Options and Futures
Finansiell ekonomi 723g28
Linköpings University
1
What is a Derivative?
• A derivative is an instrument whose value
depends on, or is derived from, the value
of another asset.
• Examples: futures, forwards, swaps,
options, exotics…
Obs: you may jump to slide #21 to start direct with options.
2
How Derivatives are Used
• To hedge risks
• To speculate (take a view on the future
direction of the market)
• To lock in an arbitrage profit
• To change the nature of a liability
• To change the nature of an investment
without incurring the costs of selling
one portfolio and buying another
3
Options vs. Futures/Forwards
• A futures/forward contract gives the holder
the obligation to buy or sell at a certain
price at a certain date in the future
• An option gives the holder the right, but
not the obligation to buy or sell at a certain
price at a certain date in the future
4
Foreign Exchange Quotes for GBP, (£)
May 24, 2010
The forward price may be different for contracts of different maturities
(as shown by the table)
Spot
Bid
1.4407
Offer
1.4411
1-month forward
1.4408
1.4413
3-month forward
1.4410
1.4415
6-month forward
1.4416
1.4422
5
Long position and short position
• The party that has agreed to buy
has a long position
• The party that has agreed to sell
has a short position
6
Example
• On May 24, 2010 the treasurer of a
corporation enters into a long forward
contract to buy £1 million in six months at
an exchange rate of 1.4422
• This obligates the corporation to pay
$1,442,200 for £1 million on November 24,
2010
• What are the possible outcomes?
7
Svenska termer
Forwards och Terminer
• Spotkontrakt: en överenskommelse mellan två
parter att utbyta något idag för ett specificerat
pris, spotpriset. à vista marknad.
• Terminskontrakt: en överenskommelse
(skyldighet) mellan två parter att utbyta något
för ett specificerat pris, terminspriset, vid en
specifik framtida tidpunkt, lösendagen.
8
Profit from a Long Forward Position
(K= delivery price=forward price at the time contract
is entered into)
Payoff diagram
Profit
K
Price of Underlying at
Maturity, ST
9
Profit from a Short Forward Position
(K= delivery price=forward price at the time contract is
entered into)
Profit
K
Price of Underlying
at Maturity, ST
10
Futures Contracts
• Agreement to buy or sell an asset for a certain
price at a certain time
• Similar to forward contract
• a forward contract is traded over the counter
(OTC) (Skräddarsydd)
• a futures contract is standardized and traded
on an exchange.
CME Group
NYSE Euronext,
BM&F (Sao Paulo, Brazil)
TIFFE (Tokyo)
11
Key Points About Futures
• They are settled daily
• Closing out a futures position
involves entering into an offsetting
trade
• Most contracts are closed out
before maturity
12
Margins
• A margin is cash or marketable securities
deposited by an investor with his or her
broker
• The balance in the margin account is
adjusted to reflect daily settlement
• Margins minimize the possibility of a loss
through a default on a contract
13
Pricing of forward
• Guld (commodities): F = (1 + rf + s) · S0
• Finansiella tillgångar: F = (1 + rf) · S0
S is the spot price.
S is the storage cost
rf is risk free interest rate
F is the forward price
0
14
Examples of Futures Contracts
Agreement to:
– Buy 100 oz. of gold @ US$1400/oz. in
December
– Sell £62,500 @ 1.4500 US$/£ in March
– Sell 1,000 bbl. of oil @ US$90/bbl. in
April
Oz: ounce
Bbl: barrel
15
Example : An Arbitrage Opportunity?
Suppose that:
The spot price of gold is US$1,400
The 1-year forward price of gold is US$1,500
The 1-year US$ interest rate is 5% per annum
Q: What should be the 1-year forward
price? Is there an arbitrage opportunity?
16
The Forward Price of Gold
If the spot price of gold is S and the
forward price for a contract deliverable in
T years is F, then
F = S (1+r )T
where r is the 1-year (domestic
currency) risk-free rate of interest.
