Transcript Risk Management - DePaul University

Risk Management
Dr. Keith M. Howe
Summer 2008
Definition
Risk and uncertainty
Risk management
Risk aversion
The process of formulating the benefit-cost tradeoffs of risk reduction and deciding on the course of
action to take (including the decision to take no
action at all).
Two more definitions
• Derivatives
• financial assets (e.g., stock option, futures, forwards, etc)
whose values depend upon the value of the underlying
assets.
• Hedge
• the use of financial instruments or of other tools to reduce
exposure to a risk factor.
Figure 1.2. Gains and losses from buying shares and a call option on Risky
Upside Inc.
Gain
+$6,000
Risky Upsid
-$3,000
Panel A. Gain from buying shares of Risky
Upside Inc. at $50 per share.
20
50
110
Gain
$5,000
0
-$1,000
Risky Upside
Panel B. Gain from buying a call option on
shares of Risky Upside Inc. with exercise price
20
50 per share. 110
of $50 for a premium
of $10
Figure 1.3. Hedging with forward contract. Garman’s
income is in dollars and the exchange rate is the dollar price
of one euro.
Unhe
Income to firm
$100 million
$90 million
Panel A. Income to Garman if it does not hedge.
$0.90
$1
E
Gain from contract
to firm
$10 million
Forward
gain
Forward
loss
Panel B. Forward contract payoff.
$0.9
Forward rate $1
Unhe
Income to firm
Forward
loss
$100 million
Forward
gain
Panel C. Hedged firm income.
Income to firm
Unhedged
income
Gain with
option
$100 million
Loss with option
Panel D. Comparison of income with put contract and income with forward contract.
Excha
Risk management irrelevance
proposition
• Bottom line: hedging a risk does not increase firm value
when the cost of bearing the risk is the same whether the risk
is borne within the firm or outside the firm by the capital
markets.
• This proposition holds when financial markets are perfect.
Risk management irrelevance
proposition
• Allows us to find out when homemade risk management is not equivalent
to risk management by the firm.
• This is the case whenever risk management by a firm affects firm value
in a way that investors cannot mimic.
• For risk management to increase firm value, it must be more expensive to
take a risk within the firm than to pay the capital markets to take it.
Role of risk management
Risk management can add value to the firm by:
• Decreasing taxes
• Decreasing transaction costs (including
bankruptcy costs)
• Avoiding investment decision errors
Bankruptcy costs and
costs of financial distress
• Costs incurred as a result of a bankruptcy filing are
called bankruptcy costs.
• The extent to which bankruptcy costs affect firm
value depends on their extent and on the probability
that the firm will have to file for bankruptcy.
• The probability that a firm will be bankrupt is the
probability that it will not have enough cash flow to
repay the debt.
• Direct bankruptcy costs
• Average ratio of direct bankruptcy costs to total assets: 2.8%
• Indirect bankruptcy costs
• Many of these indirect costs start accruing as soon as a firm’s
financial situation becomes unhealthy, called costs of financial
distress
• Managers of a firm in bankruptcy lose control of some
decisions. They might not allowed to undertake costly new
projects, for example.
Figure 3.1. Cash flow to shareholders and operating cash flow.
Cash flow to shareholders
Unhedged cash flow
$450M
Expected cash
flow $350M
$250M
Expected cash
$250M flow $350M $450M
Cash flow to the
firm
Figure 3.2. Creating the unhedged firm out of the hedged firm.
Unhedged cash flow
Cashflowto
shareholders
Forward
loss Hedged firm cash
$350M
(hedged)
flow
Forward
gain
$350M (gold sold at forward)
Figure 3.3. Cash flow to claimholders and bankruptcy costs.
Cash flow to claimholders
Unhedged cash flow
$450M
Expected cash
flow hedged
$350M
Unhedged
$340M
$230M
Bankruptcy
cost
Expected cash
$250M flow $350M $450M
Cash flow to the
firm
Analysis of decreasing
transaction cost by hedging
Value of firm unhedged = PV (C – Bankruptcy costs)
= PV (C) – PV (Bankruptcy costs)
= value of firm without bankruptcy costs – PV (bankruptcy costs)
Gain from risk management
= value of firm hedged – value of firm unhedged
= PV( bankruptcy costs)
Value of firm unhedged
+
gain from risk management
= value of firm hedged
= value of firm without bankruptcy costs
Taxes and risk
management
Tax rationale for risk management: If it moves a dollar away from a possible
outcome in which the taxpayer is subject to a high tax rate and shifts it to a
possible outcome where the taxpayer incurs a low tax rate, a firm or an
investor reduces the present value of taxes to be paid. It applies whenever
income is taxed differently at different levels.
- Carrybacks and carryforwards
- Tax shields
- Personal taxes
Example
The firm pays taxes at the rate of 50 percent on cash flow in excess of
$300 per ounce. For simplicity, the price of fold is either $250 or $450 with
Equal probability. The forward price is $350.
Optimal capital
structure and risk
management
• In general, firms cannot eliminate all risk, debt is
risky.
• By having more debt, firms increase their tax shield from debt but
increase the present value of costs of financial distress.
• The optimal capital structure of a firm:
• Balances the tax benefits of debt against the costs of financial
distress.
• Through risk management:
• A firm can reduce the present value of the costs of financial distress
by making financial distress less likely.
• As a result, it can take on more debt.
Should the firm hedge to reduce the risk of
large undiversified shareholders?
• Large undiversified shareholders can increase firm
value
• Risk and the incentives of managers
• Large shareholders, managerial incentives, and
homestake
Figure 3.6. Firm after-tax cash flow and debt issue.
