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ISBN 10: 0132993341
ISBN 13: 9780132993340
Author: John Hull
Directed primarily toward undergraduate finance students, this text also provides practical content to current and aspiring industry professionals.
Based on Hull’s Options, Futures and Other Derivatives, Fundamentals of Futures and Options Markets presents an accessible overview of the topic without the use of calculus. Packed with numerical examples and accounts of real-life situations, this text effectively guides readers through the material while helping them prepare for the working world.
Solution manual for Fundamentals of Futures and Options Markets 8th Editon Table of contents:
CHAPTER 1 Introduction
1.1 FUTURES CONTRACTS
Figure 1.1 A futures contract (assuming it is held to maturity)
1.2 HISTORY OF FUTURES MARKETS
The Chicago Board of Trade
The Chicago Mercantile Exchange
Electronic Trading
Business Snapshot 1.1 The Lehman Bankruptcy
1.3 THE OVER-THE-COUNTER MARKET
Business Snapshot 1.2 Systemic risk
Market Size
Figure 1.2 Size of over-the-counter and exchange-traded derivatives markets
1.4 FORWARD CONTRACTS
Table 1.1 Spot and forward quotes for the USD/GBP exchange rate, June 22, 2012 (GBP = British pound; USD = U.S. dollar; quote is number of USD per GBP)
1.5 OPTIONS
Table 1.2. Prices of call options on Google, June 25, 2012; stock price: bid $561.32; offer $561.51
Table 1.3 Prices of put options on Google, June 25, 2012; stock price: bid $561.32; offer $561.51
Figure 1.3 Net profit from (a) purchasing a contract consisting of 100 Google December call options with a strike price of $580 and (b) selling a contract consisting of 100 Google September put options with a strike price of $540
1.6 HISTORY OF OPTIONS MARKETS
Put and Call Brokers and Dealers Association
The Formation of Options Exchanges
The Over-the-Counter Market for Options
1.7 TYPES OF TRADER
1.8 HEDGERS
Hedging Using Forward Contracts
Business Snapshot 1.3 Hedge funds
Example 1.1 Hedging with forward contracts
Hedging Using Options
Example 1.2 Hedging with options
Figure 1.4 Value in Example 1.2 of the investor’s holding in two months
A Comparison
1.9 SPECULATORS
Speculation Using Futures
Table 1.4 Speculation using spot and futures contracts. One futures contract is on £62,500. Initial margin for four futures contracts = $20,000
Speculation Using Options
Table 1.5 Comparison of profits from two alternative strategies for using $2,000 to speculate on a stock worth $20 in October
Figure 1.5 Profit or loss from two alternative strategies for speculating on a stock currently worth $20
A Comparison
1.10 ARBITRAGEURS
Example 1.3 An arbitrage opportunity
1.11 DANGERS
SUMMARY
Business Snapshot 1.4 SocGen’s big loss in 2008
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 2 Mechanics of Futures Markets
2.1 OPENING AND CLOSING FUTURES POSITIONS
2.2 SPECIFICATION OF A FUTURES CONTRACT
The Asset
Business Snapshot 2.1 The unanticipated delivery of a futures contract
The Contract Size
Delivery Arrangements
Delivery Months
Price Quotes
Price Limits and Position Limits
2.3 CONVERGENCE OF FUTURES PRICE TO SPOT PRICE
Figure 2.1 Relationship between futures price and spot price as the delivery month is approached: (a) futures price above spot price; (b) futures price below spot price
2.4 THE OPERATION OF MARGIN ACCOUNTS
Daily Settlement
Table 2.1 Operation of margin account for a long position in two gold futures contracts. The initial margin is $6,000 per contract, or $12,000 in total; the maintenance margin is $4,500 per contract, or $9,000 in total. The contract is entered into on Day 1 at $1,650 and closed out on Day 16 at $1,626.90
Further Details
The Clearing House and Clearing Margin
Credit Risk
2.5 OTC MARKETS
Collateral
Business Snapshot 2.2 Long-Term Capital Management’s big loss
The Use of Clearing Houses in OTC Markets
Figure 2.2 (a) The traditional way in which OTC markets have operated: a series of bilateral agreements between market participants; (b) how OTC markets would operate with a single central clearing house.
Table 2.2 Futures quotes for a selection of CME Group contracts on commodities on July 13, 2012
2.6 MARKET QUOTES
Prices
Settlement Price
Trading Volume and Open Interest
Patterns of Futures
2.7 DELIVERY
Cash Settlement
2.8 TYPES OF TRADER AND TYPES OF ORDER
Orders
2.9 REGULATION
Trading Irregularities
2.10 ACCOUNTING AND TAX
Accounting
Example 2.1 Accounting treatment of a futures transaction
Tax
2.11 FORWARD vs. FUTURES CONTRACTS
Profits from Forward and Futures Contracts
Table 2.3 Comparison of forward and futures contracts
Foreign Exchange Quotes
Figure 2.3 Profit from (a) long and (b) short forward or futures position on £1 million
Example 2.2 Futures vs. forwards
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 3 Hedging Strategies Using Futures
3.1 BASIC PRINCIPLES
Short Hedges
Example 3.1 A short hedge
Long Hedges
Example 3.2 A long hedge
3.2 ARGUMENTS FOR AND AGAINST HEDGING
Hedging and Shareholders
Hedging and Competitors
Table 3.1 Danger in hedging when competitors do not hedge
Hedging Can Lead to a Worse Outcome
Business Snapshot 3.1 Hedging by gold mining companies
3.3 BASIS RISK
The Basis
Figure 3.1 Variation of basis over time
Choice of Contract
Illustrations
Example 3.3 Basis risk in a short hedge
Example 3.4 Basis risk in a long hedge
3.4 CROSS HEDGING
Calculating the Minimum Variance Hedge Ratio
Figure 3.2 Regression of change in spot price against change in futures price
Figure 3.3 Dependence of variance of hedger’s position on hedge ratio
Optimal Number of Contracts
Example 3.5 Calculation of the minimimum variance hedge ratio
Tailing the Hedge
Table 3.2 Data to calculate minimum variance hedge ratio when heating oil futures contract is used to hedge purchase of jet fuel
3.5 STOCK INDEX FUTURES
Stock Indices
Table 3.3 Futures quotes for a selection of CME Group contracts on stock indices on July 13, 2012
Hedging an Equity Portfolio
Table 3.4 Performance of stock index hedge
Reasons for Hedging an Equity Portfolio
Changing the Beta of a Portfolio
Locking in the Benefits of Stock Picking
3.6 STACK AND ROLL
Table 3.5 Data for the example on rolling oil hedge forward
SUMMARY
Business Snapshot 3.2 Metallgesellschaft: Hedging gone awry
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
APPENDIX Review of Key Concepts in Statistics and the CAPM
Standard Deviation
Correlation
Table 3A.1 Weights of finance textbooks and number of pages. S.D. = standard deviation
Linear Regression
Figure 3A.1 Best-fit relationship for data in Table 3A.1
Capital Asset Pricing Model
CHAPTER 4 Interest Rates
4.1 TYPES OF RATES
Treasury Rates
LIBOR
Repo Rates
The Risk-Free Rate
Business Snapshot 4.1 What Is the Risk-Free Rate?
