Options
Global Financial Management
Campbell R. Harvey
Fuqua School of Business
Duke University
charvey@mail.duke.edu
http://www.duke.edu/~charvey
1
Overview




Options:
» Uses, definitions, types
Put-Call Parity
» Futures and Forwards
Valuation
» Binomial
» Black Scholes
Applications
» Portfolio Insurance
» Hedging
2
Definitions
Call Option
is a right (but not an obligation) to buy an asset at a prearranged price (=exercise price) on or until a pre-arranged date
(=maturity).
Put Option
is a right (but not an obligation) to sell an asset at a prearranged price (=exercise price) on or until a pre-arranged date
(=maturity).
European Options
can be exercised at maturity only.
American Options
can be exercised at any time before maturity
3
Examples of Options
Securities
 Equity options
Warrants
Underwriting
Call provisions
Convertible bonds
Caps
Interest rate options
Insurance
Loan guarantees
 Risky bonds
 Equity
Real Options
Options to expand
Abandonment options
Options to delay investment
Model sequences
Options are everywhere!
4
Values of Options at Expiry
Buying a Call
Payoff
Buy Call Option
Payoff = max[0, ST - X]
0
X
Stock Price
5
Values of Options at Expiry
Writing a Call
Payoff
0
Sell Call Option
Payoff = - max[0, ST - X]
X
Stock Price
6
Values of Options at Expiry
Buying a Put
Payoff
Buy Put Option
Payoff = max[0, X - ST ]
X
0
X
Stock Price
7
Values of Options at Expiry
Selling a Put
Payoff
0
Sell Put Option
Payoff = - max[0, X - ST]
X
Stock Price
-X
8
Example

What are the payoffs to the buyer of a call option and a put option if the
exercise price is X=$50?
Stock
Price
20
Buy Call Write Call Buy Put Write Put
0
0
30
-30
40
0
0
10
-10
60
10
-10
0
0
80
30
-30
0
0
9
Valuation of Options:
Put-Call Parity


Principle:
» Construct two portfolios
» Show they have the same payoffs
» Conclude they must cost the same
Portfolio I: Buy a share of stock today for a price of S0 and
simultaneously borrowed an amount of PV(X)=Xe-rT.
» How much would your portfolio be worth at the end of T years?
– Assume that the stock does not pay a dividend.
Position
Buy Stock
Borrow
Portfolio I
0
-S0
PV(X)
PV(X) - S0
T
ST
-X
ST - X
10
Payoff of Portfolio I
Payoff
Payoff on
Stock
ST
ST - X
Net Payoff
0
X
Stock Price
Payoff on
Borrowing
-X
11
Put-Call Parity

Portfolio II: Buy a call option and sell a put option with a maturity date
of T and an exercise price of X. How much will your options be worth
at the end of T years?
Position
Buy Call
Sell Put
Net Position


0
-CE
PE
PE-CE
T
max[0,ST-X]
-max[0,X-ST]
ST - X
Since the two portfolios have the same payoffs at date T, they must
have the same price today.
The put-call parity relationship is:
CE - PE = S0 - PV(X)

This implies: Call - Put = Stock - Bond
12
Put-Call Parity
Payoff
Payoff on
long call
ST - X
Net
Payoff
X
0
Stock Price
Payoff on
short put
-X
13
Put-Call Parity and Arbitrage



A stock is currently selling for $100. A call option with an exercise price
of $90 and maturity of 3 months has a price of $12. A put option with
an exercise price of $90 and maturity of 3 months has a price of $2.
The one-year T-bill rate is 5.0%. Is there an arbitrage opportunity
available in these prices?
From Put-Call Parity, the price of the call option should be equal to:
» CE = PE+ S0 - Xe-rT=$13.12
Since the market price of the call is $12, it is underpriced by $1.12. We
would want to buy the call, sell the put, sell the stock, and invest
PV($90)= 88.88 for 3 months.
14
Put-Call Parity and Arbitrage


The cash flows for this investment are outlined below:
Position
Buy call
Sell put
Sell stock
Buy T-bill
0
-12.00
2.00
100.00
-90e-(0.05)0.25
ST<X
0
ST-90
-ST
90
ST>X
ST-90
0
-ST
90
Net Position
1.12
0
0
Hence, realize an arbitrage profit of 1.12
» This is independent of the value of the stock price!
15
Options and Futures

Compare this with a futures contract that specicifies that you buy
a stock at X at time T. The futures contract trades today at F0.
» What is the price of the futures if there is no arbitrage?
– Construct zero-payoff portfolio: Buy a Put, Write a Call,
and buy the futures contract
Position
Write Call
Buy Put
Buy Futures
Net Position
0
CE
-PE
-F0
CE-PE-F0
T
-Max[0,ST - X]
Max[0,ST - X]
ST - X
0
» Hence, the relationship between futures and options is:
F0  CE  PE
16
Options and Futures
Payoff
Payoff on
long call
X
0
ST - X