In our examples, S = 1400, T = 1, and r
=0.05 so that
F = 1400(1+0.05) = 1470
17
Hedging Examples
1. An investor owns 1,000 Microsoft
shares currently worth $28 per share.
A two-month put with a strike price of
$27.50 costs $1. The investor decides
to hedge by buying 10 contracts
18
Value of Microsoft Shares with and
without Hedging
40,000
Value of Holding ($)
35,000
No Hedging
30,000
Hedging
25,000
Stock Price ($)
20,000
20
25
30
35
40
19
Some Terminology
• Open interest: the total number of
contracts outstanding
– equal to number of long positions or number
of short positions
• Settlement price: the price just before the
final bell each day
– used for the daily settlement process
• Trading Volume : the number of trades in
one day
20
Forward Contracts vs Futures Contracts
FORWARDS
FUTURES
Private contract between 2 parties
Exchange traded
Non-standard contract
Standard contract
Usually 1 specified delivery date
Range of delivery dates
Settled at end of contract
Delivery or final cash
settlement usually occurs
Some credit risk
Settled daily
Contract usually closed out
prior to maturity
Virtually no credit risk
21
Options
The right but not the obligation…
Options
• A call option is an option to buy a certain
asset by a certain date for a certain price
(the strike price)
• A put option is an option to sell a certain
asset by a certain date for a certain price
(the strike price)
23
Option Obligations: the writer of the
option
Call option
Put option
Buyer
Right to buy asset
Right to sell asset
Writer
Obligation to sell asset
Obligation to buy asset
24
American vs. European Options
• An American option can be exercised at
any time during its life
• A European option can be exercised only
at maturity
• The time value will be lost when you
exercise prematurely.
25
Option Value: Example
Option values given an exercise price of $720
Stock Price
$600
660
720
780
840
Call Value
Put Value
$0
$120
0
60
0
0
60
0
120
0
What are the payoff limits for call option buyers? Sellers?
What are the payoff limits for put option buyers? Sellers?
Call Option Value
Call option value
Call option value (buyer) given a $720 exercise price.
$120
720
Share Price
840
Call Option Profit
Profit (buyer): Current Price - Exercise Price - Cost of Call
$20 call option (buyer) given a $720 exercise price
Call option value
Profit = ($840  $720)  $20  $100
$100
720
Share Price
23-28
840
Call Option Value
Call option $ payoff
Call option payoff (seller) given a $720 exercise price.
720
$-120
Share Price
23-29
840
Call Option Profit
Profit (Seller): Exercise Price - Current Price + Cost of Call
Call option $ payoff
$20 call option (seller) given a $720 exercise price:
720
840
$-100
$-120
Share Price
Profit = $720  $840  $20  $100
Call Option: Example
How much must the stock be worth at expiration in order for a call
holder to break even if the exercise price is $50 and the call premium
was $4?
Put Option Value
Put option value
Put option value (buyer) given a $720 exercise price:
$120
600 720
Share Price
Put Option Profit
Profit (buyer): Exercise Price - Current Price - Cost of Put
Put option value
$30 put option (buyer) given a $720 exercise price:
Profit = $720  $600  $30  $90
$90
600 720
Share Price
Put Option Value
Put option $ payoff
Put option payoff (seller) given a $720 exercise price.
Share Price
-$120
600 720
23-34
Put Option Profit
Profit (Seller): Current Price - Exercise Price + Cost of Put
Put option $ payoff
$30 put option (seller) given a $720 exercise price.
Share Price
-$90
Profit = $600  $720  $30  $90
600 720
Put Options: Example
What is your return on exercising a put option which was purchased
for $10 with an exercise price of $85? The stock price at expiration is
$81.
Options Value
Stock Price
Upper Limit
Lower Limit
(Stock price - exercise price) or 0
which ever is higher
37
Option Value
38
Option Value
• Point A -When the stock is worthless, the option is
worthless.
• Point B -When the stock price becomes very high, the
option price approaches the stock price less the
present value of the exercise price.