After tax cash flow of hedged
firm
330
325
320
315
Principal amount of debt
310
305
Optimal amount of
debt, $317.073M
100
200
300
400
Risk management process
Risk identification
Risk assessment
Review
Selection of risk-mgt
techniques
Implementation
The rules of risk management
Risk Management
• There is no return without risk
• Be transparent
• Seek experience
• Know what you don’t know
• Communicate
• Diversify
• Show discipline
• Use common sense
• Get a RiskGrade
Source: Riskmetrics Group (www.riskmetrics.com)
Types of risks firms face
Market risk
- interest rate
- foreign exchange
- commodity price
Operational risk
- industry sectors
- geographical regions
Hazard risk
- physical damage
- liabilities
- business interruption
Strategic risk
- competition
- reputation
- investor support
Assignment of risk responsibilities
CEO
Strategic risk
management
CRO
Market risk
management
Hazard risk
management
Operational risk
management
Hedgeable
Insurable
Diversifiable
Three dimensions of risk transfer
•Hedging
•Insuring
•Diversifying
A new concept of risk management
(VAR)
• Value-at-risk (VAR) is a category of risk measures that
describe probabilistically the market risk of mostly a trading
portfolio.
• It summarizes the predicted maximum loss (or worst loss) over
a target horizon within a given confidence interval.
• If the portfolio return is normally distributed, has zero mean,
and has volatility s over the measurement period, the 5 percent
VAR of the portfolio is:
VAR = 1.65 X s X Portfolio value
Example of VAR
• The US bank J.P. Morgan states in its 2000
annual report that its aggregate VAR is about
$22m.
• The bank, one of the pioneers in risk management,
may say that for 95 percent of the time it does not
expect to lose more than $22m on a given day.
More on VAR
• The main appeal of VAR was to describe risk in dollars - or
whatever base currency is used - making it far more
transparent and easier to grasp than previous measures.
• VAR also represents the amount of economic capital necessary
to support a business, which is an essential component of
“economic value added” measures.
• VAR has become the standard benchmark” for measuring
financial risk.
Instruments used in risk
management
•
•
•
•
•
•
•
Forward contracts
Futures contracts
Hedging
Interest rate futures contracts
Duration hedging
Swap contracts
Options
Forward Contracts
• A forward contract specifies that a certain commodity will be
exchanged for another at a specified time in the future at prices
specified today.
• Its not an option: both parties are expected to hold up their end of the deal.
• If you have ever ordered a textbook that was not in stock, you have entered into
a forward contract.
Example
Suppose S&P index price is $1050 in 6 months. A
holder who entered a long position at a forward price of
$1020 is obligated to pay $1020 to acquire the index,
and hence earns $1050 - $1020 = $30 per unit of the
index. The short is likewise obligated to sell for $1020,
and thus loses $30.
Payoff after 6 months
If the index price in 6 months = $1020, both the long and short have a 0 payoff.
If the index price > $1020, the long makes money and the short loses money.
If the index price < $1020, the long loses money and the short makes money.
S&R Index
S&R Forward
in 6 months
long
short
900
-$120
$120
950
-70
70
1000
-20
20
1020
0
0
1050
30
-30
1100
80
-80
Problem: The current S&P index is $1000. You have just
purchased a 6- month forward with a price of $1100. If
the index in 6 months has appreciated by 7%, what is
the payoff of this position?
Solution: F0=1100
S1=1000*1.07=1070
Payoff: 1070-1100= - $30.
Example: Valuing a Forward Contract
on a Share of Stock
Consider the obligation to buy a share of Microsoft stock one
year from now for $100. Assume that the stock currently sells
for $97 per share and that Microsoft will pay no dividends
over the coming year. One-year zero-coupon bonds that pay
$100 one year from now currently sell for $92. At what price
are you willing to buy or sell this obligation?
Valuing a forward contract
Strategy 1---- the forward contract
Today
Buy a forward contract
One year from now
Buy stock at a price of $100.
Sell the share for cash at market
Strategy 2 ---- the portfolio strategy
Today
Buy stock today
Sell short $100 in face value
of 1-year zero-coupon bonds
One year from now
Sell the stock
Buyback the zero-coupon
bonds of $100
Valuing a forward contract
Cost
Today
Strategy 1
Strategy 2
?
$97-$92
Cash flow one
year from now
S1- $100
S1- $100
Since strategies 1 and 2 have identical cash flows in the future,
they should have the same cost today to prevent arbitrage.
? = $97 - $92 = $5
In strategy 1, the obligation to buy the stock for $100 one year
from now, should cost $5.
Valuing a forward contract
The no-arbitrage value of a forward contract on a share of stock (the
obligation to buy a share of stock at a price of K, T years in the future),
assuming the stock pays no dividends prior to T, is
S0 
where
S0 = current price of the stock
K
(1  rf )T
K
(1  rf )T = the current market price of a default-free zero-coupon bond
paying K, T years in the future
At no arbitrage: S0 
K
0
T
(1  rf )
F0  K  S0 (1  rf )
T
Currency Forward Rates
• Currency forward rates are a variation on forward price of stock.
• In the absence of arbitrage, the forward currency rate F0 (for example,
Euros/dollar) is related to the current exchange rate (or spot rate) S0, by
the equation
F0 1  rforeign

S0 1  rdomestic
• where r = the return (unannualized) on a domestic or foreign risk-free
security over the life of the forward agreement, as measured in the
respective country's currency
Forward Currency Rates
Example: The Relation Between Forward Currency
Rates and Interest Rates
Assume that six-month LIBOR on Canadian funds is 4 percent
and the US$ Eurodollar rate (six-month LIBOR on U.S. funds)
is 10 percent and that both rates are default free. What is the
six-month forward Can$/US$ exchange rate if the current spot
rate is Can$1.25/US$? Assume that six months from now is
182 days.
Currency Forward Rates
Answer: (LIBOR is a zero-coupon rate based on an
actual/360 day count.) So
Canada
Six-month interest
Rate (unannualized):
The forward rate is
2.02% 
United States
182
182
 4% 5.06 % 
 10 %
360
360
Can$1.21 1.0202

1.25.
US $
1.0506
Futures Contracts: Preliminaries
• A futures contract is like a forward contract:
• It specifies that a certain commodity will be exchanged for another at
a specified time in the future at prices specified today.