4.2 MEASURING INTEREST RATES
Continuous Compounding
Table 4.1 Effect of the compounding frequency on the value of $100 at the end of one year when the interest rate is 10% per annum
4.3 ZERO RATES
Example 4.1 Changing the compounding frequency
4.4 BOND PRICING
Table 4.2 Treasury zero rates
Bond Yield
Par Yield
4.5 DETERMINING TREASURY ZERO RATES
Table 4.3 Data for bootstrap method
Table 4.4 Continuously compounded zero rates determined from data in Table 4.3
Figure 4.1 Zero rates given by the bootstrap method
4.6 FORWARD RATES
Table 4.5 Calculation of forward rates
Business Snapshot 4.2 Orange County’s yield curve plays
4.7 FORWARD RATE AGREEMENTS
Example 4.2 Cash flows from an FRA
Valuation
Example 4.3 Valuation of an FRA
4.8 THEORIES OF THE TERM STRUCTURE OF INTEREST RATES
The Management of Net Interest Income
Table 4.6 Example of rates offered by a bank to its customers
Table 4.7 Five-year rates are increased in an attempt to match maturities of assets and liabilities
Business Snapshot 4.3 Liquidity and the 2007–2009 Financial Crisis
Liquidity
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
APPENDIX Exponential and Logarithmic Functions
CHAPTER 5 Determination of Forward and Futures Prices
5.1 INVESTMENT ASSETS vs. CONSUMPTION ASSETS
5.2 SHORT SELLING
Table 5.1 Cash flows from short sale and purchase of shares
5.3 ASSUMPTIONS AND NOTATION
5.4 FORWARD PRICE FOR AN INVESTMENT ASSET
Table 5.2 Arbitrage opportunities when forward price is out of line with spot price for asset providing no income (asset price = $40; interest rate = 5%; maturity of forward contract = 3 months)
A Generalization
Example 5.1 Forward price of an asset providing no income
Business Snapshot 5.1 Kidder Peabody’s embarrassing mistake
What If Short Sales Are Not Possible?
5.5 KNOWN INCOME
Table 5.3 Arbitrage opportunities when 9-month forward price is out of line with spot price for asset providing known cash income (asset price = $900; income of $40 occurs at 4 months; 4-month and 9-month rates are 3% and 4% per annum)
A Generalization
Example 5.2 Forward price of an asset providing a known income
Example 5.3 Forward price of an asset providing a known yield
5.6 KNOWN YIELD
5.7 VALUING FORWARD CONTRACTS
Example 5.4 Valuing a forward contract
Business Snapshot 5.2 A systems error?
5.8 ARE FORWARD PRICES AND FUTURES PRICES EQUAL?
5.9 FUTURES PRICES OF STOCK INDICES
Business Snapshot 5.3 The CME Nikkei 225 futures contract
Example 5.5 Calculation of index futures price
Business Snapshot 5.4 Index arbitrage in October 1987
Index Arbitrage
5.10 FORWARD AND FUTURES CONTRACTS ON CURRENCIES
Figure 5.1 Two ways of converting 1,000 units of a foreign currency to dollars at time T. S0 is spot exchange rate; F0 is forward exchange rate; r and rf are dollar and foreign risk-free rates
Example 5.6 Arbitrage in forward and spot foreign exchange markets
Table 5.4 Futures quotes for a selection of CME Group contracts on foreign currencies on July 13, 2012
Example 5.7 Estimating interest rate differentials
A Foreign Currency as an Asset Providing a Known Yield
5.11 FUTURES ON COMMODITIES
Income and Storage Costs
Consumption Commodities
Example 5.8 Gold futures price
Convenience Yields
5.12 THE COST OF CARRY
5.13 DELIVERY OPTIONS
5.14 FUTURES PRICES AND EXPECTED SPOT PRICES
Keynes and Hicks
Risk and Return
The Risk in a Futures Position
Table 5.5 Relationship between futures price and expected future spot price
Normal Backwardation and Contango
SUMMARY
Table 5.6 Summary of results for a contract with time to maturity T on an investment asset with price S0 when the risk-free interest rate for a T-year period is r
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 6 Interest Rate Futures
6.1 DAY COUNT AND QUOTATION CONVENTIONS
Day Counts
Business Snapshot 6.1 Day counts can be deceptive
Price Quotations of U.S. Treasury Bills
Price Quotations of U.S. Treasury Bonds
6.2 TREASURY BOND FUTURES
Quotes
Conversion Factors
Table 6.1 Futures quotes for a selection of CME Group contracts on interest rates on July 13, 2012
Cheapest-to-Deliver Bond
Example 6.1 Choosing the cheapest-to-deliver bond
Business Snapshot 6.2 The wild card play
Determining the Futures Price
Example 6.2 Calculation of Treasury bond futures price
6.3 EURODOLLAR FUTURES
Table 6.2 Possible sequence of prices for December 2012 Eurodollar futures contract
Example 6.3 Use of Eurodollar futures for hedging
Forward vs. Futures Interest Rates
