.2 Transition Probabilities from One Rating to Another: The Transition Matrix
28.1.3 Risk-Neutral Default Probabilities Implicit in the Spread Curve and Discounting Methods in the Presence of Credit Risk
28.1.3.1 Risk-Neutral or Forward-Neutral Default Probabilities Implied in Credit Spreads
Example 1
28.1.3.2 Cash-Flow Discounting of a Fixed-Income Security Affected by Credit Risk
28.1.3.3 Discounting of a Random Cash-Flow Bearing Default or Counterparty Risk: Valuation of Derivatives Affected by Counterp...
Example 2
28.2 Modeling Default Events and Valuation of Securities
28.2.1 Reduced-Form Approach (Intensity Models)
28.2.1.1 Mathematical Tool: Generalized Poisson Process, and Default and Survival Probabilities
28.2.1.2 The Jarrow and Turnbull Model (1995)
28.2.1.3 Default Model with Nonconstant Recovery Rate: Duffie and Singleton model (1999)
28.2.2 Structural Approach: Merton´s Model and Barrier Models
28.2.2.1 The ``Seminal´´ Model (Merton´s Model (1974))
Example 3
28.2.2.2 Merton´s Model with Bankruptcy Costs
Example 4
28.2.2.3 Barrier Models (Dynamic Models)
28.2.2.4 Comparison, Merits, and Limitations of Default Models
28.2.3 A Practical Application: the Valuation of Convertible Bonds
28.2.3.1 Structural Approach
28.2.3.2 Intensity Model, with a Trinomial Tree Representing the Dynamics of S(t)
28.2.3.3 Evaluation with Monte Carlo simulations
Example 5
28.3 Summary
Appendix
Suggestions for Further Reading
Books
Articles
Website
29: Modeling Credit Risk (2): Credit-VaR and Operational Methods for Credit Risk Management
29.1 Determining the Credit-VaR of an Asset: Overview and General Principles
29.2 Empirical Credit-VaR of an Asset Based on the Migration Matrix
29.2.1 Computation of the Credit-VaR of an Individual Asset
Example 1 (Simplified)
29.2.2 Limitations of the Empirical Approach
29.3 Credit-VaR of an Individual Asset: Analytical Approaches Based on Asset Price Dynamics (MKMV) and on Structural Models
29.3.1 Asset Dynamics, Standardized Return, Default Probabilities, and Distance to Default
Example 2
29.3.2 Derivation of the Rating Migration Quantiles Associated with the Standardized Return
Example 3
29.3.3 Computation of the Distance to Default and Expected Default Frequency (MKMV-Moody´s Analytics Method)
29.3.4 Comparing the Two Approaches
29.3.5 Estimation of the Credit-VaR of an Asset Using EDF and a Valuation Model Based on RN-FN Probabilities
29.3.6 Relationship between Historical and RN Default Probabilities
29.4 Credit-VaR of an Entire Portfolio (Step 3) and Factor Models
29.4.1 Marked-to-Market (MTM) Models Involving Simulations
29.4.2 A Single-Factor DM Model of the Credit Risk of a Perfectly Diversified Portfolio (The Asymptotic Granular Vasicek-Gordy...
Example 4
29.4.3 Extensions of the Asymptotic Single-Factor Granular Model
29.4.4 Alternative Approach: Modeling the Default Dependence Structure with a Copula
29.4.5 Probability Distribution of the Default Dates Affecting a Portfolio
29.4.6 Portfolio Comprising Several Positions on the Same Obligor: Netting
29.5 Credit-VaR, Unexpected Loss and Economic Capital
29.5.1 Definition of Unexpected Loss (UL)
Example 5
29.5.2 Probability Threshold and Rating
29.6 Control and Regulation of Banking Risks
29.6.1 Regulators and the Basel Committee: General Presentation
29.6.2 Capital and liquidity Rules under Basel 3
29.6.3 Pillar 1 Capital Requirements under Basel 3
29.6.3.1 From Basel 2 to Basel 3
29.6.3.2 Specific Improvements on Required Capital Achieved by Basel 3: Buffers and Leverage Ratio
29.6.4 Details on Pillar 1 Liquidity Requirements
29.6.5 Additional Basel 3 Reflections and Reforms
29.6.5.1 Additional Improvements Sought for and Basel 3 Reforms
29.6.5.2 Limits Inherent in the Modeling of Economic Phenomena
29.7 Summary
Appendix 1. Correlation of Defaults in a Portfolio of Debt Assets
Example 6
Appendix 2. Regulatory Capital, Market VaR, and Backtesting
Appendix 3. Calculation of Regulatory Capital under the IRB Approach: Adjustment to the Infinitely Grained One-Factor Model
Suggestion for Further Reading
Books
Articles and Documentation
Websites
30: Credit Derivatives, Securitization, and Introduction to xVA
30.1 Credit Derivatives
30.1.1 General Principles and Description of Credit Default Swaps
30.1.1.1 Single-Name CDS: Basic Pay-off and Risk Transfer Mechanism
Example 1 Numerical Illustration of a Single-name CDS Mechanism
30.1.1.2 Common Contractual Terminology Regarding the CDS Market
30.1.2 Single-Name CDS Valuation Techniques
30.1.2.1 The Valuation of a Single-Name CDS
Basic Principle: Breaking Down the CDS into Two Legs
CDS Pricing at Inception
Example 2
Example 3
Par Spread and Value of a Single-Name CDS at any Time T
Example 4
JPMorgan Model: ISDA
Additional Provisions Regarding the CDS Recovery Rate
Specificities of CDS Hedging
30.1.2.2 Additional Elements on the Credit Derivatives Market
CDS Market: Some Key Contemporaneous Figures
Other Types of Credit Derivatives
CDS Index
Example 5
CDS Index Futures
Options on CDS (Credit Default Swaptions)
Total Return Swaps
Example 6 A TRS
Unfunded and Funded Credit Derivatives
30.2 Securitization
30.2.1 Introduction to Securitization and ABS
Example 7 Simple Securitization (Without Tranche Structuring)
30.2.2 ABS Tranching Structuration
Example 8 Securitization Structured in Tranches
30.3 The ``xVA´´ Framework
30.3.1 Counterparty Risk Exposure Measurement and Risk Mitigation Techniques
Example 9 Threshold and Minimum Transfer Amount
30.3.2 Counterparty Risk Exposure Modeling Techniques
30.3.3 Collateralized vs Non-collateralized Trades: Some Statistics
30.3.4 Introduction to CVA
30.3.5 Introduction to DVA
30.3.6 The FVA Puzzle
30.4 Summary
Appendix 1
Asset Swap Analysis
Example 10
Suggestion for Further Reading
Books
Articles
Website: defaultrisk.com.
Index
