Wiley FRM Exam Review Study Guide 2016
Part I, Volume 2
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Wiley FRM Exam Review Study Guide 2016
Part I, Volume 2
Financial Markets and Products,
Valuation and Risk Models
Christian H. Cooper, CFA, FRM
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Cover image: Loewy Design
Cover design: Loewy Design
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved.
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Contents
How to Study for the Exam
ix
About the Instructor
x
Financial Markets and Products (FMP)
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 1. Introduction 3
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 2. Mechanics of Futures Markets 13
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 3. Hedging Strategies Using Futures 17
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 4. Interest Rates 23
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 5. Determination of Forward and Futures Prices 35
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 6. Interest Rate Futures 45
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 7. Swaps 55
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 10. Mechanics of Options Markets 67
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 11. Properties of Stock Options 69
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 12. Trading Strategies Involving Options 77
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Contents
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York:
Pearson, 2014). Chapter 26. Exotic Options 83
Lesson: Robert McDonald, Derivatives Markets, 3rd Edition (Boston: Pearson, 2012).
Chapter 6. Commodity Forwards and Futures 87
Lesson: Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk
Management Approach, 8th Edition (New York: McGraw-Hill, 2014).
Chapter 13. Foreign Exchange Risk 93
Lesson: Jon Gregory, Central Counterparties: Mandatory Clearing and Bilateral Margin Requirements
for OTC Derivatives (West Sussex, UK: John Wiley & Sons, 2014). Chapter 1. Introduction 99
Lesson: Jon Gregory, Central Counterparties: Mandatory Clearing and Bilateral Margin Requirements
for OTC Derivatives (West Sussex, UK: John Wiley & Sons, 2014).
Chapter 2. Exchanges, OTC Derivatives, DPCs and SPVs 101
Lesson: Jon Gregory, Central Counterparties: Mandatory Clearing and Bilateral Margin Requirements
for OTC Derivatives (West Sussex, UK: John Wiley & Sons, 2014). Chapter 3. Basic Principles of
Central Clearing 105
Lesson: Jon Gregory, Central Counterparties: Mandatory Clearing and Bilateral Margin Requirements
for OTC Derivatives (West Sussex, UK: John Wiley & Sons, 2014). Chapter 14 (section 14.4 only).
Risks Caused by CCPs: Risks Faced by CCPs 109
Lesson: Frank Fabozzi (editor), Steve Mann, and Adam Cohen, The Handbook of Fixed Income Securities,
8th Edition (New York: McGraw-Hill, 2012). Chapter 12. Corporate Bonds 111
Lesson: Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today’s Markets, 3rd Edition
(Hoboken, NJ: John Wiley & Sons, 2011). Chapter 20. Mortgages and Mortgage-Backed Securities 115
Lesson: John B. Caouette, Edward I. Altman, Paul Narayanan, and Robert W.J. Nimmo, Managing
Credit Risk: The Great Challenge for Global Financial Markets, 2nd Edition (New York:
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123
Valuation and Risk Models (VRM)
Lesson: Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and
Operational Risk: The Value at Risk Approach (New York: Wiley-Blackwell, 2004). Chapter 2.
Quantifying Volatility in VaR Models 127
Lesson: Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and
Operational Risk: The Value at Risk Approach (New York: Wiley-Blackwell, 2004).
Chapter 3. Putting VaR to Work 133
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Contents
Lesson: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England: John Wiley & Sons,
2005). Chapter 2. Measures of Financial Risk 137
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York: Pearson, 2014).
Chapter 13. Binomial Trees
141
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York: Pearson, 2014).
Chapter 15. The Black-Scholes-Merton Model 151
Lesson: John Hull, Options, Futures, and Other Derivatives, 9th Edition (New York: Pearson, 2014).
Chapter 19. Greek Letters 157
Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 1. Prices, Discount Factors, and Arbitrage
163
Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 2. Spot, Forward and Par Rates
169
Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 3. Returns, Spreads and Yields
173
Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 4. One-Factor Risk Metrics and Hedges
181
Lesson: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011).
Chapter 5. Multi-Factor Risk Metrics and Hedges 191
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Lesson: Aswath Damodaran, “Country Risk: Determinants, Measures and Implications - The 2015
Edition” (July 2015).