Example 6.4 Calculation of convexity adjustment
Convexity Adjustment
6.4 DURATION
Table 6.3 Convexity adjustment for the futures rate in Example 6.4
Table 6.4 Calculation of duration
Example 6.5 Testing the duration relationship
Modified Duration
Example 6.6 Testing the modified duration relationship
Bond Portfolios
Hedging Portfolios of Assets and Liabilities
Figure 6.1 Two bond portfolios with the same duration
Business Snapshot 6.3 Asset–liability management by banks
6.5 DURATION-BASED HEDGING STRATEGIES USING FUTURES
Hedging a Bond Portfolio
Example 6.7 Hedging a bond portfolio
Hedging a Floating-Rate Loan
Example 6.8 Hedging a floating-rate loan
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 7 Swaps
7.1 MECHANICS OF INTEREST RATE SWAPS
LIBOR
Illustration
Figure 7.1 Interest rate swap between Microsoft and Intel
Table 7.1 Cash flows (millions of dollars) to Microsoft in a $100 million three-year interest rate swap when a fixed rate of 5% is paid and LIBOR is received
Table 7.2 Cash flows (millions of dollars) from Table 7.1 when there is a final exchange of principal
Example 7.1 The versatility of swaps
Using the Swap to Transform a Liability
Figure 7.2 Microsoft and Intel use the swap to transform a liability
Figure 7.3 Microsoft and Intel use the swap to transform an asset
Using the Swap to Transform an Asset
Figure 7.4 Interest rate swap from Figure 7.2 when financial institution is involved
Role of Financial Intermediary
Market Makers
Figure 7.5 Interest rate swap from Figure 7.3 when financial institution is involved
Table 7.3 Bid and offer fixed rates in the swap market and swap rates (percent per annum); payments exchanged semiannually
7.2 DAY COUNT ISSUES
7.3 CONFIRMATIONS
7.4 THE COMPARATIVE-ADVANTAGE ARGUMENT
Business Snapshot 7.1 Extract from hypothetical swap confirmation
Table 7.4 Borrowing rates that provide a basis for the comparative-advantage argument
Figure 7.6 Swap agreement between AAACorp and BBBCorp when the rates in Table 7.4 apply
Figure 7.7 Swap agreement between AAACorp and BBBCorp when rates in Table 7.4 apply and a financial intermediary is involved
Criticism of the Argument
7.5 THE NATURE OF SWAP RATES
7.6 OVERNIGHT INDEXED SWAPS
7.7 VALUATION OF INTEREST RATE SWAPS
7.8 ESTIMATING THE ZERO CURVE FOR DISCOUNTING
Example 7.2 Valuing an interest rate swap using FRAs
Determining Zero Rates for LIBOR Discounting
Example 7.3 Determining zero rates from swaps
Determining Zero Rates for OIS Discounting
7.9 FORWARD RATES
Forward Rates When LIBOR Discounting Is Used
Forward Rates When OIS Discounting Is Used
Example 7.4 Bootstrapping LIBOR forward rates with LIBOR discounting
7.10 VALUATION IN TERMS OF BONDS
Example 7.5 Bootstrapping LIBOR forward rates with OIS discounting
Figure 7.8 Valuation of floating-rate bond when bond principal is L and next payment is k* at t*
Example 7.6 Valuation in terms of bonds for LIBOR discounting
7.11 TERM STRUCTURE EFFECTS
7.12 FIXED-FOR-FIXED CURRENCY SWAPS
Illustration
Figure 7.9 Value of forward rate agreements underlying a swap as a function of maturity. In (a) the term structure of interest rates is upward sloping and we receive fixed, or it is downward sloping and we receive floating; in (b) the term structure of interest rates is upward sloping and we receive floating, or it is downward sloping and we receive fixed.
Use of a Currency Swap to Transform Liabilities and Assets
Figure 7.10 A currency swap
Table 7.5 Cash flows to IBM in currency swap
Comparative Advantage
Table 7.6 Borrowing rates providing basis for currency swap
Figure 7.11 A currency swap motivated by comparative advantage
Figure 7.12 Alternative arrangement for currency swap: Qantas Airways bears some foreign exchange risk
Figure 7.13 Alternative arrangement for currency swap: General Electric bears some foreign exchange risk
7.13 VALUATION OF FIXED-FOR-FIXED CURRENCY SWAPS
Valuation in Terms of Bond Prices
Valuation as Portfolio of Forward Contracts
Example 7.7 Currency swap valuation in terms of bonds
7.14 OTHER CURRENCY SWAPS
Example 7.8 Currency swap valuation in terms of forward contracts
7.15 CREDIT RISK
Figure 7.14 The credit exposure in a swap
Business Snapshot 7.2 The Hammersmith and Fulham story
Central Clearing
Credit Default Swaps
7.16 OTHER TYPES OF SWAP
Variations on the Standard Interest Rate Swap
Other Currency Swaps
Equity Swaps
Options
Commodity Swaps, Volatility Swaps, etc.