Payoff on
Future


Call is right to
purchase
Short Put is obligation
to sell
Future combines both
When is F0=0?
Stock
Price
Payoff on
short put
X
17
Debt and Equity as Options

Suppose a firm has debt with a face value of $1m outstanding
that matures at the end of the year. What is the value of debt
and equity at the end of the year?
0.3m
Payoff to
Shareholders
0
Payoff to
Debtholders
0.3m
0.6m
0
0.6m
0.9m
0
0.9m
1.2m
0.2m
1.0m
1.5m
0.5m
1.0m
Asset Value
18
Debt and Equity




Consider a firm with zero coupon debt outstanding with a face value of
F. The debt will come due in exactly one year.
The payoff to the equityholders of this firm one year from now will be
the following:
Payoff to Equity = max[0, V-F]
where V is the total value of the firm’s assets one year from now.
Similarly, the payoff to the firm’s bondholders one year from now will be:
Payoff to Bondholders = V - max[0,V-F]
Equity has a payoff like that on a call option. Risky debt has a payoff
that is equal to the total value of the firm, less the payoff on a call
option.
19
Debt and Equity
Payoffs
Equityholders
Bondholders
0
F
Firm Value
20
Valuing Options
Establish bounds for Options


Upper bound on European call:
» Compare to following portfolio: buy one share, borrow PV of
exercise price
» Consider value at maturity:
S<X
S>X
Call
0
S-X
Share
S
S
Borrow
X
X
Portfolio
S-X<0
S-X
Hence, since the call is worth more at maturity, CE>S-PV(X)
before maturity
21
Bounds on Option Values
S, C


C=S
CE>S-PV(X); dominates
portfolio of stock and
borrowing X.
CE<S, otherwise buy
stock straightaway
C=S-PV(X)
PV(X)
Stock Price
PV(X)
22
Example on Option Bounds I


Suppose a stock is selling for $50 per share. The riskfree interest rate is
8%. A call option with an exercise price of $50 and 6 months to maturity
is selling for $1.50. Is there an arbitrage opportunity available?
» CE > max[ 0, S0 - Xe-rT ]
» CE > max[ 0, 50 - 50e-(0.08)0.5 ] = 1.96
Since the price is only $1.50, the call is underpriced by at least $0.46.
Position
Buy call
Sell stock
Buy T-bill
0
-1.50
50
-50e-(0.08)0.5
ST<X
0
-ST
50
ST>X
ST-50
-ST
50
Net Postion
0.46
50-ST>0
0
23
Example on Option Bounds II


Now suppose you observe a put option with an exercise price of $55
and 6 months to maturity selling for $2.50. Does this represent an
arbitrage opportunity?
» PE > max[ 0, Xe-rT - S0 ]
» PE > max[ 0, 55e-(0.08)0.5 - 50] = 2.84
Since the price is only $2.50, the put is underpriced by at least
$0.34
Position
Buy put
Buy stock
Borrow
0
-2.50
-50
55e-(0.08)0.5
ST<X
55-ST
ST
-55
ST>X
0
ST
-55
Net Postion
0.34
0
ST-55>0
24
Valuing Options as Contingent
Claims
Idea:
 Investors attach different values to states in which assets pay
off: $1 is worth more in bad times than in good times.
 Values depend on preferences for insuring against bad times
and discounting (time value of money).
 Value of $1 in good times or bad times (or a continuum of
states) can be inferred from prices of stocks and bonds.
125
High State
80
Low State
Stock Price = 100
r=10%
Procedure:
» Determine value of $1 in good and in bad state
» Use the value to infer the value of the option
25
Pricing Contingent Claims
Step 1: Determine the value of states
Method
 Break up payment to shareholders into two components:
» Shareholders receive at least 80 for sure (in good and bad state).
» Shareholders receive an additional 45 if the share price is high,
otherwise nothing.
Steps:
1. The present value of a safe payment of 80 is simply:
80
 72.73
110
.
2. The value shareholders attach to the uncertain 45=125-80 must be the
difference between the current share price and the value of the safe
payment:
100 - 72.73 = 27.27
3. The present value of $1 in the good state is 27.27/45=0.606.
26
Pricing Contingent Claims
Step 2: Value an Option

Consider the following option:
Maturity:
1 year
Exercise price:
110
Type:
European
15
High State
0
Low State
Option Value = ?
How does the option value develop?
27
Why does this work?
Contingent Claim Pricing and Arbitrage

Compare two portfolios:
Portfolio 1: 1 Call option
Asset/State
Stock Price = 80 Stock Price = 125
Portfolio 2: 1/3 share; 1 loan which pays off 80/3 at the end
Loan
-80/3
-80/3
Call Option
0
15
Portfolio 1
0
15
Portfolio 2
0
15
$100 $80 / 3

 $9.09
3
110
.
28
Arbitrage: The General Idea
General Rule:

Use arbitrage principle by constructing portfolio with same
payoffs as option (this is called replication).