197
Lesson: Arnaud de Servigny and Olivier Renault, Measuring and Managing Credit Risk
(New York: McGraw-Hill, 2004). Chapter 2. External and Internal Ratings
203
Lesson: Gerhard Schroeck, Risk Management and Value Creation in Financial Institutions
(New York: John Wiley & Sons, 2002). Chapter 5. Capital Structure in Banks (pp. 170-186 only) 207
Lesson: John Hull, Risk Management and Financial Institutions, 4th Edition (Hoboken, NJ:
John Wiley & Sons, 2015). Chapter 23. Operational Risk 211
Lesson: Philippe Jorion, Value-at-Risk: The New Benchmark for Managing Financial Risk,
3rd Edition (New York: McGraw Hill, 2007). Chapter 14. Stress Testing. 215
Lesson: “Principles for Sound Stress Testing Practices and Supervision” (Basel Committee on
Banking Supervision Publication,” May 2009).
219
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How to Study for the Exam
The FRM Exam Part I curriculum covers the tools used to assess financial risk:
•
•
•
•
Foundations of risk management—20%
Quantitative analysis—20%
Financial markets and products—30%
Valuation and risk models—30%
It is important to focus only on the learning objectives as you are asked and pay close
attention to the percentages of each section. That is the core of my focus through the text,
the online lecture sessions, and with the practice questions. A study hour doesn’t count
unless you are laser focused on specifically how GARP asks a learning objective.
Consistency is also key. Making a regular weekly study time is going to be important to
staying on track. There is a reason: Only ~50% of the candidates pass the exam every year.
It’s a tough exam. It also tests intuition, not just memorization. That is why I attempt at
every opportunity to connect the dots across readings, teach how changing environments
change both markets and the models we use to model them, as well as help you with the
questions where GARP specifically wants you to calculate an outcome.
Calculator policy:
It is best to begin your study with one of the approved calculators. You will not be admitted
to the exam without one of these approved calculators!
• Hewlett Packard 12C (including the HP 12C Platinum and the Anniversary
Edition)
• Hewlett Packard 10B II
• Hewlett Packard 10B II+
• Hewlett Packard 20B
• Texas Instruments BA II Plus (including the BA II Plus Professional)
Every year, candidates are turned away from the exam site because of wrong calculators.
Make sure you aren’t one of them.
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About the Instructor
Christian H. Cooper is an author and trader based in New York City. He initially created
the FRM program because, as a candidate, he was frustrated with the quality of study
programs available. Writing from a practitioner’s point of view, Christian drew on his
experience as a trader across fixed income and equity markets, most recently as head of
derivatives trading at a bank in New York, to create a program that is very focused on exam
results while connecting the dots across topics to increase intuition and understanding.
Christian is active with the Aspen Institute, a Truman National Security Fellow, and a term
member at the Council on Foreign Relations.
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15 December 2015 4:31 PM
Wiley FRM Exam Review Study Guide 2016
Part I, Volume 2
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Financial Markets and Products (FMP) This area tests your knowledge of financial products and the markets in which they trade,
more specifically, the following knowledge areas:
•
•
•
•
•
•
•
•
Structure and mechanics of OTC and exchange markets
Structure, mechanics, and valuation of forwards, futures, swaps, and options
Hedging with derivatives
Interest rates and measures of interest rate sensitivity
Foreign exchange risk
Corporate bonds
Mortgage‐backed securities
Rating agencies
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Hull, Chapter 1
John Hull, Options, Futures, and Other Derivatives, 9th Edition
(New York: Pearson Prentice Hall, 2014). Chapter 1. Introduction
After completing this reading you should be able to:
• Describe the over‐the‐counter market, distinguish it from trading on an exchange,
and evaluate its advantages and disadvantages.
• Differentiate between options, forwards, and futures contracts.
• Identify and calculate option and forward contract payoffs.
• Calculate and compare the payoffs from hedging strategies involving forward
contracts and options.
• Calculate and compare the payoffs from speculative strategies involving futures
and options.
• Calculate an arbitrage payoff and describe how arbitrage opportunities are
temporary.
• Describe some of the risks that can arise from the use of derivatives.
Learning objective: Describe the over‐the‐counter market, distinguish
it from trading on an exchange, and evaluate its advantages and
disadvantages.