• A futures contract is different from a forward:
• Futures are standardized contracts trading on organized exchanges
with daily resettlement (“marking to market”) through a
clearinghouse.
Futures Contracts: Preliminaries
• Standardizing Features:
• Contract Size
• Delivery Month
• Daily resettlement
• Minimizes the chance of default
• Initial Margin
• About 4% of contract value, cash or T-bills held in
a street name at your brokerage.
Daily Resettlement: An Example
Suppose you want to speculate on a rise in the $/¥ exchange
rate (specifically you think that the dollar will appreciate).
Japan (yen)
1-month forward
3-months forward
6-months forward
U.S. $ equivalent
Wed
Tue
0.007142857 0.007194245
0.006993007 0.007042254
0.006666667 0.006711409
0.00625 0.006289308
Currency per
U.S. $
Wed
Tue
140
139
143
142
150
149
160
159
Currently $1 = ¥140.
The 3-month forward price is $1=¥150.
Daily Resettlement: An Example
• Currently $1 = ¥140 and it appears that the dollar is
strengthening.
• If you enter into a 3-month futures contract to sell ¥ at the
rate of $1 = ¥150 you will make money if the yen
depreciates. The contract size is ¥12,500,000
• Your initial margin is 4% of the contract value:
$1
$3,333.33  .04  ¥12,500,000 
¥150
Daily Resettlement: An Example
If tomorrow, the futures rate closes at $1 = ¥149, then
your position’s value drops.
Your original agreement was to sell ¥12,500,000 and
receive $83,333.33:
$1
$83,333.33  ¥12,500,000 
¥150
But ¥12,500,000 is now worth $83,892.62:
$1
$83,892.62  ¥12,500,000 
¥149
You have lost $559.28 overnight.
Daily Resettlement: An Example
• The $559.28 comes out of your $3,333.33 margin account,
leaving $2,774.05
• This is short of the $3,355.70 required for a new position.
$1
$3,355.70  .04  ¥12,500,000 
¥149
Your broker will let you slide until you run through
your maintenance margin. Then you must post
additional funds or your position will be closed out.
This is usually done with a reversing trade.
Selected Futures Contracts
Contract
Agricultural
Contract Size
Exchange
Corn
Wheat
Cocoa
OJ
Metals & Petroleum
Copper
Gold
Unleaded gasoline
Financial
British Pound
Japanese Yen
Eurodollar
5,000 bushels
5,000 bushels
10 metric tons
15,000 lbs.
Chicago BOT
Chicago & KC
CSCE
CTN
25,000 lbs.
100 troy oz.
42,000 gal.
CMX
CMX
NYM
£62,500
¥12.5 million
$1 million
IMM
IMM
LIFFE
Futures Markets
• The Chicago Mercantile Exchange (CME) is
by far the largest.
• Others include:
• The Philadelphia Board of Trade (PBOT)
• The MidAmerica Commodities Exchange
• The Tokyo International Financial Futures Exchange
• The London International Financial Futures Exchange
The Chicago Mercantile Exchange
• Expiry cycle: March, June, September, December.
• Delivery date 3rd Wednesday of delivery month.
• Last trading day is the second business day preceding
the delivery day.
• CME hours 7:20 a.m. to 2:00 p.m. CST.
CME After Hours
• Extended-hours trading on GLOBEX runs from 2:30
p.m. to 4:00 p.m dinner break and then back at it from
6:00 p.m. to 6:00 a.m. CST.
• Singapore International Monetary Exchange
(SIMEX) offer interchangeable contracts.
• There’s other markets, but none are close to CME and
SIMEX trading volume.
Wall Street Journal Futures Price Quotes
Highest price that day
Open
High
Low
Settle
Change
Lifetime
High
Low
Open
Interest
Highest and lowest prices over the lifetime of the contract.
July
Sept
Dec
Corn (CBT) 5,000 bu.; cents per bu.
179
180
178¼
178½
-1½
186
186½
184
186
-¾
196
197
194
196½
-¼
Sept
Dec
TREASURY BONDS (CBT) - $1,000,000; pts. 32nds of 100%
117-05 117-21 116-27 117-05
+5
131-06 111-15 647,560
116-19 117-05 116-12 116-21
+5
128-28 111-06
13,857
Opening price
Sept
Dec
Closing price
312
280
291¼
177
184
194
2,837
104,900
175,187
Daily Change
DJ INDUSTRIAL AVERAGE (CBOT) - $10 times average
11200
11285 11145
11241
-17
11324
7875
11287
11385 11255
11349
-17
11430
7987
18,530
1,599
Lowest price that day
Number of open contracts
Expiry month
Basic Currency Futures Relationships
• Open Interest refers to the number of contracts
outstanding for a particular delivery month.
• Open interest is a good proxy for demand for a
contract.
• Some refer to open interest as the depth of the
market. The breadth of the market would be how
many different contracts (expiry month, currency) are
outstanding.
Hedging
• Two counterparties with offsetting risks can eliminate
risk.
• For example, if a wheat farmer and a flour mill enter into a forward
contract, they can eliminate the risk each other faces regarding the
future price of wheat.
• Hedgers can also transfer price risk to speculators and
speculators absorb price risk from hedgers.
• Speculating: Long vs. Short
Hedging and Speculating Example
You speculate that copper will go up in price, so you go long
10 copper contracts for delivery in 3 months. A contract is
25,000 pounds in cents per pound and is at $0.70 per
pound or $17,500 per contract.
If futures prices rise by 5 cents, you will gain:
Gain = 25,000 × .05 × 10 = $12,500
If prices decrease by 5 cents, your loss is:
Loss = 25,000 × -.05 × 10 = -$12,500
Hedging: How many contacts?
You are a farmer and you will harvest 50,000 bushels of corn in
3 months. You want to hedge against a price decrease. Corn is
quoted in cents per bushel at 5,000 bushels per contract. It is
currently at $2.30 cents for a contract 3 months out and the
spot price is $2.05.