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 8 Securitization and the Credit Crisis of 2007
8.1 SECURITIZATION
ABSs
Figure 8.1 An asset-backed security (simplified); bp = basis points (1bp = 0.01%)
Figure 8.2 The waterfall in an asset-backed security
ABS CDOs
Figure 8.3 Creation of ABSs and an ABS CDO from portfoloios of assets (simplified)
Table 8.1 Estimated losses to AAA-rated tranches of ABS CDO in Figure 8.3
8.2 THE U.S. HOUSING MARKET
The Relaxation of Lending Standards
Figure 8.4 The S&P/Case–Shiller Composite-10 index of U.S. real estate prices, 1987–2012
Subprime Mortgage Securitization
The Bubble Bursts
The Losses
The Credit Crisis
8.3 WHAT WENT WRONG?
Regulatory Arbitrage
Incentives
8.4 THE AFTERMATH
Business Snapshot 8.1 The Basel Committee
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 9 Mechanics of Options Markets
9.1 TYPES OF OPTION
Call Options
Put Options
Example 9.1 Profit from call option
Figure 9.1 Profit from buying a European call option on one share of a stock. Option price = $5; strike price = $100
Early Exercise
Example 9.2 Profit from put option
Figure 9.2 Profit from buying a European put option on one share of a stock. Option price = $7; strike price = $70
9.2 OPTION POSITIONS
Figure 9.3 Profit from writing a European call option on one share of a stock. Option price = $5; strike price = $100
Figure 9.4 Profit from writing a European put option on one share of a stock. Option price = $7; strike price = $70
Figure 9.5 Payoffs from positions in European options: (a) long call, (b) short call, (c) long put, (d) short put. Strike price = K; price of asset at maturity = ST
9.3 UNDERLYING ASSETS
Stock Options
Foreign Currency Options
Index Options
Futures Options
9.4 SPECIFICATION OF STOCK OPTIONS
Expiration Dates
Strike Prices
Terminology
Flex Options and Other Nonstandard Products
Business Snapshot 9.1 Gucci Group’s large dividend
Dividends and Stock Splits
Example 9.3 Impact on option terms of a stock split
Example 9.4 Impact on option terms of a stock dividend
Position Limits and Exercise Limits
9.5 TRADING
Market Makers
Offsetting Orders
9.6 COMMISSIONS
Table 9.1 A sample commission schedule for a discount broker
9.7 MARGIN REQUIREMENTS
Writing Naked Options
Example 9.5 Margin calculations for a naked call option
Other Rules
9.8 THE OPTIONS CLEARING CORPORATION
Exercising an Option
9.9 REGULATION
9.10 TAXATION
Wash Sale Rule
Constructive Sales
9.11 WARRANTS, EMPLOYEE STOCK OPTIONS, AND CONVERTIBLES
Business Snapshot 9.2 Tax planning using options
9.12 OVER-THE-COUNTER OPTIONS MARKETS
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 10 Properties of Stock Options
10.1 FACTORS AFFECTING OPTION PRICES
Table 10.1 Summary of the effect on the price of a stock option of increasing one variable while keeping all others fixed*
Stock Price and Strike Price
Time to Expiration
Volatility
Figure 10.1 Effect of changes in stock price, strike price, and expiration date on option prices when S0 = 50, K = 50, r = 5%, σ = 30%, and T = 1
Figure 10.2 Effect of changes in volatility and risk-free interest rate on option prices when S0 = 50, K = 50, r = 5%, σ = 30%, and T = 1
Risk-Free Interest Rate
Dividends
10.2 ASSUMPTIONS AND NOTATION
10.3 UPPER AND LOWER BOUNDS FOR OPTION PRICES
Upper Bounds
Lower Bound for Calls on Non-Dividend-Paying Stocks
Example 10.1 Call option price too low
Lower Bound for Puts on Non-Dividend-Paying Stocks
Example 10.2 Lower bound for call option
Example 10.3 Put option price too low
Example 10.4 Lower bound for put option
10.4 PUT–CALL PARITY
Table 10.2 Values of Portfolio A and Portfolio C at time T
Table 10.3 Arbitrage opportunities when put–call parity does not hold. Stock price = $31; interest rate = 10%; call price = $3. Both put and call have strike price of $30 and three months to maturity
Business Snapshot 10.1 Put–call parity and capital structure
American Options
Example 10.5 Relation between American call and put options
10.5 CALLS ON A NON-DIVIDEND-PAYING STOCK
Bounds
10.6 PUTS ON A NON-DIVIDEND-PAYING STOCK
Figure 10.3 Bounds for European and American call options when there are no dividends
Figure 10.4 Variation of price of an American or European call option on a non-dividend-paying stock with the stock price S0. Curve moves in the direction of the arrows when there is an increase in the interest rate, the time to maturity, or the stock price volatility
Bounds
Figure 10.5 Bounds for European and American put options when there are no dividends
Figure 10.6 Variation of price of an American put option with stock price. Curve moves in the direction indicated by the arrows when the time to maturity or stock price volatility increases or when the interest rate decreases
Figure 10.7 Variation of price of a European put option with the stock price
10.7 EFFECT OF DIVIDENDS
Lower Bound for Calls and Puts
Early Exercise
Put–Call Parity
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 11 Trading Strategies Involving Options
11.1 PRINCIPAL-PROTECTED NOTES
Example 11.1 Creation of a principal-protected note
11.2 STRATEGIES INVOLVING A SINGLE OPTION AND A STOCK
Figure 11.1 Profit patterns (a) long position in a stock combined with short position in a call; (b) short position in a stock combined with long position in a call; (c) long position in a put combined with long position in a stock; (d) short position in a put combined with short position in a stock
Figure 11.2 Profit from bull spread created using call options
11.3 SPREADS
Bull Spreads
Table 11.1 Payoff from a bull spread created using calls
Example 11.2 Bull spread using call options
Figure 11.3 Profit from bull spread created using put options
Figure 11.4 Profit from bear spread created using put options
Bear Spreads
Table 11.2 Payoff from a bear spread created with put options
Example 11.3 Bear spread using put options
Figure 11.5 Profit from bear spread created using call options
Box Spreads
Table 11.3 Payoff from a box spread
Business Snapshot 11.1 Losing money with box spreads
Butterfly Spreads
Figure 11.6 Profit from butterfly spread using call options
