Capital Market Finance: An Introduction to Primitive Assets, Derivatives, Portfolio Management and Risk

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