The key advantages of over‐the‐counter instruments is the fact that you can almost customize
the instrument to anything that you need. In other words, if you need to have a particular
instrument that starts and settles and extends to a particular date and time for cash flow
management purposes, that can be structured within the over‐the‐counter market. However,
the over‐the‐counter market exposes you to counterparty risk that you are not exposed to
while trading on an exchange. By contrast, exchange‐traded contracts are not customizable.
Learning objective: Differentiate between options, forwards, and futures
contracts.
A derivative instrument is a security whose value is derived from, and therefore depends
upon, the value of some underlying asset(s). The major derivatives considered here are
forward contracts, futures contracts, options contracts, and swap contracts. Financial
derivatives offer numerous benefits to risk managers compared with the underlying assets.
Examples of underlying assets are a stock, commodity, or index.
A financial derivative is a financial contract. The payoffs from this contract depend on
another financial instrument, security, asset, or contract, known as the underlying asset.
Typically, the underlying asset trades in a market in which its price is determined on a
regular basis. The current price of the underlying asset is known as the cash or spot price.
Also, a derivative has a specific and limited life. The contract begins on one date and ends
on a future date.
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Financial Markets and Products (FMP)
For example, IBM shares trade on the New York Stock Exchange and the spot price of
IBM shares is determined by investors buying and selling on the exchange. The value of an
option on a share of IBM depends on the value of the IBM share. A share of IBM stock has
an unlimited life, while an option on a share of IBM stock has a fixed life.
Derivatives trade in two types of markets. They can be exchange‐traded contracts or
they can trade over‐the‐counter. Contracts that trade on an organized exchange have
standardized terms that are set by the exchange or by the clearinghouse. Over‐the‐counter
derivatives result from agreements between two parties who can negotiate whatever
contract terms are mutually acceptable. Derivatives also can be divided into two types,
forward commitments and contingent claims.
When two entities choose to transact with each other, one party agrees to pay a negotiated
price while the other party agrees to deliver the asset underlying the transaction. Typically,
the transaction occurs immediately. However, one can envision situations where the
entities do not wish to transact immediately. Instead, they negotiate the terms of the
transaction, the price, and the underlying asset, but perform the transaction at a later date.
Such an agreement, where the transaction is delayed until some pre‐specified future point
in time, is designated a forward commitment. In contrast, a contingent claim is an asset,
such as an option, whose payoff and, therefore, whose value is contingent or depends on
the value of some other underlying asset
Forwards and Futures
The three primary types of forward commitments are forward contracts, futures contracts,
and swaps, and are defined as follows:
A forward contract is a commitment whereby the two parties to the contract negotiate
the price, underlying asset, and future point in time at which the transaction will occur.
A futures contract is a type of forward contract distinguished by having highly
standardized contract terms.
In a swap contract, two parties agree to exchange sets of cash flows at several future
points in time.
A forward contract is the most basic form of a forward commitment. An example of a
forward contract would be as follows: A mining company and jewelry manufacturer
agree in March that the mining company, in October, will sell 1,000 ounces of gold to the
jewelry company at a price of $325 per ounce with delivery to take place at the jewelry
manufacturer’s plant. Both parties in a forward commitment face the possibility that the
other party will default if the price of the underlying asset moves against them. The mining
company might be tempted to default if the spot price of gold in October is more than $325
per ounce. If, for example, the spot price of gold is $350 per ounce in October, then the
mining company would prefer to sell its gold in the spot market for $350 per ounce rather
than to the jewelry company at $325 per ounce. On the other hand, if the spot price of gold in
October is only $300 per ounce, then the jewelry manufacturer might be tempted to default.
Although there are well‐developed forward markets with standardized market features
for many goods, forward contracts are typically unique. An attractive feature of a forward
contract is the flexibility afforded by custom tailoring the terms of the agreement. Because
all of the terms of a forward contract are negotiable, forward contracts are customized
agreements.
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Hull, Chapter 1
The parties to a futures contract also agree to the exchange of some underlying asset at
a future date for cash. Like a forward contract, the agreement takes place today, but the
payment occurs at that future date. The purchaser of a futures contract agrees to take
delivery of the good and pay for it. It is the obligation of the seller to deliver the good and
receive payment.
Two characteristics distinguishing futures contracts from other forward contracts are:
1. Futures contracts are exchange traded. Forward contracts are not. Exchange traded
means the contracts have standardized terms such as expiration date, strike, and
settlement.