To hedge you will sell 10 corn futures contracts:
50,000 bushels
 10 contracts
5,000 bushels per contract
Now you can quit worrying about the price of corn
and get back to worrying about the weather.
Interest Rate Futures
Contracts
Pricing of Treasury Bonds
Consider a Treasury bond that pays a semiannual coupon of $C
for the next T years:
• The yield to maturity is r
0
C
C
C
1
2
3
…
CF
2T
Value of the T-bond under a flat term structure
= PV of face value + PV of coupon payments
F
C
1 
PV 
 1 
T
T 
(1  r )
r  (1  r ) 
Pricing of Treasury Bonds
If the term structure of interest rates is not flat, then
we need to discount the payments at different rates
depending upon maturity
0
C
C
C
1
2
3
…
CF
2T
= PV of face value + PV of coupon payments
C
C
C
CF
PV 



2
3
T
(1  r1 ) (1  r2 ) (1  r3 )
(1  r2T )
Pricing of Forward Contracts
An N-period forward contract on that T-Bond
 Pforward C
C
C
CF
…
0
N N+1 N+2 N+3
N+2T
Can be valued as the present value of the forward price.
PV 
Pforward
(1  rN )
N
C
C
C
CF



2
3
(1  rN 1 ) (1  rN  2 ) (1  rN 3 )
(1  rN  2T )T
PV 
(1  rN ) N
Futures Contracts
• The pricing equation given above will be
a good approximation.
• The only real difference is the daily
resettlement.
Hedging in Interest Rate Futures
• A mortgage lender who has agreed to loan money in
the future at prices set today can hedge by selling
those mortgages forward.
• It may be difficult to find a counterparty in the
forward who wants the precise mix of risk, maturity,
and size.
• It’s likely to be easier and cheaper to use interest rate
futures contracts however.
Duration Hedging
• As an alternative to hedging with futures or forwards,
one can hedge by matching the interest rate risk of
assets with the interest rate risk of liabilities.
• Duration is the key to measuring interest rate risk.
Duration Hedging
• Duration measures the combined effect of maturity,
coupon rate, and YTM on bond’s price sensitivity
• Measure of the bond’s effective maturity
• Measure of the average life of the security
• Weighted average maturity of the bond’s cash flows
Duration Formula
PV (C1 ) 1  PV (C2 )  2    PV (CT )  T
D
PV
N
Ct  t

t
(
1

r
)
D  tN1
Ct

t
(
1

r
)
t 1
Calculating Duration
Calculate the duration of a three-year bond that
pays a semi-annual coupon of $40, has a $1,000
par value when the YTM is 8% semiannually?
Calculating Duration
Years
Discount
Cash flow factor
0.5
$40.00
1
$40.00
1.5
$40.00
2
$40.00
2.5
$40.00
3 $1,040.00
0.96154
0.92456
0.88900
0.85480
0.82193
0.79031
Present Years x PV
value / Bond price
$38.46
0.0192
$36.98
0.0370
$35.56
0.0533
$34.19
0.0684
$32.88
0.0822
$821.93
2.4658
$1,000.00
2.7259 years
Bond price Bond duration
Duration is expressed in units of time; usually years.
Duration
The key to bond portfolio management
• Properties:
•
•
•
•
Longer maturity, longer duration
Duration increases at a decreasing rate
Higher coupon, shorter duration
Higher yield, shorter duration
• Zero coupon bond: duration = maturity
Swaps Contracts: Definitions
• In a swap, two counterparties agree to a contractual
arrangement wherein they agree to exchange cash
flows at periodic intervals.
• There are two types of interest rate swaps:
• Single currency interest rate swap
• “Plain vanilla” fixed-for-floating swaps are often just called interest rate
swaps.
• Cross-Currency interest rate swap
• This is often called a currency swap; fixed for fixed rate debt service in
two (or more) currencies.
The Swap Bank
• A swap bank is a generic term to describe a financial
institution that facilitates swaps between
counterparties.
• The swap bank can serve as either a broker or a
dealer.
• As a broker, the swap bank matches counterparties but does not assume
any of the risks of the swap.
• As a dealer, the swap bank stands ready to accept either side of a
currency swap, and then later lay off their risk, or match it with a
counterparty.
An Example of an Interest Rate Swap
• Consider this example of a “plain vanilla” interest rate swap.
• Bank A is a AAA-rated international bank located in the U.K.
and wishes to raise $10,000,000 to finance floating-rate
Eurodollar loans.
• Bank A is considering issuing 5-year fixed-rate Eurodollar bonds at 10 percent.
• It would make more sense to for the bank to issue floating-rate notes at LIBOR
to finance floating-rate Eurodollar loans.
An Example of an Interest Rate Swap
• Firm B is a BBB-rated U.S. company. It needs
$10,000,000 to finance an investment with a five-year
economic life.
• Firm B is considering issuing 5-year fixed-rate Eurodollar bonds at
11.75 percent.
• Alternatively, firm B can raise the money by issuing 5-year floating-
rate notes at LIBOR + ½ percent.
• Firm B would prefer to borrow at a fixed rate.
An Example of an Interest Rate Swap
The borrowing opportunities of the two firms are:
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
An Example of an Interest Rate Swap
The swap bank makes
this offer to Bank A: You
pay LIBOR – 1/8 % per
year on $10 million for 5
years and we will pay
you 10 3/8% on $10
million for 5 years
Swap
10 3/8%
Bank
LIBOR – 1/8%
Bank
A
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
An Example of an Interest Rate Swap
½% of $10,000,000 =
$50,000. That’s quite
a cost savings per year
for 5 years.
10 3/8%
Here’s what’s in it for Bank A:
They can borrow externally at
10% fixed and have a net
borrowing position of
Swap
Bank
-10 3/8 + 10 + (LIBOR – 1/8) =
LIBOR – 1/8%
LIBOR – ½ % which is ½ %
better than they can borrow
floating without a swap.
