Download Lecture 6 - IEI: Linköping University

Options and Futures:
Risk Management
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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…
2
Size of OTC and Exchange-Traded Markets
Source: Bank for International Settlements. Chart shows total principal amounts for
OTC market and value of underlying assets for exchange market
3
The Lehman Bankruptcy
Lehman’s filed for bankruptcy on September 15,
2008. one of the biggest bankruptcy in US
history
• Lehman was an active participant in the OTC derivatives
markets and got into financial difficulties because it took
high risks and found it was unable to roll over its short
term funding
• It had hundreds of thousands of transactions outstanding
with about 8,000 counterparties
• Unwinding these transactions has been challenging for
both the Lehman liquidators and their counterparties
4
How Derivatives are Used
• To hedge risks
• To speculate (take a bet on the future
direction of the market)
• To lock in an arbitrage profit (forwards)
• 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
5
Foreign Exchange Quotes for GBP,
($/£) May 24, 2010
The forward price may be different for contracts of different maturities (as
shown by the table)
Bid
Offer
Spot
1.4407
1.4411
1-month forward
1.4408
1.4413
3-month forward
1.4410
1.4415
6-month forward
1.4416
1.4422
6
Long position and short position
• The party that has agreed to buy
has a long position
• The party that has agreed to sell
has a short position
7
Example
• On May 24, 2010 the treasurer of a
corporation enters into a long forward
contract to buy £1 million in six months at
an exchange rate of 1.4422$
• This obligates the corporation to pay
$1442200 for £1 million on November 24,
2010
• What are the possible outcomes?
8
Profit from a Long Forward Position
(K= delivery price = forward price at the time
contract is entered into)
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
• Whereas a forward contract is traded over the
counter, a futures contract is traded on an
exchange.
CME Group
NYSE Euronext,
BM&F (Sao Paulo, Brazil)
TIFFE (Tokyo)
…
11
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
12
Forward price
• If the spot price of a commodity is S and
the forward price for a contract deliverable
in T years is F, then
F = S (1+r )T
• Essentially a forward is a implicit loan.
13
Gold price: An Arbitrage Opportunity?
Suppose you observe:
the spot price of gold is US$1400
The 1-year forward price of gold is
US$1500
The 1-year US$ interest rate is 5% per
annum,
Is there an arbitrage opportunity?
14
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.
S = 1400, T = 1, and r =0.05 so that
F = 1400(1+0.05) = 1470$
Yes, there is an arbitrage opportunity. Borrow at
5%, buy gold now and cover it with the forward
contract. You net 1500-1470=30 $ per contract!
15
Options
• A call option is the right but not the
obligation to buy a certain asset by a
certain date for a certain price (the strike
price)
• A put option is the right but not the
obligation to sell a certain asset by a
certain date for a certain price (the strike
price)
16
The writer of the option has
obligations to buy and sell
Call option
Put option
Buyer
Right to buy asset
Right to sell asset
Writer
Obligation to sell asset
Obligation to buy asset
17
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
18
Google Call Option Prices
(June 15, 2010; Stock Price is bid 497.07, offer 497.25)
Strike
Price
Jul 2010
Bid
Jul
2010
Offer
Sep 2010
Bid
Sep 2010
Offer
Dec
2010 Bid
Dec 2010
Offer
460
43.30
44.00
51.90
53.90
63.40
64.80
480
28.60
29.00
39.70
40.40
50.80
52.30
500
17.00
17.40
28.30
29.30
40.60
41.30
520
9.00
9.30
19.10
19.90
31.40
32.00
540
4.20
4.40
12.70
13.00
23.10
24.00
560
1.75
2.10
7.40
8.40
16.80
17.70
Source: CBOE
19
Google Put Option Prices
(June 15, 2010; Stock Price is bid 497.07, offer 497.25);
Strike
Price
Jul 2010
Bid
Jul 2010
Offer
Sep 2010
Bid
Sep 2010
Offer
Dec 2010
Bid
Dec 2010
Offer
460
6.30
6.60