2. Participation in a futures exchange is government regulated. There is no similar
regulation for forward contracts because the contracts are negotiated between two
parties. This is what is called the OTC market.
A party to a futures contract does not face significant credit risk, because if the
counterparty fails to satisfy the counterparty’s obligations, these obligations are
guaranteed. The guarantor is a financial clearinghouse associated with the exchange on
which the futures contracts trade. There is no similar guarantee associated with a forward
contract; hence a party to a forward contract can face significant credit risk.
The terms associated with futures contracts are standardized. The terms associated with
forward contracts are based on the negotiations between the two parties.
Parties to a futures contract are required to place collateral into an account to ensure
that they have the funds available to cover any obligations they may face in the future.
The name of this account is the margin account. Further, adjustments are made to this
account on a frequent basis, such as daily, to reflect any losses that the party may have
incurred. The process through which these adjustments are made is designated “marking‐
to‐market.”
Forward contracts are structured so that each party’s obligation to the other lasts until the
contract expires. However, with futures contracts, a party’s obligation can be terminated
early by entering into an offsetting transaction.
Learning objective: Identify and calculate option and forward contract
payoffs.
A forward is a contract between two parties, where one (the long position) has
the obligation to buy, and the other (the short position) an obligation to sell the
underlying asset at a specified price (established at the inception of the contract) at a
future date.
Pricing and Valuation of Forward Contracts
The price of a forward contract is the fixed price or rate at which the underlying transaction
will occur at contract expiration. The forward price is agreed upon at initiation of the
forward contract. Pricing a forward contract means determining this forward price. The
value of a forward contract is the amount that a counterparty would need to pay, or would
expect to receive, to get out of the (already‐assumed) forward position. We will first work
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Financial Markets and Products (FMP)
with a generic example to introduce you to the concepts and mechanics behind pricing and
valuing forward contracts.
• The contract initiation date is denoted by t = 0.
• The contract expiration date is denoted by t = T.
• Any point in time between the contract initiation and expiration dates is denoted by
t = t.
The forward price (F) is determined at contract initiation. It does not change over the term
of the contract. The term F(0,T) is used to refer to the forward price for a contract that was
initiated at
t = 0 and expires at t = T. See Example 1.
The value (V) of the forward contract changes over the term of the contract as the price of
the underlying asset changes:
• V0(0,T) refers to the value of a forward contract (initiated at t = 0 and expiring at t
= T) at initiation (t = 0).
• Vt(0,T) refers to the value of a forward contract (initiated at t = 0 and expiring at t
= T) at a point in time during the term of the contract (t = t).
• Vt(0,T) refers to the value of a forward contract (initiated at t = 0 and expiring at t
= T) at expiration (t = T).
The spot price (S) of the underlying asset also changes over the term of the contract.
• S0 refers to the spot price at initiation of the forward contract (t = 0).
• St refers to the spot price at a point in time during the term of the forward contract (t = t).
• St refers to the spot price at expiration of the forward contract (t = T).
In the following sections, we illustrate how forward contracts are valued at various points
in time and I will keep this notation through the program.
Note that we will be taking the perspective of the long position on the contract when
valuing a forward. Once the value of the long position has been determined, the value of
the short can be determined by simply changing the sign. Further, we will be assuming that
the underlying asset entails no storage or carrying costs, and makes no payments to the
owner of the asset over the term of the forward contract.
Valuing a Forward Contract at Expiration (t = T)
The long position on the forward has an obligation to buy the underlying asset for the
agreed upon (at contract initiation) price of F(0,T) at contract expiration. The price of the
underlying asset at expiration of the forward equals ST. Therefore, the value of the long
position on the forward contract at expiration equals the difference between:
• The current worth of the asset, St, which represents the price at which the
underlying asset can be sold; and
• The price that the long position in the forward must pay to acquire the asset F(0,T).
VT(0, T) = ST − F(0, T)
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Hull, Chapter 1
If the value at expiration does not equal this amount, arbitrage profits can be made. For
example, if the forward price established at contract initiation equals $30 and the spot price
at contract expiration equals $35, then the value of the forward contract must equal $35 –
$30 = $5 at expiration. See Example 1.
• If the contract value at expiration were greater than $5, it would mean that
someone is willing to pay more than $5 to obtain an obligation to buy something
worth $35 for $30, which wouldn’t make sense.