Table 11.4 Payoff from a butterfly spread
Calendar Spreads
Figure 11.7 Profit from butterfly spread using put options
Figure 11.8 Profit from calendar spread created using two calls when T2 > T1, calculated at the time when the short maturity call expires
Figure 11.9 Profit from calendar spread created using two puts when T2 > T1, calculated at the time when the short maturity put expires
Diagonal Spreads
11.4 COMBINATIONS
Straddle
Figure 11.10 Profit from a straddle
Table 11.5 Payoff from a straddle
Strips and Straps
Figure 11.11 Profit from a strip and a strap
Business Snapshot 11.2 How to make money from trading straddles
Strangles
Figure 11.12 Profit from a strangle
Table 11.6 Payoff from a strangle
11.5 OTHER PAYOFFS
SUMMARY
Figure 11.13 “Spike payoff” from a butterfly spread that can be used as a building block to create other payoffs
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 12 Introduction to Binomial Trees
12.1 A ONE-STEP BINOMIAL MODEL AND A NO-ARBITRAGE ARGUMENT
Figure 12.1 Stock price movements in numerical example
A Generalization
Figure 12.2 Stock and option prices in a general one-step tree
Irrelevance of the Stock’s Expected Return
12.2 RISK-NEUTRAL VALUATION
The One-Step Binomial Example Revisited
Real World vs. Risk-Neutral World
12.3 TWO-STEP BINOMIAL TREES
Figure 12.3 Stock prices in a two-step tree
Figure 12.4 Stock and option prices in a two-step tree. The upper number at each node is the stock price; the lower number is the option price
Figure 12.5 Evaluation of option price at node B of Figure 12.4
A Generalization
Figure 12.6 Stock and option prices in general two-step tree
12.4 A PUT EXAMPLE
Figure 12.7 Use of two-step tree to value European put option. At each node the upper number is the stock price; the lower number is the option price
12.5 AMERICAN OPTIONS
Figure 12.8 Use of two-step tree to value American put option. At each node the upper number is the stock price; the lower number is the option price
12.6 DELTA
12.7 DETERMINING u AND d
Figure 12.9 Two-step tree to value a two-year American put option when stock price is 50, strike price is 52, risk-free rate is 5%, and volatility is 30%
12.8 INCREASING THE NUMBER OF TIME STEPS
12.9 USING DerivaGem
12.10 OPTIONS ON OTHER ASSETS
Options on Stocks Paying a Continuous Dividend Yield
Options on Stock Indices
Options on Currencies
Options on Futures
Example 12.1 Option on a stock index
Example 12.2 Option on a foreign currency
Example 12.3 Option on futures
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
APPENDIX Derivation of Black–Scholes–Merton Option Pricing Formula from Binomial Tree
CHAPTER 13 Valuing Stock Options: The Black–Scholes–Merton Model
13.1 ASSUMPTIONS ABOUT HOW STOCK PRICES EVOLVE
The Lognormal Distribution
Figure 13.1 A lognormal distribution
Figure 13.2 A normal distribution
Example 13.1 Confidence limits, mean, and variance for a future stock price
Example 13.2 Confidence limits for stock price return
13.2 EXPECTED RETURN
Business Snapshot 13.1 Mutual fund returns can be misleading
13.3 VOLATILITY
13.4 ESTIMATING VOLATILITY FROM HISTORICAL DATA
Example 13.3 Calculation of volatility from historical data
Trading Days vs. Calendar Days
Business Snapshot 13.2 What causes volatility?
13.5 ASSUMPTIONS UNDERLYING BLACK–SCHOLES–MERTON
Table 13.1 Computation of volatility
13.6 THE KEY NO-ARBITRAGE ARGUMENT
Figure 13.3 Relationship between call price and stock price. Current stock price is S0
13.7 THE BLACK–SCHOLES–MERTON PRICING FORMULAS
Properties of the Black–Scholes–Merton Formulas
Figure 13.4 Shaded area represents N(x)
Example 13.4 Using the Black–Scholes–Merton formulas
Understanding N(d1) and N(d2)
13.8 RISK-NEUTRAL VALUATION
Application to Forward Contracts
13.9 IMPLIED VOLATILITIES
The VIX Index
Example 13.5 Trading VIX Futures
Figure 13.5 The VIX index: January 2004 to September 2012
13.10 DIVIDENDS
European Options
Example 13.6 Using Black–Scholes–Merton when there are dividends
American Call Options
Example 13.7 Using Black’s approximation for an American call
Black’s Approximation
SUMMARY
FURTHER READING
On the Black–Scholes–Merton model and its extensions
On the causes of volatility
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
APPENDIX The Early Exercise of American Call Options on Dividend-Paying Stocks
Example 13A.1 Test of whether a call option should ever be exercised early
CHAPTER 14 Employee Stock Options
14.1 CONTRACTUAL ARRANGEMENTS
The Early Exercise Decision
14.2 DO OPTIONS ALIGN THE INTERESTS OF SHAREHOLDERS AND MANAGERS?
14.3 ACCOUNTING ISSUES
Nontraditional Option Plans
14.4 VALUATION
Example 14.1 A popular approach for valuing employee stock options
Business Snapshot 14.1 Employee stock options and dilution
Dilution
14.5 BACKDATING SCANDALS
Figure 14.1 Erik Lie’s results providing evidence of backdating (reproduced, with permission, from www.biz.uiowa.edu/faculty/elie/backdating.htm)
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 15 Options on Stock Indices and Currencies
15.1 OPTIONS ON STOCK INDICES
Portfolio Insurance
Example 15.1 Protecting the value of a portfolio that mirrors the S&P 500
When the Portfolio’s Beta Is Not 1.0
Table 15.1 Calculation of expected value of portfolio when the index is 1,040 in three months and β = 2.0
Table 15.2 Relationship between value of index and value of portfolio for β = 2.0
Example 15.2 Protecting the value of a portfolio that has a beta of 2.0
15.2 CURRENCY OPTIONS
Range Forwards
Figure 15.1 Payoffs from (a) short and (b) long range forward contract
Figure 15.2 Exchange rate realized when either (a) a short range forward contract is used to hedge a future foreign currency inflow or (b) a long range forward contract is used to hedge a future foreign currency outflow
15.3 OPTIONS ON STOCKS PAYING KNOWN DIVIDEND YIELDS
Lower Bounds for Option Prices
Put–Call Parity
Pricing Formulas
15.4 VALUATION OF EUROPEAN STOCK INDEX OPTIONS
Example 15.3 Valuation of stock index option
Using Forward Prices
Business Snapshot 15.1 Can we guarantee that stocks will beat bonds in the long run?