15.70
16.20
26.00
27.30
480
11.30
11.70
22.20
22.70
33.30
35.00
500
19.50
20.00
30.90
32.60
42.20
43.00
520
31.60
33.90
41.80
43.60
52.80
54.50
540
46.30
47.20
54.90
56.10
64.90
66.20
560
64.30
66.70
70.00
71.30
78.60
80.00
Source: CBOE
20
Options vs. Futures/Forwards
• A futures/forward contract gives the holder
the obligation to buy or sell at a certain
price
• An option gives the holder the right but not
the obligation to buy or sell at a certain
price
21
Hedging Examples
1. A German company will pay £10 million
for imports from Britain in 3 months and
decides to hedge using a long position in
a forward contract
2. An investor owns 1,000 Microsoft shares
currently worth $28 per share. A twomonth put with a strike price of $27.50
costs $1. The investor decides to hedge
by buying 10 contracts
22
Value of Microsoft Shares with and
without Hedging
40,000
Value of
Holding ($)
35,000
No
Hedging
30,000
25,000
Stock Price ($)
20,000
20
25
30
35
40
The investor will do better when the price goes below 26,5$.
23
Example: Speculation vs. hedging
An investor with $2,000 to invest feels that a
stock price will increase over the next 2
months. The current stock price is $20 and
the price of a 2-month call option with a
strike of 22.50 is $1
• What are the alternative strategies?
24
Speculation
1. He buys 100 shares of the stock. 100*20$=2000
2. He buys 2000 call option 2000*1=2000$. If the
market price increases to more than 22,5 during
the 2 months, the call option will have a higher
payoff than the plain stocks.
3. Suppose the stock price is 25 at the expiration
date: the call option has gained (2522,5)*2000=5000$ or a 150% profit, while the
first case gives 500/2000=25%
4. Suppose the market price is lower than 22,5$, the
investor loses all, that is, -100%.
25
Hedge Funds
•
Hedge funds are not subject to the same rules as
mutual funds and cannot offer their securities
publicly.
Mutual funds must
•
–
–
–
disclose investment policies,
makes shares redeemable at any time,
limit use of leverage
Hedge funds are not subject to these constraints.
• Hedge funds use complex trading strategies are big
users of derivatives for hedging, speculation and
arbitrage
26
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
• Margin call: when the margin is below the
required minimum, it is subject to margin call.
The client is obliged to increase the margin to
the minimun.
27
Example of a Futures Trade
• An investor takes a long position
in 2 December gold futures contracts on
June 5
• contract size is 100 oz.
• futures price is US$1250
• initial margin requirement is
US$6,000/contract (US$12,000 in total)
• maintenance margin is US$4,500/contract
(US$9,000 in total)
28
Options 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
29
Key Points About Futures
• They are settled daily. (marked to the
market)
• Closing out a futures position involves
entering into an offsetting trade
• Most contracts are closed out before
maturity
30
Delivery
• If a futures contract is not closed out before maturity,
it is usually settled by delivering the assets underlying
the contract. When there are alternatives about what
is delivered, where it is delivered, and when it is
delivered, the party with the short position chooses.
• A few contracts (for example, those on stock indices
and Eurodollars) are settled in cash.
• When there is cash settlement, contracts are traded
until a predetermined time. All are then declared to be
closed out.
31
Forward Contracts vs Futures Contracts
FORWARDS
Private contract between 2 parties
FUTURES
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
32
Option Value
• The value of an option at expiration is a function of the
stock price and the exercise price (S-X for call and X-S
for put).
Example Option values (exercise price = $720)
Stock Price $600
Call Value
0
Put Value
120
660
0
60
720
0
0
780
60
0
840
120
0
33
Option Value
Call
option value
Call option value (graphic) on Google Stock on option
expiration date, exercise price=$720.