Bank
10%
A
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
An Example of an Interest Rate Swap
The swap bank
makes this offer to
company B: You
pay us 10½% per
year on $10 million
for 5 years and we
will pay you
LIBOR – ¼ % per
year on $10 million
for 5 years.
Swap
Bank
10 ½%
LIBOR – ¼%
Company
B
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
An Example of an Interest Rate Swap
Here’s what’s in it for B:
Swap
½ % of $10,000,000 =
$50,000 that’s quite a cost
savings per year for 5
years.
Bank
10 ½%
They can borrow externally at
LIBOR + ½ % and have a net
LIBOR – ¼%
Company
borrowing position of
10½ + (LIBOR + ½ ) - (LIBOR - ¼ ) = 11.25%
which is ½% better than they can borrow floating.
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
B
LIBOR
+ ½%
An Example of an Interest Rate Swap
The swap bank makes money too.
¼% of $10 million
= $25,000 per year
for 5 years.
Swap
10 3/8%
Bank
10 ½%
LIBOR – 1/8%
Bank
LIBOR – ¼%
LIBOR – 1/8 – [LIBOR – ¼ ]= 1/8
A
Company
10 ½ - 10 3/8 = 1/8
¼
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
B
An Example of an Interest Rate Swap
The swap bank makes ¼%
Swap
10 3/8%
Bank
10 ½%
LIBOR – 1/8%
LIBOR – ¼%
Bank
Company
A
B
A saves ½%
B saves ½%
COMPANY
Fixed rate
Floating rate
B
BANK A
11.75%
10%
LIBOR + .5%
LIBOR
An Example of a Currency Swap
• Suppose a U.S. MNC wants to finance a £10,000,000
expansion of a British plant.
• They could borrow dollars in the U.S. where they are well
known and exchange for dollars for pounds.
• This will give them exchange rate risk: financing a sterling project with
dollars.
• They could borrow pounds in the international bond market,
but pay a premium since they are not as well known abroad.
An Example of a Currency Swap
• If they can find a British MNC with a mirrorimage financing need they may both benefit
from a swap.
• If the spot exchange rate is S0($/£) = $1.60/£,
the U.S. firm needs to find a British firm
wanting to finance dollar borrowing in the
amount of $16,000,000.
An Example of a Currency Swap
Consider two firms A and B: firm A is a U.S.–based multinational
and firm B is a U.K.–based multinational.
Both firms wish to finance a project in each other’s country of the
same size. Their borrowing opportunities are given in the table
below.
$
£
Company A
8.0%
11.6%
Company B
10.0% 12.0%
An Example of a Currency Swap
Swap
Bank
$8%
$9.4%
£11%
$8%
£12%
Firm
Firm
A
B
$
£
Company A
8.0%
11.6%
Company B
10.0% 12.0%
£12%
An Example of a Currency Swap
A’s net position is to borrow at £11%
Swap
Bank
$8%
$9.4%
£11%
$8%
£12%
Firm
Firm
A
B
A saves £.6%
$
£
Company A
8.0%
11.6%
Company B
10.0% 12.0%
£12%
An Example of a Currency Swap
B’s net position is to borrow at $9.4%
Swap
Bank
$8%
$9.4%
£11%
$8%
£12%
Firm
Firm
A
B
$
£
Company A
8.0%
11.6%
Company B
10.0% 12.0%
£12%
B saves $.6%
An Example of a Currency Swap
The swap bank makes money too:
Swap
Bank
$8%
£11%
$8%
Firm
1.4% of $16 million
financed with 1% of
£10 million per year
for 5 years.
$9.4%
£12%
Firm £12%
At S0($/£) = $1.60/£, that
is a gain of $124,000 per
A
B
year for 5 years.
The swap bank
$
£
faces exchange rate
Company A 8.0% 11.6% risk, but maybe
Company B 10.0% 12.0% they can lay it off
(in another swap).
Variations of Basic Swaps
• Currency Swaps
• fixed for fixed
• fixed for floating
• floating for floating
• amortizing
• Interest Rate Swaps
• zero-for floating
• floating for floating
• Exotica
• For a swap to be possible, two humans must like the idea. Beyond
that, creativity is the only limit.
Risks of Interest Rate and
Currency Swaps
• Interest Rate Risk
• Interest rates might move against the swap bank after it has only gotten
half of a swap on the books, or if it has an unhedged position.
• Basis Risk
• If the floating rates of the two counterparties are not pegged to the same
index.
• Exchange Rate Risk
• In the example of a currency swap given earlier, the swap bank would
be worse off if the pound appreciated.
Risks of Interest Rate and
Currency Swaps
• Credit Risk
• This is the major risk faced by a swap dealer—the risk that a counter
party will default on its end of the swap.
• Mismatch Risk
• It’s hard to find a counterparty that wants to borrow the right amount of
money for the right amount of time.
• Sovereign Risk
• The risk that a country will impose exchange rate restrictions that will
interfere with performance on the swap.
Pricing a Swap
• A swap is a derivative security so it can be priced in
terms of the underlying assets:
• How to:
• Plain vanilla fixed for floating swap gets valued just like a bond.
• Currency swap gets valued just like a nest of currency futures.
Options
• Many corporate securities are similar to the stock options
that are traded on organized exchanges.
• Almost every issue of corporate stocks and bonds has
option features.
• In addition, capital structure and capital budgeting
decisions can be viewed in terms of options.
Options Contracts: Preliminaries
• An option gives the holder the right, but not the obligation, to
buy or sell a given quantity of an asset on (or perhaps before) a
given date, at prices agreed upon today.
• Calls versus Puts
• Call options gives the holder the right, but not the obligation, to buy a
given quantity of some asset at some time in the future, at prices agreed
upon today. When exercising a call option, you “call in” the asset.
• Put options gives the holder the right, but not the obligation, to sell a
given quantity of an asset at some time in the future, at prices agreed
upon today. When exercising a put, you “put” the asset to someone.
Options Contracts: Preliminaries
• Exercising the Option
• The act of buying or selling the underlying asset through the option contract.