Implied Dividend Yields
15.5 VALUATION OF EUROPEAN CURRENCY OPTIONS
Example 15.4 Implied volatility for a currency option
Using Forward Exchange Rates
15.6 AMERICAN OPTIONS
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 16 Futures Options
16.1 NATURE OF FUTURES OPTIONS
Example 16.1 Mechanics of call futures options
Example 16.2 Mechanics of put futures options
Expiration Months
16.2 REASONS FOR THE POPULARITY OF FUTURES OPTIONS
16.3 EUROPEAN SPOT AND FUTURES OPTIONS
16.4 PUT–CALL PARITY
Example 16.3 Put–call parity using futures prices
16.5 BOUNDS FOR FUTURES OPTIONS
16.6 VALUATION OF FUTURES OPTIONS USING BINOMIAL TREES
Figure 16.1 Futures price movements in the numerical example
A Generalization
Figure 16.2 Futures price and option price in a general situation
Multistep Trees
16.7 A FUTURES PRICE AS AN ASSET PROVIDING A YIELD
16.8 BLACK’S MODEL FOR VALUING FUTURES OPTIONS
16.9 USING BLACK’S MODEL INSTEAD OF BLACK–SCHOLES–MERTON
Example 16.4 Valuation of a European futures option
Example 16.5 Valuing a spot option using futures prices
16.10 AMERICAN FUTURES OPTIONS vs. AMERICAN SPOT OPTIONS
16.11 FUTURES-STYLE OPTIONS
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 17 The Greek Letters
17.1 ILLUSTRATION
17.2 NAKED AND COVERED POSITIONS
17.3 A STOP-LOSS STRATEGY
Figure 17.1 A stop-loss strategy
Table 17.1 Performance of stop-loss strategy. (The performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the theoretical price of the option.)
17.4 DELTA HEDGING
Figure 17.2 Calculation of delta
Example 17.1 Use of delta hedging
Delta of European Stock Options
Example 17.2 Delta of a stock option
Figure 17.3 Variation of delta with stock price for (a) call option and (b) put option on a non-dividend-paying stock
Dynamic Aspects of Delta Hedging
Figure 17.4 Typical patterns for variation of delta with time to maturity for a call option
Table 17.2 Simulation of delta hedging. Option closes in the money and cost of hedging is $263,300
Table 17.3 Simulation of delta hedging. Option closes out of the money and cost of hedging = $256,600
Table 17.4 Performance of delta hedging. The performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the theoretical price of the option
Where the Cost Comes From
Delta of a Portfolio
Transaction Costs
17.5 THETA
Example 17.3 Theta of a stock option
Figure 17.5 Variation of theta of a European call option with stock price
Figure 17.6 Typical patterns for variation of theta of a European call option with time to maturity
17.6 GAMMA
Figure 17.7 Hedging error introduced by nonlinearity
Example 17.4 Impact of gamma on change in value of a delta-neutral portfolio
Making a Portfolio Gamma Neutral
Figure 17.8 Relationship between ΔΠ and ΔS in time t for a delta-neutral portfolio: (a) slightly positive gamma, (b) large positive gamma, (c) slightly negative gamma, and (d) large negative gamma
Example 17.5 Making a portfolio gamma and delta neutral
Example 17.6 Gamma of a stock option
Calculation of Gamma
17.7 RELATIONSHIP BETWEEN DELTA, THETA, AND GAMMA
Figure 17.9 Variation of gamma with stock price for an option
Figure 17.10 Variation of gamma with time to maturity for a stock option
17.8 VEGA
Example 17.7 Making a portfolio delta, gamma, and vega neutral
Example 17.8 Vega of a stock option
Figure 17.11 Variation of vega with stock price for an option
17.9 RHO
Example 17.9 Rho of a stock option
17.10 THE REALITIES OF HEDGING
17.11 SCENARIO ANALYSIS
Business Snapshot 17.1 Dynamic hedging in practice
Table 17.5 Profit or loss realized in two weeks under different scenarios (millions of dollars)
17.12 EXTENSION OF FORMULAS
Delta of Forward Contracts
Table 17.6 Greek letters for European options on an asset that provides a yield at rate q
Delta of a Futures Contract
Example 17.10 Using futures to hedge a currency portfolio
17.13 CREATING OPTIONS SYNTHETICALLY FOR PORTFOLIO INSURANCE
Example 17.11 Portfolio insurance trading strategy
Use of Index Futures
Example 17.12 Portfolio insurance using futures
17.14 STOCK MARKET VOLATILITY
Business Snapshot 17.2 Was portfolio insurance to blame for the crash of 1987?