$120
720
840
Share Price
34
Option Value
Put option value
Put option value (graphic) on Google stock on option
expiration date. ( exercise price=$720 )
$120
600 720
Share Price
35
Option Value
Call option payoff (to the writer) on Google stock
Call option $ payoff
($720 exercise price)
720
Share Price
36
Option Value
Put option payoff (to the writer ) on Google Stock
Put option $ payoff
exercise price=$720 .
720
Share Price
37
Option Value
Protective Put - Long stock and long put
Long Stock
Position Value
Protective Put
Long Put
Share Price
38
Option Value
Protective Put - Long stock and long put
Position Value
Protective Put
Share Price
39
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
40
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.
41
Combinations of Options (cont'd)
• Strangle
– A portfolio that is long a call option and a put
option on the same stock with the same exercise
date but the strike price on the call exceeds the
strike price on the put
Textbook Example 20.5
Textbook Example 20.5
Combinations of Options
• Butterfly Spread
A portfolio that is long two call options with
differing strike prices, and short two call
options with a strike price equal to the
average strike price of the first two calls
• While a straddle strategy makes money when the stock
and strike prices are far apart, a butterfly spread makes
money when the stock and strike prices are close.
Figure 20.6 Butterfly Spread
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).
47
Figure 20.7 Portfolio Insurance
The plots show two different ways to insure against the possibility of the price of Amazon stock falling below $45. The
orange line in (a) indicates the value on the expiration date of a position that is long one share of Amazon stock and one
European put option with a strike of $45 (the blue dashed line is the payoff of the stock itself). The orange line in (b) shows
the value on the expiration date of a position that is long a zero-coupon riskfree bond with a face value of $45 and a
European call option on Amazon with a strike price of $45 (the green dashed line is the bond payoff).
Option Value
Call buyer profit diagram on Google stock–
strike price = $720 and option price= $80.50
Long call
Position Value
Break even
-80.50
720
800.50
Share Price
49
Option Value
Position Value
Put seller profit diagram with
strike price=$720 and option price of $71.20
Break even
Short put
+71.20
648.80
720
Share Price
50
Options Value
Stock Price
Upper Limit
Lower Limit
(Stock price - exercise price) or 0
which ever is higher
51
Option Value
52
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.
53
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)
54
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
55
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
56
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
57
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
58
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
59
Market Makers
• Most exchanges use market makers to
facilitate options trading
• A market maker quotes both bid and
ask prices when requested.
60
Margins
• Margins are required when options are written.
• When a naked option is written the margin is the greater
of:
– A total of 100% of the proceeds of the sale plus 20% of
the underlying share price less the amount (if any) by
which the option is out of the money
– A total of 100% of the proceeds of the sale plus 10% of
the underlying share price (for call) or exercise price
(for put)
• For other trading strategies there are special rules
61
Put-Call Parity: No Dividends
• Consider the following 2 portfolios:
Portfolio A: call option on a stock + a
deposit that pays K at time T
Portfolio B: Put option on the stock +
the stock
62
Values of Portfolios at the expiration
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
63
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 at t=0. This means that
c + Ke -rT = p + S0
64
Ex: put-call parity
Suppose that
c= 3
T = 0.25
K =30
S0= 31
r = 10%
• 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
65
Bounds for European and American Put Options
(No Dividends)
66
The Black-Scholes-Merton Formulas
 rT
c  S0 N (d1 )  K e N (d 2 )
pKe
 rT
N (d 2 )  S0 N (d1 )
ln( S0 / K )  (r  2 / 2)T
where d1 
 T
2
ln( S0 / K )  (r   / 2)T
 d1   T
d2 
 T
67
Using Equity Prices: Merton’s Model
• Merton’s model regards the equity as an
option on the assets of the firm
• In a simple situation the equity value is
max(VT −D, 0)
where VT is the value of the firm and D
is the debt repayment required
68
Risk Management
• Understand why companies hedge to reduce risk.