• Strike Price or Exercise Price
• Refers to the fixed price in the option contract at which the holder can buy or
sell the underlying asset.
• Expiry
• The maturity date of the option is referred to as the expiration date, or the
expiry.
• European versus American options
• European options can be exercised only at expiry.
• American options can be exercised at any time up to expiry.
Options Contracts: Preliminaries
• In-the-Money
• The exercise price is less than the spot price of the underlying asset.
• At-the-Money
• The exercise price is equal to the spot price of the underlying asset.
• Out-of-the-Money
• The exercise price is more than the spot price of the underlying asset.
Options Contracts: Preliminaries
• Intrinsic Value
• The difference between the exercise price of the option and the spot
price of the underlying asset.
• Speculative Value
• The difference between the option premium and the intrinsic value of
the option.
Option
Premium
=
Intrinsic
Value
+ Speculative
Value
Call Options
• Call options gives the holder the right, but not the
obligation, to buy a given quantity of some asset
on or before some time in the future, at prices
agreed upon today.
• When exercising a call option, you “call in” the
asset.
Basic Call Option Pricing
Relationships at Expiry
• At expiry, an American call option is worth the same as a
European option with the same characteristics.
• If the call is in-the-money, it is worth ST - E.
• If the call is out-of-the-money, it is worthless.
CaT = CeT = Max[ST - E, 0]
• Where
ST is the value of the stock at expiry (time T)
E is the exercise price.
CaT is the value of an American call at expiry
CeT is the value of a European call at expiry
Call Option Payoffs
60
Option payoffs ($)
40
Buy a call
20
0
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
Call Option Payoffs
60
Option payoffs ($)
40
20
0
-20
0
10
20
30
40
50
60
70
80
Stock price ($)
Write a call
-40
-60
Exercise price = $50
90
100
Call Option Profits
60
Option profits ($)
40
Buy a call
20
0
-20
0
10
20
30
40
50
60
70
80
90
Stock price ($)
Write a call
-40
-60
Exercise price = $50; option premium = $10
100
Put Options
• Put options gives the holder the right, but
not the obligation, to sell a given quantity of
an asset on or before some time in the
future, at prices agreed upon today.
• When exercising a put, you “put” the asset
to someone.
Basic Put Option Pricing
Relationships at Expiry
• At expiry, an American put option is worth
the same as a European option with the same
characteristics.
• If the put is in-the-money, it is worth E - ST.
• If the put is out-of-the-money, it is
worthless.
PaT = PeT = Max[E - ST, 0]
Put Option Payoffs
60
Option payoffs ($)
40
Buy a put
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
Put Option Payoffs
60
Option payoffs ($)
40
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
write a put
-60
Exercise price = $50
Option profits ($)
Put Option Profits
60
40
20
10
0
-10
-20
Write a put
0
10
20
30
Stock price ($)
40
50
60
70 80
Buy a put
90
-40
-60
Exercise price = $50; option premium = $10
100
Selling Options
($)
Option profitsOption
profits ($)
• The seller (or writer) of an
option has an obligation.
• The purchaser of an option
has an option.
60
40
20
10
0
-10
-20
Buy a call
Write a put
0
10
20
30
Stock price ($)
40
50
60
70 80
Buy a put
Write a call
-40
-60
90
100
Reading The Wall Street Journal
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
Reading The Wall Street Journal
This option has a strike price of $135;
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
a recent price for the stock is $138.25
July is the expiration month
Reading The Wall Street Journal
This makes a call option with this exercise price in-themoney by $3.25 = $138¼ – $135.
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
Puts with this exercise price are out-of-the-money.
Reading The Wall Street Journal
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
On this day, 2,365 call options with this exercise price were
traded.
Reading The Wall Street Journal
The CALL option with a strike price of $135 is trading for
$4.75.
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
Since the option is on 100 shares of stock, buying this option
would cost $475 plus commissions.
Reading The Wall Street Journal
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
On this day, 2,431 put options with this exercise price were
traded.
Reading The Wall Street Journal
The PUT option with a strike price of $135 is trading for
$.8125.
Option/Strike Exp.
130 Oct
IBM
130 Jan
138¼
135 Jul
138¼
135 Aug
138¼
140 Jul
138¼
140 Aug
138¼
--Put---Call-Vol. Last Vol. Last
5¼
107
364 15¼
9¼
420
112 19½
4¾ 2431 13/16
2365
5½
94
9¼
1231
2¾
427
1¾
1826
7½
58
6½
2193
Since the option is on 100 shares of stock, buying this
option would cost $81.25 plus commissions.
Combinations of Options
• Puts and calls can serve as the building
blocks for more complex option contracts.
• If you understand this, you can become a
financial engineer, tailoring the risk-return
profile to meet your client’s needs.
Protective Put Strategy: Buy a Put and Buy
the Underlying Stock: Payoffs at Expiry
Value at
expiry
Protective Put strategy has
downside protection and
upside potential
$50
Buy the
stock
Buy a put with an exercise
price of $50
$0
$50
Value of
stock at
expiry
Protective Put Strategy Profits
Value at
expiry
$40
Buy the stock at $40
Protective Put
strategy has
downside protection
and upside potential
$0
$40 $50
-$40
Buy a put with
exercise price of
$50 for $10
Value of
stock at
expiry
Covered Call Strategy
Value at
expiry
$40
Buy the stock at $40
Covered call
$10
$0
Value of stock at expiry
$30 $40 $50
-$30
-$40
Sell a call with
exercise price of
$50 for $10
Long Straddle: Buy a Call and a Put
Value at
expiry
Buy a call with an
exercise price of
$50 for $10
$40
$30
$0
-$10
-$20
$30 $40 $50 $60
Buy a put with an
$70 exercise price of
$50 for $10
A Long Straddle only makes money if the
stock price moves $20 away from $50.
Value of
stock at
expiry
Short Straddle: Sell a Call and a Put
Value at
expiry
$20
$10
$0
A Short Straddle only loses money if the stock
price moves $20 away from $50.