• Point C -The option price always exceeds its minimum
value (except at maturity or when stock price is zero).
• The value of an option increases with both the variability of
the share price and the time to expiration.
39
Option Value
Components of the Option Price
1 - Underlying stock price
2 - Strike or Exercise price
3 - Volatility of the stock returns (standard deviation of annual
returns)
4 - Time to option expiration
5 - Time value of money (discount rate)
40
Call Option Value
Put-Call Parity: No Dividends
• Consider the following 2 portfolios:
– Portfolio A: call option on a stock + zero-coupon
bond (or a deposit) that pays K at time T
– Portfolio B: Put option on the stock + the stock
42
Values of Portfolios are the same at
expiration (förfalldag)
Portfolio A
Portfolio B
ST > K
ST < K
ST − K
0
Zero-coupon bond
K
K
Total
ST
K
Put Option
0
K− ST
Share
ST
ST
Total
ST
K
Call option
43
The Put-Call Parity Result
• Both are worth max(ST , K ) at the maturity
of the options
• They must therefore be worth the same
today. This means that
c + Ke -rT = p + S0
44
Ex: put-call parity
Suppose that
c= 3
T = 0.25
S0= 31
r = 10%
K =30
• What are the put option price?
c + Ke -rT = p + S0
p = c-S0 +Ke -rT
=3-31+30*EXP(-0,1*0,25)
= 1,259
45
Bounds for European and American Put Options
(No Dividends)
46
Synthetic options
Two or more options combines together creates exotic options
47
Option Value: profit diagram for a
straddle
Straddle - Long call and long put
- Strategy for profiting from high volatility
Long put
Position Value
Long call
Straddle
Share Price
48
Option Value
Position Value
Straddle - Long call and long put
- Strategy for profiting from high volatility
Straddle
Share Price
An investor may take a long straddle position if he thinks the market is
highly volatile, but does not know in which direction it is going to
move.
49
Exotic options: a butterfly option
• A butterfly
x1
x2
x3
A long butterfly position will make profit if the future
volatility is lower than the implied volatility.
The spread is created by buying a call with a relatively low
strike (x1), buying a call with a relatively high strike (x3), and
shorting two calls with a strike in between (x2).
50
Long Call
Profit from buying one European call option: option
price = $5, strike price = $100, option life = 2 months
30 Profit ($)
20
10
70
0
-5
80
90
100
Terminal
stock price ($)
110 120 130
51
Short Call
Profit from writing one European call option: option
price = $5, strike price = $100
Profit ($)
5
0
-10
110 120 130
70
80
90
100
Terminal
stock price ($)
-20
-30
52
Long Put
Profit from buying a European put option: option
price = $7, strike price = $70
30 Profit ($)
20
10
0
-7
Terminal
stock price ($)
40
50
60
70
80
90
100
53
Short Put
Profit from writing a European put option: option
price = $7, strike price = $70
Profit ($)
7
0
40
50
Terminal
stock price ($)
60
70
80
90
100
-10
-20
-30
54
Payoffs from Options
What is the Option Position in Each Case?
K = Strike price, ST = Price of asset at maturity
Payoff
Payoff
K
K
ST
Payoff
ST
Payoff
K
K
ST
ST
55
The Black-Scholes-Merton Formulas
c  S 0 N (d1 )  K e
 rT
N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
ln( S 0 / K )  (r   2 / 2)T
where d1 
 T
ln( S 0 / K )  (r   2 / 2)T
d2 
 d1   T
 T
56
Real options
• With the limited liability of the modern corporations,
the shareholders´ equity can be regarded as a real
option on the assets of the firm.
• The shareholder value of equity value is
max(VT −D, 0)
where VT is the value of the firm and D is the debt
repayment required.
Thus the company can be considered as a call option
on the firm value V at the strike price of D.
57
Options on Real Assets
Real Options - Options embedded in real assets
Option to Abandon
Option to Expand
Options on Financial Assets
Executive Stock Options
Warrants
Convertible Bonds
Callable Bonds