Portfolio
has
delta shares
loan
Spread
of Option
Valuesand15
 0 which
1 pays exactly the
  value of the delta shares.

 is called the option delta:
lowest
delta
Spread of Stock Prices 125  80 3

If portfolio replicates option, then it must have the same value as
the option.
29
Options with Many States
Suppose there are more than two possible states at the
end of the period. Then: subdivide period.
Example:
117
3 states at the end of the period:
Divide movement into two
100
periods with two-states
in each.

137
100
85
Solution:
 Value the option for each of the mid-period nodes and
then fold it backwards into the first node.
 Repeat this for ever smaller intervals to cover larger
numbers of states.
73
30
The Black-Scholes Formula
Alternative Solution:

Repeat the above process until infinity;
Continuum of different states.

Use mathematical theory to determine result of this process.
ln S / PV  X   T
d1 

Black-Scholes Formula:
 T
2
d 2  d1   T
Option value=
[delta x share
price]
N (d )  Cumulative
Normal
Density -
N(d1) x P
-
[bank loan]
N(d2) x PV(X)
31
Call Option Sensitivities

The Option Pricing formula gives the following sensitivies for a
call option:
Increase In:
Effect on Call Price
S

T
r
X
32
Intuition for Black-Scholes
C0  e  rC T E ST | ST  X Pr ST  X
 e  rC T X Pr ST  X
 rf T
E * ST | ST  X Pr * ST  X
 rf T
X Pr * ST  X
C0  e
e
e
 rf T
E * ST | ST  X Pr * ST  X  SN(d1 )
Pr * ST  X  N(d2 )
33
Black-Scholes Put Option Formula

We can use the put-call parity relationship to derive the Black-Scholes
put option formula:
PE = CE - S + Xe-rT

Use Put-Call Parity and the fact that the normal distribution is
symmetric around the mean:
PE = -SN(-d1) + Xe-rTN(-d2)
34
Put Option Sensitivities

The Option Pricing formula gives the following sensitivies for a
put option:
Increase In:
Effect on Put Price
S

T
r
X
35
Example


On February 2, 1996, Microsoft stock closed at a price of $93 per
share.
» Annual standard deviation is about 32%.
» The one-year T-bill rate is 4.82%.
What are the Black-Scholes prices for both calls and puts with:
» An exercise price of $100 and
» a maturity of April 1996 (77 days)?
» How do these prices compare to the actual market prices of
these options?
36
How to Use Black-Scholes




The inputs for the Black-Scholes formula are:
» S = $93.00
s r = 4.82%
» X = $100.00
s  = 32%
» T = 77/365
This gives:
d1 = -0.351
d2 = -0.498.
The cumulative normal density for these values are
N(d1) = 0.3628
N(d2) = 0.3103.
Plugging these values into the Black-Scholes formula gives:
c = $3.02
p = $9.02.
37
How to Use Black-Scholes

Microsoft Put and Call Options
Option
B-S Prices
Actual Prices
Apr. call 100
$3.02
$3.25
Apr. put 100
$9.02
$9.125
38
Implied Volatility
39
Implied Volatilities



It is common for traders to quote prices in terms of implied
volatilities.
This is the volatility () that sets the Black-Scholes price equal to
the market price.
This can be computed using SOLVER in EXCEL.
40
Applications of Options I:
Volatility Bets


Suppose you have no information about the return of the stock,
but you believe that the market underrates the volatility of the
stock:
» Give an example!
– How can you trade?
Buy Straddle:
» Buy a call and a put on the same stock
– same exercise price
– same time to maturity..
41
Option Trading Strategies:
The Straddle
Payoff
X
Straddle
Payoff
Put
Payoff
0
Call
Payoff
X
Stock Price
42
Hedging with Options


Initial investment (option premium) is required
You eliminate downside risks, while retaining upside potential
Example
» It is the end of August and we will receive 1m DM at the end of
October.
» At this point, we will sell DM, converting them back into dollars.
» We are concerned about the price at which we will be able to sell
DM.
» We can lock in a minimum sale price by buying put options.
– Since the total exposure is for 1m DM and each contract is
for 62,500 DM we buy 16 put option contracts.
– Suppose we choose the puts struck at 0.66 - locking in a
lower bound of 0.66 $/DM.
43
Heding with Currency Options



Scenario I:
Deutschmark falls to $0.30
We have the right to sell 1m
DM for $0.66 each by
exercising the put options.
Since DM’s are only worth
$0.30 each we do choose to
exercise.
Our cash inflow is therefore
$660,000




Scenario II:
Deutschemark rises to $0.90
We have the right to sell 1m
DM for $0.66 each by
exercising the put options.
Since DM’s are worth $0.90
each we do not choose to
exercise.
We sell the DM on the open
market for $0.90 each.
Our cash inflow is therefore
$900,000
44
Portfolio Insurance

Reconsider the case of a fund manager who wishes to insure
his portfolio
45
Summary





Options are derivative securities:
» Replicate payoffs with combinations of underlying assets
Put and Call prices are linked
Valuation as contingent claims
» Use Black-Scholes as approximation
Value of option increases with volatility of underlying assets
Use options for
» Volatility bets
» Portfolio Insurance
» Hedging
46