• If the contract value were less than $5, it would mean that someone is willing to
accept less than $5 to give up an obligation to buy something worth $35 for $30,
which wouldn’t make sense either.
Valuing a Forward Contract at Initiation
Now let’s work with a forward contract that has a term of 1 year. The price (S0) of the asset
underlying the contract is currently $100 and the risk‐free rate (r) is 8%. Determine the
value of the contract at initiation, V0(0,1), given that:
1. The forward price, F(0,1) equals $110
2. The forward price, F(0,1) equals $106
Note that no money changes hands at origination of the forward contract.
Scenario 1: F(0,1) = $110
In this scenario, arbitrage profits can be made through cash and carry arbitrage by
undertaking the following steps:
• Borrow $100 at 8%.
• Purchase the underlying asset at the current spot price of $100.
• Sell the underlying asset forward by taking the short position on the forward
contract at a forward price of $110.
The table below illustrates the computation of arbitrage profits from this strategy:
t = 0 (At Contract Origination)
Transaction
Cash Flow
Borrow $100 @8%
Buy asset at current price S0
Take the short position in a
forward contract on the asset with
a forward price, F(0,1), of $110
Net Cash Flow
$100
($100)
t = T (At Contract Expiration)
Transaction
Cash Flow
Deliver the asset under the terms
of the forward contract in return
for F(0,1)
Repay loan plus interest
$110
($108)
$0
0
Net Cash Flow
$2
Scenario 2: F(0,1) = $106
In this scenario arbitrage profits can be made through reverse cash and carry arbitrage by
undertaking the following steps:
• Short the underlying asset at the current spot price of $100.
• Invest the proceeds at 8%.
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Financial Markets and Products (FMP)
• Buy the underlying asset forward by taking the long position on the forward
contract at the forward price of $106.
The table below illustrates the computation of arbitrage profits from this strategy:
t = 0 (At Contract Origination)
Transaction
Cash Flow
$100
Short the underlying asset at S0
Invest $100 @8%
($100)
Take the long position in a
forward contract on the asset with
a forward price, F(0,1), of $106
$0
Net Cash Flow
0
t = T (At Contract Expiration)
Transaction
Cash Flow
Take delivery of the asset under
the terms of the forward contract
by paying F(0,1)
($106)
Receive investment amount
plus interest
$108
Net Cash Flow
$2
Now let’s understand the value of a forward contract. Recall that we are taking the
perspective of the long position.
• The long position has the obligation to pay the forward price, F(0,T), and take
delivery of the underlying asset at contract expiration.
○○ The value of this obligation at contract initiation equals F(0,T) / (1 + r)T
• At expiration, the long position will receive the underlying asset, which will be
worth ST.
○○ The value of this underlying asset at contract initiation equals S0
• Therefore, the value of the forward contract to the long position at contract
initiation equals the current worth of the asset minus the present value of the
obligation.
○○ V0 (0, T) = S0 − [ F(0, T) / (1+r )T ]
In Scenario 1, the value of the forward contract at initiation is calculated as:
V0 (0, T) = S0 − [ F(0, T) / (1+ r )T ]
V0 (0,1) = 100 − [110/ (1+ 0.08)1 ] = − $1.85
In Scenario 2, the value of the forward contract at initiation is calculated as:
V0 (0, T) = S0 − [ F(0, T) / (1+ r )T ]
V0 (0,1) = 100 − [106 / (1+ 0.08)1 ] = $1.85
These non‐zero values at initiation would entice traders to engage in arbitrage until the
price of the forward contract equals the no‐arbitrage forward price.
Forward contracts are typically priced to have zero value at origination, V0(0,T) = 0.
Therefore, we can express the forward price in terms of the spot price of the asset as:
V0 (0, T) = S0 − [ F(0, T) / (1+ r )T ] = 0
F(0, T) = S0 (1+ r )T
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Hull, Chapter 1
Valuing a Forward Contract During Its Life
By now you should have digested that the value of the forward contract to the long position
equals the asset’s current price minus the present value of the forward price. We now use
this logic to derive the expression for the value of the forward contract at any point in time
(t) during its life. See Example 1.
• The long position has an obligation to pay the forward price, F(0,T), and take
delivery of the underlying asset at contract expiration.