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 18 Binomial Trees in Practice
18.1 THE BINOMIAL MODEL FOR A NON-DIVIDEND-PAYING STOCK
Risk-Neutral Valuation
Figure 18.1 Stock price movements in time Δt under the binomial model
Determination of p, u, and d
The Tree of Stock Prices
Figure 18.2 Tree used to value a stock option
Working Backward through the Tree
Illustration
Figure 18.3 Binomial tree from DerivaGem for American put on non-dividend-paying stock
Expressing the Approach Algebraically
Figure 18.4 Convergence of option price calculated from a binomial tree
Estimating Delta and Other Greek Letters
18.2 USING THE BINOMIAL TREE FOR OPTIONS ON INDICES, CURRENCIES, AND FUTURES CONTRACTS
Example 18.1 Tree for option on index futures
Example 18.2 Tree for option on currency
18.3 THE BINOMIAL MODEL FOR A DIVIDEND-PAYING STOCK
Known Dividend Yield
Known Dollar Dividend
Figure 18.5 Tree when stock pays a known dividend yield at one particular time
Figure 18.6 Tree when dollar amount of dividend is assumed known and volatility is assumed constant
Example 18.3 Tree for option on a dividend-paying stock
18.4 EXTENSIONS OF THE BASIC TREE APPROACH
Time-Dependent Interest Rates and Volatilities
The Control Variate Technique
Figure 18.7 Tree produced by DerivaGem for European version of option in Figure 18.3. At each node, the upper number is the stock price and the lower number is the option price
18.5 ALTERNATIVE PROCEDURE FOR CONSTRUCTING TREES
18.6 MONTE CARLO SIMULATION
Example 18.4 Alternative tree construction
Example 18.5 Using Monte Carlo simulation with a tree
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 19 Volatility Smiles
19.1 FOREIGN CURRENCY OPTIONS
Figure 19.1 Volatility smile for foreign currency options
Figure 19.2 Implied distribution and lognormal distribution for foreign currency options
Empirical Results
Table 19.1 Percent of days when daily exchange rate moves are greater than one, two, . . ., six standard deviations (S.D. = standard deviation of daily change)
Business Snapshot 19.1 Making money from foreign currency options
Reasons for the Smile in Foreign Currency Options
19.2 EQUITY OPTIONS
Figure 19.3 Volatility smile for equities
Figure 19.4 Implied distribution and lognormal distribution for equity options
Business Snapshot 19.2 Crashophobia
The Reason for the Smile in Equity Options
19.3 THE VOLATILITY TERM STRUCTURE AND VOLATILITY SURFACES
Table 19.2 Volatility surface
The Role of the Model
19.4 WHEN A SINGLE LARGE JUMP IS ANTICIPATED
Figure 19.5 Effect of a single large jump. The solid line is the true distribution; the dashed line is the lognormal distribution
Figure 19.6 Change in stock price in one month
SUMMARY
Table 19.3 Implied volatilities in situation where it is known that the stock price will move from $50 to either $42 or $58
Figure 19.7 Volatility smile for situation in Table 19.3
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
APPENDIX Why the Put Volatility Smile Is the Same As the Call Volatility Smile
Example 19A.1 Implied volatilities for puts and calls
CHAPTER 20 Value at Risk
20.1 THE VaR MEASURE
Business Snapshot 20.1 How bank regulators use VaR
Figure 20.1 Calculation of VaR from the probability distribution of changes in the portfolio value; confidence level is X percent. Gains in portfolio value are positive; losses are negative
Figure 20.2 Alternative situation to Figure 20.1; VaR is the same, but the potential loss is larger
The Time Horizon
20.2 HISTORICAL SIMULATION
Illustration: Investment in Four Stock Indices
Table 20.1 Investment portfolio used for VaR calculations
Table 20.2 U.S. dollar equivalent of stock indices for historical simulation calculation
Table 20.3 Scenarios generated for September 26, 2008, using data in Table 20.2
Figure 20.3 Histogram of losses for the scenarios considered between September 25 and September 26, 2008
Table 20.4 Losses ranked from highest to lowest for 500 scenarios
20.3 MODEL-BUILDING APPROACH
Daily Volatilities
Single-Asset Case
Two-Asset Case
Example 20.1 Calculation of VaR in a simple situation
The Benefits of Diversification
20.4 GENERALIZATION OF LINEAR MODEL
Correlation and Covariance Matrices
Table 20.5 A correlation matrix: ρij is the correlation between variable i and variable j
Table 20.6 A variance–covariance matrix: covij is the covariance between variable i and variable j. Diagonal entries are variance: covii = vari
Handling Interest Rates
Applications of the Linear Model
The Linear Model and Options
Example 20.2 Using the linear model for options
20.5 QUADRATIC MODEL
Figure 20.4 Probability distribution for value of portfolio: (a) positive gamma, (b) negative gamma
Figure 20.5 Translation of normal probability distribution for asset into probability distribution for value of a long call on asset
Figure 20.6 Translation of normal probability distribution for asset into probability distribution for value of a short call on asset
20.6 ESTIMATING VOLATILITIES AND CORRELATIONS
EWMA
Example 20.3 Updating volatility using EWMA
Correlations
Example 20.4 Updating correlation using EWMA
Example Involving Four Stock Indices
Table 20.7 Correlation matrix on September 25, 2008, calculated by giving equal weight to the last 500 daily returns: variable 1 is DJIA; variable 2 is FTSE 100; variable 3 is CAC 40; variable 4 is Nikkei 225
Table 20.8 Covariance matrix on September 25, 2008, calculated by giving equal weight to the last 500 daily returns: variable 1 is DJIA; variable 2 is FTSE 100; variable 3 is CAC 40; variable 4 is Nikkei 225
Use of EWMA
Table 20.9 Covariance matrix on September 25, 2008, calculated using the EWMA method with λ = 0.94: variable 1 is DJIA; variable 2 is FTSE 100; variable 3 is CAC 40; variable 4 is Nikkei 225
Table 20.10 Volatilities (% per day) using equal weighting and EWMA
20.7 COMPARISON OF APPROACHES
20.8 STRESS TESTING AND BACK TESTING