Sell a put with exercise price of
$50 for $10
Value of stock at
expiry
-$30
-$40
$30 $40 $50 $60 $70
Sell a call with an
exercise price of $50 for $10
Long Call Spread
Value at
expiry
Buy a call with an
exercise price of
$50 for $10
$5
$0
-$5
-$10
long call spread
$50 $60
Value of
stock at
expiry
$55
Sell a call with exercise
price of $55 for $5
Put-Call Parity
In market equilibrium, it mast be the case that option prices
are set such that:
C  Xe rT  P  S
0
0
0
Otherwise, riskless portfolios with positive payoffs exist.
Value at
expiry
Buy a call option with
an exercise price of $40
Buy the
Buy the stock at $40 stock at $40
financed with some
debt: FV = $X
P0
Sell a put with an
exercise price of $40
$0
 C0
-[$40-P0]
 rT
 ($40  Xe )
-$40
$40-P0
$40
$40  Xe rT
$40  C0
Value of
stock at
expiry
Valuing Options
• The last section
concerned itself with the
value of an option at
expiry.
• This section considers
the value of an option
prior to the expiration
date.
• A much more
interesting question.
Option Value Determinants
1.
2.
3.
4.
5.
Stock price
Exercise price
Interest rate
Volatility in the stock price
Expiration date
+
Call Put
+
–
–
+
–
+
+
+
+
The value of a call option C0 must fall within
max (S0 – E, 0) < C0 < S0.
The precise position will depend on these factors.
Market Value, Time Value and Intrinsic Value for an
American Call
Profit
The value of a call option C0 must fall within
max (S0 – E, 0) < C0 < S0.
CaT > Max[ST - E, 0]
Market Value
Time value
Intrinsic value
loss
Out-of-the-money
E
In-the-money
ST
An Option-Pricing Formula
• We will start with a
binomial option pricing
formula to build our
intuition.
• Then we will graduate
to the normal
approximation to the
binomial for some realworld option valuation.
Binomial Option Pricing Model
Suppose a stock is worth $25 today and in one period will either be worth
15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or
$21.25. The risk-free rate is 5%. What is the value of an at-the-money call
option?
S0
S1
$28.75
$25
$21.25
Binomial Option Pricing Model
1.
2.
A call option on this stock with exercise price of $25 will have the
following payoffs.
We can replicate the payoffs of the call option. With a levered position in
the stock.
S0
S1
C1
$28.75
$3.75
$21.25
$0
$25
Binomial Option Pricing Model
Borrow the present value of $21.25 today and buy 1 share.
The net payoff for this levered equity portfolio in one period is either $7.50
or $0.
The levered equity portfolio has twice the option’s payoff so the portfolio is
worth twice the call option value.
S0
( S1 - debt ) = portfolio C1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
$0
$0
Binomial Option Pricing Model
The levered equity portfolio value today is
today’s value of one share less the present value
of a $21.25 debt:
$21.25
$25 
S0
(1  rf )
( S1 - debt ) = portfolio C1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
$0
$0
Binomial Option Pricing Model
We can value the option today as
half of the value of the levered
equity portfolio:
S0
1 
$21.25 
C0  $25 
2 
(1  rf ) 
( S1 - debt ) = portfolio C1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
$0
$0
The Binomial Option Pricing Model
If the interest rate is 5%, the call is worth:
1
$21.25  1
  $25  20.24  $2.38
C0   $25 
2
(1.05)  2
S0
( S1 - debt ) = portfolio C1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
$0
$0
The Binomial Option Pricing Model
If the interest rate is 5%, the call is worth:
1
$21.25  1
  $25  20.24  $2.38
C0   $25 
2
(1.05)  2
S0
C0
( S1 - debt ) = portfolio C1
$28.75 - $21.25 = $7.50
$3.75
$25 $2.38
$21.25 - $21.25 =
$0
$0
Binomial Option Pricing Model
The most important lesson (so far) from the binomial
option pricing model is:
the replicating portfolio intuition.
Many derivative securities can be valued by
valuing portfolios of primitive securities
when those portfolios have the same
payoffs as the derivative securities.
The Risk-Neutral Approach to Valuation
S(U), V(U)
q
S(0), V(0)
1- q
S(D), V(D)
We could value V(0) as the value of the replicating portfolio.
An equivalent method is risk-neutral valuation
V (0) 
q  V (U )  (1  q)  V ( D)
(1  rf )
The Risk-Neutral Approach to Valuation
S(U), V(U)
q
q is the risk-neutral
probability of an “up”
move.
S(0), V(0)
1- q
S(0) is the value of the
S(D), V(D)
underlying asset today.
S(U) and S(D) are the values of the asset in the next
period following an up move and a down move,
respectively.
V(U) and V(D) are the values of the asset in the next period following an
up move and a down move, respectively.
The Risk-Neutral Approach to
Valuation
S(U), V(U)
q
V (0) 
S(0), V(0)
q  V (U )  (1  q)  V ( D)
(1  rf )
1- q
S(D), V(D)
• The key to finding q is to note that it is already impounded into
an observable security price: the value of S(0):
q  S (U )  (1  q)  S ( D)
S (0) 
(1  rf )
A minor bit of algebra yields: q 
(1  rf )  S (0)  S ( D)
S (U )  S ( D)
Example of the Risk-Neutral Valuation of a Call:
Suppose a stock is worth $25 today and in one period will either
be worth 15% more or 15% less. The risk-free rate is 5%. What
is the value of an at-the-money call option?
The binomial tree would look like this:
$28.75  $25  (1.15)
q
$25,C(0)
$28.75,C(D)
$21.25  $25  (1  .15)
1- q
$21.25,C(D)
Example of the Risk-Neutral Valuation of a Call:
The next step would be to compute the risk neutral
probabilities
q
q
(1  rf )  S (0)  S ( D)
S (U )  S ( D)
(1.05)  $25  $21.25
$5

2 3
$28.75  $21.25
$7.50
2/3
$28.75,C(D)
$25,C(0)
1/3
$21.25,C(D)
Example of the Risk-Neutral Valuation of a Call:
After that, find the value of the call in the up state
and down state.