Bigger risks can have potential disastrous effect on
firms.
• Use options, futures, and forward contracts to devise
simple hedging strategies: Managing currency
exposure
• Explain how companies can use swaps to change the
risk of securities that they have issued. Interest rate
swaps, exchange rate swaps, etc
69
Illustration of an Interest Rate Swap
We consider a floating and a fixed interest rate
swap
1. Suppose Company A can borrow at a floating
rate of ERIBOR plus 1% or at a fixed rate of 10%
2. Company B can borrow at a floating rate of
ERIBOR plus 2% or at a fixed rate of 9,5%
3. Company A desires a fixed rate, company B
desires a floating rate.
4. Through a swap dealer company A and B can get
a better rate than they could independently.
ERIBOR: the Euro Interbank Offered Rate. The EURIBOR rates are based on the average
interest rates at which a panel of more than 50 European banks borrow funds from
one another. There are different maturities, ranging from one week to one year.
70
An illustration of an Interest Rate
Swap
Here the prime is ERIBOR: the Euro
Interbank Offered Rate
71
Reasons for Interest rate SWAP
• Note the interest rate SWAP has fulfilled two
companies´ needs to hedge their interest rate
exposure on their balance sheet. Other
reasons for the swap can be:
• To reduce the cost of borrowing.
• The company has a vision of where the
interest rate might go in the short run and
tries to take advantage of it.
72
Exercise
You observe the spot exchange rate is 2,45$/£, the 3 Month
forward rate is 2,60$/£. 3 Month Call option premium for strike
price 2,6 is 2 cents.
• A US company will pay £10 million for imports from Britain in 3
months and decides to hedge using a long position in a
forward contract, he fixed his cost of £ 10 million with the
forward exchange rate of 2,60$/£, total cost $26 M.
• If the 3-M call option premium is 0,02 dollars per call option,
Then, the cost of buying call option on British Pounds is
0,02*10M=0,2M. In this case, you have fixed your cost of £10M
to 26+0,2=$26,2M, but with a freedom. If the market
exchange rate is less than your strike price, you can take fully
advantage of the lower market price. So you have effectively
capped your payment. Total Cost $M
2,6
E$/£
73
Quick Quiz
1. Why is risk management more important now than it was in the 1960s?
Interest rates, energy prices, and exchange rates are more volatile
today than in the 1960s due to structural changes in their respective
markets.
2. What is a short-run exposure? A long-run exposure?
A short-run exposure involves risk due to temporary price changes; a
long-run exposure arises from permanent changes in economic
fundamentals.
3. How does a forward contract differ from a futures contract?
A forward contract is not standardized, not generally traded on an
organized exchange, and doesn’t involve margin.
A futures contract is highly standardized, traded on an organized
exchange, and requires margin.
74
The payoff to a long call and a short put is the
same as owning the underlying assets by
borrowed funds
• Ex: Suppose a financial manager buys call
options on 50,000 barrels of oil with an exercise
price of $30 per barrel. She simultaneously sells
a put option on 50,000 barrels of oil with the
same exercise price of $30 per barrel. Consider
her gains and losses if oil prices are $25, $28,
$30, $32, and $34. What do you notice about
the payoff profile?
75
Solution to Problem
Solution:
The call options give the manager the right to
purchase oil futures contracts at a futures price
of $30 per barrel. The manager will exercise the
option if the price rises above $30.
• Writing put options obligates the manager to
buy oil futures contracts at a futures price of
$30 per barrel. The put holder will exercise the
option if the price falls below $30. The payoffs
are:
76
Solution to Problem
Oil price:
$25 $28 $30 $32 $34
Value of call option position:
0 0 0 2 4
Value of put option position: -5 -2
0 0 0
Total value
-$5 -$2 $0 $2 $4
The payoff profile is identical to that of a forward
contract with a $30 strike price. Or purchased the
assets at 30$ by borrowed funds
77