Table 20.11 Correlation matrix on September 25, 2008, calculated using the EWMA method: variable 1 is DJIA; variable 2 is FTSE 100; variable 3 is CAC 40; variable 4 is Nikkei 225
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 21 Interest Rate Options
21.1 EXCHANGE-TRADED INTEREST RATE OPTIONS
Example 21.1 A Eurodollar futures option trade
Example 21.2 A Treasury bond futures option trade
21.2 EMBEDDED BOND OPTIONS
21.3 BLACK’S MODEL
Extension of Black’s Model
How the Model Is Used
21.4 EUROPEAN BOND OPTIONS
Example 21.3 Valuation of a bond option
Yield Volatilities
Example 21.4 Bond option valuation using yield volatilities
21.5 INTEREST RATE CAPS
Figure 21.1 Effect of a cap in providing insurance against LIBOR rising above the cap rate
Example 21.5 Use of an interest rate cap
The Cap as a Portfolio of Interest Rate Options
Floors and Collars
Business Snapshot 21.1 Put–call parity for caps and floors
Valuation of Caps and Floors
Example 21.6 Valuation of a caplet
Figure 21.2 The implied volatility hump
Using DerivaGem
Table 21.1 Typical broker flat volatility quotes for U.S. dollar caps and floors (percent per annum)
Business Snapshot 21.2 Swaptions and bond options
21.6 EUROPEAN SWAP OPTIONS
Valuation of European Swaptions
Example 21.7 Valuation of a swaption
Table 21.2 Typical broker quotes for U.S. European swaptions (mid-market volatilities % per annum)
21.7 TERM STRUCTURE MODELS
Figure 21.3 Mean reversion
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 22 Exotic Options and Other Nonstandard Products
22.1 EXOTIC OPTIONS
Packages
Nonstandard American Options
Gap Options
Example 22.1 A gap option in insurance
Forward Start Options
Cliquet Options
Compound Options
Chooser Options
Barrier Options
Binary Options
Lookback Options
Shout Options
Asian Options
Options to Exchange One Asset for Another
Options Involving Several Assets
22.2 AGENCY MORTGAGE-BACKED SECURITIES
Collateralized Mortgage Obligations
IOs and POs
22.3 NONSTANDARD SWAPS
Variations on the Vanilla Deal
Business Snapshot 22.1 Hypothetical confirmation for nonstandard swap
Compounding Swaps
Business Snapshot 22.2 Hypothetical confirmation for compounding swap
Currency Swaps
Valuation and Convexity Adjustments
LIBOR-in-Arrears Swap
CMS Swaps
Differential Swaps
Business Snapshot 22.3 Hypothetical confirmation for equity swap
Equity Swaps
Accrual Swaps
Cancelable Swaps
Index Amortizing Swaps
Commodity Swaps
Volatility and Variance Swaps
Other Swaps
Business Snapshot 22.4 Procter & Gamble’s bizarre deal
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 23 Credit Derivatives
Business Snapshot 23.1 Who Bears the Credit Risk?
23.1 CREDIT DEFAULT SWAPS
Figure 23.1 Credit default swap
Recovery Rate
Business Snapshot 23.2 The CDS Market
Credit Default Swaps and Bond Yields
Table 23.1 Recovery rates on corporate bonds as a percentage of face value, 1982–2011 (Source: Moody’s)
The Cheapest-to-Deliver Bond
23.2 VALUATION OF CREDIT DEFAULT SWAPS
Table 23.2 Unconditional annual default probabilities and survival probabilities.
Table 23.3 Calculation of the present value of expected payments. Payment = s per annum.
Marking to Market a CDS
Table 23.4 Calculation of the present value of expected payoff. Notional principal = $1.
Table 23.5 Calculation of the present value of accrual payment
Default Probabilities
Binary Credit Default Swaps
Table 23.6 Calculation of the present value of expected payoff from a binary credit default swap. Principal = $1.
Basket Credit Default Swaps
The Future of the CDS Market
23.3 TOTAL RETURN SWAPS
Business Snapshot 23.3 Is the CDS market a fair game?
Figure 23.2 Total return swap
23.4 CDS FORWARDS AND OPTIONS
23.5 CREDIT INDICES
23.6 THE USE OF FIXED COUPONS
Example 23.1 How fixed coupons work
23.7 COLLATERALIZED DEBT OBLIGATIONS
Synthetic CDOs
Figure 23.3 The structure of a synthetic CDO
Standard Portfolios and Single-Tranche Trading
Table 23.7 Mid-market quotes for five-year tranches of iTraxx Europe. Quotes are in basis points except for the 0–3% tranche where the quote equals the percent of the tranche principal that must be paid up front in addition to 500 basis points per year (Source: Creditex Group Inc.)
The Role of Default Correlation
SUMMARY
FURTHER READING
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Questions
CHAPTER 24 Weather, Energy, and Insurance Derivatives
24.1 WEATHER DERIVATIVES
24.2 ENERGY DERIVATIVES
Crude Oil
Natural Gas
Electricity
Characteristics of Energy Prices
How an Energy Producer Can Hedge Risks
24.3 INSURANCE DERIVATIVES
SUMMARY
FURTHER READING
On weather derivatives
On energy derivatives
On insurance derivatives
Quiz (Answers at End of Book)
Practice Questions (Answers in Solutions Manual/Study Guide)
Further Question
CHAPTER 25 Derivatives Mishaps and What We Can Learn from Them
25.1 LESSONS FOR ALL USERS OF DERIVATIVES
Business Snapshot 25.1 Big losses by financial institutions
Business Snapshot 25.2 Big losses by nonfinancial organizations
Define Risk Limits
Take the Risk Limits Seriously
Do Not Assume You Can Outguess the Market
Do Not Underestimate the Benefits of Diversification
Carry out Scenario Analyses and Stress Tests
25.2 LESSONS FOR FINANCIAL INSTITUTIONS
Monitor Traders Carefully
Separate the Front, Middle, and Back Office
Do Not Blindly Trust Models
Be Conservative in Recognizing Inception Profits
Do Not Sell Clients Inappropriate Products
Beware of Easy Profits
Do Not Ignore Liquidity Risk
Beware When Everyone Is Following the Same Trading Strategy
Do Not Make Excessive Use of Short-Term Funding for Long-Term Needs
Market Transparency Is Important
Manage Incentives
Never Ignore Risk Management
25.3 LESSONS FOR NONFINANCIAL CORPORATIONS
Make Sure You Fully Understand the Trades You Are Doing
Make Sure a Hedger Does Not Become a Speculator
Be Cautious about Making the Treasury Department a Profit Center
SUMMARY
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