C (U )  $28.75  $25
2/3
$28.75, $3.75
C ( D)  max[$ 25  $28.75,0]
$25,C(0)
1/3
$21.25, $0
Example of the Risk-Neutral Valuation of a Call:
Finally, find the value of the call at time 0:
C (0) 
q  C (U )  (1  q)  C ( D)
(1  rf )
C (0) 
2 3  $3.75  (1 3)  $0
(1.05)
C (0) 
$2.50
 $2.38
(1.05)
2/3
$28.75,$3.75
$25,$2.38
$25,C(0)
1/3
$21.25, $0
Risk-Neutral Valuation and the Replicating Portfolio
This risk-neutral result is consistent with valuing the
call using a replicating portfolio.
2 3  $3.75  (1 3)  $0 $2.50
C0 

 $2.38
(1.05)
1.05
1
$21.25  1
  $25  20.24  $2.38
C0   $25 
2
(1.05)  2
The Black-Scholes Model
The Black-Scholes Model is
C0  S  N(d1 )  Ee rT  N(d 2 )
Where
C0 = the value of a European option at time t = 0
r = the risk-free interest rate.
σ2
ln( S / E )  (r  )T
2
d1 
s T
d 2  d1  s T
N(d) = Probability that a
standardized, normally
distributed, random
variable will be less than
or equal to d.
The Black-Scholes Model allows us to value options in the
real world just as we have done in the 2-state world.
The Black-Scholes Model
Find the value of a six-month call option on the Microsoft with
an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsic value of the option is
$10—our answer must be at least that amount.
The Black-Scholes Model
Let’s try our hand at using the model. If you have a calculator
handy, follow along.
First calculate d1 and d2
ln( S / E )  (r  .5σ 2 )T
d1 
s T
ln( 160 / 150)  (.05  .5(0.30) 2 ).5
d1 
 0.5282
0.30 .5
Then,
d 2  d1  s T  0.52815  0.30 .5  0.31602
The Black-Scholes Model
C0  S  N(d1 )  Ee
d1  0.5282
d 2  0.31602
 rT
 N(d 2 )
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
C0  $160  0.7013  150e .05.5  0.62401
C0  $20.92
Stocks and Bonds as Options
• Levered Equity is a Call Option.
• The underlying asset comprise the assets of the firm.
• The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of the firm are greater in
value than the debt, the shareholders have an in-the-money call,
they will pay the bondholders and “call in” the assets of the firm.
• If at the maturity of the debt the shareholders have an out-of-themoney call, they will not pay the bondholders (i.e. the
shareholders will declare bankruptcy) and let the call expire.
Stocks and Bonds as Options
• Levered Equity is a Put Option.
• The underlying asset comprise the assets of the firm.
• The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of the firm are less in
value than the debt, shareholders have an in-the-money put.
• They will put the firm to the bondholders.
• If at the maturity of the debt the shareholders have an out-of-
the-money put, they will not exercise the option (i.e. NOT
declare bankruptcy) and let the put expire.
Stocks and Bonds as Options
• It all comes down to put-call parity.
C0  S  P0  X e
Value of a
call on the
firm
rT
Value of a
Value of
= the firm + put on the –
firm
Stockholder’s
position in terms
of call options
Stockholder’s
position in terms
of put options
Value of a
risk-free
bond
Capital-Structure Policy and Options
• Recall some of the agency costs of debt:
they can all be seen in terms of options.
• For example, recall the incentive
shareholders in a levered firm have to take
large risks.
Balance Sheet for a Company in Distress
Assets
Cash
Fixed Asset
Total
BV
$200
$400
$600
MV
$200
$0
$200
Liabilities
LT bonds
Equity
Total
BV
$300
$300
$600
MV
?
?
$200
What happens if the firm is liquidated today?
The bondholders get $200; the shareholders get nothing.
Selfish Strategy 1: Take Large Risks
(Think of a Call Option)
The Gamble
Win Big
Lose Big
Probability
10%
90%
Payoff
$1,000
$0
Cost of investment is $200 (all the firm’s cash)
Required return is 50%
Expected CF from the Gamble = $1000 × 0.10 + $0 = $100
$100
NPV  $200 
1.50
NPV  $133
Selfish Stockholders Accept Negative NPV Project with Large
Risks
• Expected cash flow from the Gamble
• To Bondholders = $300 × 0.10 + $0 = $30
• To Stockholders = ($1000 - $300) × 0.10 + $0 = $70
•
•
•
•
PV of Bonds Without the Gamble = $200
PV of Stocks Without the Gamble = $0
PV of Bonds With the Gamble = $30 / 1.5 = $20
PV of Stocks With the Gamble = $70 / 1.5 = $47
The stocks are worth more with the high risk project because
the call option that the shareholders of the levered firm hold
is worth more when the volatility is increased.
Mergers and Options
• This is an area rich with optionality, both
in the structuring of the deals and in their
execution.
Investment in Real Projects & Options
• Classic NPV calculations typically ignore
the flexibility that real-world firms typically
have.
• The next chapter will take up this point.
Summary and Conclusions
• The most familiar options are puts and calls.
• Put options give the holder the right to sell stock at a set
price for a given amount of time.
• Call options give the holder the right to buy stock at a set
price for a given amount of time.
• Put-Call parity
C0  X e
rT
 S  P0
Summary and Conclusions
• The value of a stock option depends on six factors:
1. Current price of underlying stock.
2. Dividend yield of the underlying stock.
3. Strike price specified in the option contract.
4. Risk-free interest rate over the life of the contract.
5. Time remaining until the option contract expires.
6. Price volatility of the underlying stock.
• Much of corporate financial theory can be presented
in terms of options.
1.
2.
Common stock in a levered firm can be viewed as a call option on the
assets of the firm.
Real projects often have hidden option that enhance value.