○○ The value of this obligation at any point in time during the term of the
contract equals F(0, T) / (1+r )T − t
• At expiration, the long position will receive the underlying asset, which will be
worth ST.
○○ The value of this asset at any point in time during the term of the contract
equals St.
• Therefore, the value of the forward contract to the long position at any point in
time during the term of the contract equals the current value of the asset minus the
present value of the obligation.
○○ Vt (0, T) = St − [ F(0, T) / (1+r )T − t ]
Just one more thing that we need to reemphasize before moving on: F(0,T) represents the
forward price that is agreed upon at the inception of the contract. Both spot and forward
prices continue to fluctuate after inception of the contract, but for our purposes (to
determine the value of a particular forward contract at any point in time) we compare the
“then‐current” spot price of the underlying asset to the present value of the initially agreed‐
upon (or fixed) forward price to determine how much value a position on the particular
forward holds to the long or short.
Why We Would Need to Determine the Value of a Forward Contract
• To measure credit exposure.
• To mark‐to‐market for financial statement purposes or to adhere to the terms of the
forward agreement.
• To determine how much money would need to be paid to terminate the contract.
Table 1 summarizes what we have learned so far.
Table 1: Value of a Forward Contract
Time
Long Position Value
Short Position Value
At initiation
Zero, as the contract is
priced to prevent arbitrage
Zero, as the contract is
priced to prevent arbitrage
During life of the contract
 F(0, T ) 
St − 
T-t 
 (1+r ) 
 F(0, T) 
 (1+r )T − t  − St


At expiration
ST − F(0, T)
F(0, T) − ST
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Financial Markets and Products (FMP)
Example 1: Calculating the Forward Price
Amanda holds an asset worth $250, which she plans to sell in 6 months. To eliminate
the price risk, she decides to take the short position in a forward contract on the asset.
Given an annual risk‐free rate of 5%, calculate the no‐arbitrage forward price of the
contract.
Solution
Forwards are priced to have zero value to either party at origination. Therefore, the
forward price is calculated as:
F(0, T) = S0 (1 + r )T
F(0, 6 /12) = 250 × (1 + 0.05)0.5 = $256.17
Example 2: Calculating the Value of a Forward Contract During Its Life
In Example 1, we calculated the forward price as $256.17. Suppose that 2 months into
the term of the forward the spot price of the underlying asset is $262. Given an annual
risk‐free rate of 5%, calculate the value of the long and short positions in the forward
contract.
Solution
The value of the long position in the forward contract is calculated as:
Vt (0, T) = St − [ F(0, T) / (1 + r )T − t ]
V2/12 (0, 6 /12) = 262 − [256.17 / (1+0.05)6/12 − 2/12 ] = $9.96
The value of the short position is just the opposite of the value of the long position.
Therefore, the value of the short position equals −$9.96.
Example 3: Calculating the Value of a Forward Contract at Expiration
Continuing from Example 1, suppose that the spot price of the underlying asset at
contract expiration is actually $247. Given an annual risk‐free rate of 5%, calculate the
value of the long position.
Solution
At expiration, the value of the long position in a forward contract is calculated as:
VT (0, T) = ST − F(0, T )
VT (0, 6 /12) = 247 − 256.17 = − $9.17
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Hull, Chapter 1
For the next three learning objectives, none of this is actually covered anywhere in this
reading. This is important and we will do this, but it is out of place here.
• Calculate and compare the payoffs from hedging strategies involving forward
contracts and options.
• Calculate and compare the payoffs from speculative strategies involving futures
and options.
• Calculate an arbitrage payoff and describe how arbitrage opportunities are
temporary.
Also note that that payoff profile is not dependent upon strategy. Whether hedging,
speculation, or an arbitrage, the payoffs of futures versus options are all the same. We will
calculate these in the future. With respect to arbitrage being temporary, if “free money”
exists between two markets—for example, a large price difference in oil between London
and New York—cost and location being ignored, the market will buy one and sell the
other until the arbitrage opportunity is gone. With high‐frequency trading and electronic
markets, this almost never happens in the real world.
Learning objective: Describe some of the risks that can arise from the use
of derivatives.
Just know that derivatives can be either negotiated or exchange‐traded contracts. With
exchange‐traded contracts, it is more difficult to hide large positions when you have a
negotiated contract. You can face many counterparties, and no one knows the impact
should one of the dominoes fall.
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