730g81
Linköpings University
1

• 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

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

• 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

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
6-month forward
1.4410
1.4416
1.4415
1.4422
6

• The party that has agreed to buy has a long position
• The party that has agreed to sell has a short position
7

• 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

(K= delivery price = forward price at the time contract is entered into)
Profit
K
Price of Underlying at
Maturity, S
T
9

(K= delivery price = forward price at the time contract is entered into)
Profit
Price of Underlying at Maturity,
S
T K
10

• 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

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

• 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

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

• 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

Call option
Put option
Buyer Writer
Right to buy asset Obligation to sell asset
Right to sell asset Obligation to buy asset
17

• 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
460
Jul 2010
Bid
Jul
2010
Offer
43.30
44.00
Sep 2010
Bid
Sep 2010
Offer
Dec
2010 Bid
Dec 2010
Offer
51.90
53.90
63.40
64.80
480
500
520
28.60
29.00
17.00
17.40
9.00
9.30
39.70
28.30
19.10
40.40
29.30
19.90
50.80
40.60
31.40
52.30
41.30
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
460
Jul 2010
Bid
Jul 2010
Offer
Sep 2010
Bid
6.30
6.60
15.70
Sep 2010
Offer
Dec 2010
Bid
Dec 2010
Offer
16.20
26.00
27.30
480
500
520
11.30
11.70
19.50
20.00
31.60
33.90
22.20
30.90
41.80
22.70
32.60
43.60
33.30
42.20
52.80
35.00
43.00
54.50
540
560
46.30
47.20
64.30
66.70
54.90
70.00
56.10
71.30
64.90
78.60
66.20
80.00
Source: CBOE
20

• 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

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

40 000
35 000
30 000
Value of
Holding ($)
No
Hedging
25 000
Stock Price ($)
20 000
20 25 30 35 40
The investor will do better when the price goes below 26,5$.
23

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

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 (25-
22,5)*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 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

• 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

• 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

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

• 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

• 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
Non-standard contract
Usually 1 specified delivery date
Settled at end of contract
Delivery or final cash settlement usually occurs
Some credit risk
FUTURES
Exchange traded
Standard contract
Range of delivery dates
Settled daily
Contract usually closed out prior to maturity
Virtually no credit risk
32

• 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 $600 660 720 780 840
Call Value 0 0 0 60 120
Put Value 120 60 0 0 0
33

Call option value (graphic) on Google Stock on option expiration date, exercise price=$720.
$120
720 840
Share Price
34

Put option value (graphic) on Google stock on option expiration date. ( exercise price=$720 )
$120
600 720
Share Price
35

Call option payoff (to the writer) on Google stock
($720 exercise price)
720
Share Price
36

Put option payoff (to the writer ) on Google Stock exercise price=$720 .
720
Share Price
37

Protective Put - Long stock and long put
Long Stock
Protective Put
Share Price
Long Put
38

Protective Put - Long stock and long put
Protective Put
Share Price
39

Straddle - Long call and long put
- Strategy for profiting from high volatility
Long put
Long call
Straddle
Share Price
40

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

• 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

• 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.

• A butterfly x
1 x
2 x
3
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 (x
1
), buying a call with a relatively high strike (x
3
), and shorting two calls with a strike in between (x
2
).
47

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).

Call buyer profit diagram on Google stock– strike price = $720 and option price= $80.50
Long call
Break even
-80.50
720 800.50
Share Price
49

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

Stock Price
Upper Limit
Lower Limit
(Stock price - exercise price) or 0 which ever is higher
51

52

• 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

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

Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months
30
Profit ($)
20
10
0
-5
70 80 90 100
Terminal stock price ($)
110 120 130
55

Profit from writing one European call option: option price = $5, strike price = $100
Profit ($)
5
0
110 120 130
-10
70 80 90 100 Terminal stock price ($)
-20
-30
56

20
10
0
-7
Profit from buying a European put option: option price = $7, strike price = $70
30
Profit ($)
40 50 60 70
Terminal stock price ($)
80 90 100
57

Profit from writing a European put option: option price = $7, strike price = $70
Profit ($)
7
0
40 50 60
70
Terminal stock price ($)
80 90 100
-10
-20
-30
58

What is the Option Position in Each Case?
K = Strike price, S
T
= Price of asset at maturity
Payoff Payoff
K
Payoff
K S
T
S
T
Payoff
K
K S
T
S
T
59

• Most exchanges use market makers to facilitate options trading
• A market maker quotes both bid and ask prices when requested.
60

• 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

T
62

Portfolio A Call option
Zero-coupon bond
Total
Portfolio B Put Option
Share
Total
S
T
> K
S
T
− K
K
S
T
0
S
T
S
T
S
T
< K
0
K
K
K− S
T
S
T
K
63

• Both are worth max(
S
T of the options
, K ) at the maturity
• They must therefore be worth the same today at t=0. This means that c + Ke -rT = p + S
0
64

Suppose that c = 3
T = 0.25
S
0
= 31 r = 10%
K =30
• What are the put option price?
c + Ke -rT = p + S
0 p = cS
0
+ Ke -rT
=3-31+30*EXP(-0,1*0,25)
= 1,259
65

Bounds for European and American Put Options
(No Dividends)
66

c

S
0
N ( d
1
)

K e
 rT
N ( d
2
) p

K e
 rT
N (
 d
2
)

S
0
N (
 d
1
)
d
1
 ln( S
0
/ K )

( r
  2
/ 2 ) T

T d
2
 ln( S
0
/ K )

( r
  2
/ 2 ) T

T
 d
1
 
T
67

• Merton’s model regards the equity as an option on the assets of the firm
• In a simple situation the equity value is max( V
T
−
D , 0) where V
T is the value of the firm and is the debt repayment required
D
68

• 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

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

Here the prime is ERIBOR: the Euro
Interbank Offered Rate
71

• 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

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
73
2,6
E
$/£

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:
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

Oil price:
Value of call option position:
Value of put option position:
Total value
$25 $28 $30 $32 $34
0 0 0 2 4
-5 -2 0 0 0
-$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