F520 Options
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Options
F520 Options
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Financial options contracts





An option is a right (rather than a commitment) to buy
or sell an asset at a pre-specified price
The right to purchase is a call option; the right to sell
is a put option
The strike price (or exercise price) is the price at
which an option can be exercised
Options which can be exercised only at maturity are
“European Options”; “American Options” can be
exercise any time prior or at maturity
Options can be traded on exchanges or OTC markets.
F520 Options
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Call Options

Buying a Call Option--Gives the purchaser the
right, but not the obligation, to buy the underlying
security from the writer of the option at a prespecified price
» t=0 pay C

t=1 receive Max(0,PR-X)
Writing a Call Option—Gives the writer the
obligation to sell the underlying security at a prespecified price
» t=0 receive C
t=1 pay Max(0,PR-X)
C = Call Premium
PR = Price of underlying security
X = Exercise Price
Payoff of Call ($)
F520 Options
Net Payoff of a Call Option
(includes call premium)
Buyer of a Call
Net Payoff
0
Security Price
-C
Payoff of Call ($)
X
A
Writer of a Call
+C
Net Payoff
0
X
A
Security Price
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Value of a Call Option
Call Price
Intrinsic value
max(0,S-X)
Time Value
X
Security Price
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Put Options

Buying a Put Option - Gives the purchaser the right,
not the obligation, to sell the underlying security to
the writer of the option at a pre-specified exercise
price.
» t=0 pay P

t=1 receive Max(0,X-PR)
Writing a Put Option - Gives the writer the
obligation to buy the underlying security at a prespecified price.
» t=0 receive P
P
= Put Premium
t=1 pay Max(0,X-PR)
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Payoff of Put ($)
Net Payoff of a Put Option
+(S0-P)
Buyer of a Put
0
-P
Payoff of Put ($)
B
X
Net Payoff
Security Price
Writer of a Put
Net Payoff
+P
0
B
-(S0-P)
X
Security Price
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Value of a Put Option
Put Price
Time Value
Intrinsic value
max(0,X-S)
X
Security Price
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Caps, Floors, and Collars



A cap is a call option where the seller guarantees to
pay the buyer when the designated reference price
exceed a predetermined cap price. The buyer pays a
cap fee.
A floor is a put option where the seller guarantees to
pay the buyer when the designated reference price
falls below a predetermined floor price. The buyer
pays a floor fee.
A collar is a position that simultaneously buys a cap
and sells a floor.
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Option Price
Option
Price
=
Intrinsic
Value
Security Price (S)
Exercise Price (X)
Time
Value
+
Volatility (s)
Interest rate (r)
Time to Expiration (T)
Call intrinsic value = max(0,S - X)
Put intrinsic value = max(0,X - S)
F520 – Futures
Copper Cash and Forward Prices
LME Official Prices (US$/tonne) for 13 September 2013
COPPER
Cash Buyer
Cash Seller &
Settlement
3-months Buyer
3-months Seller
15-months Buyer
15-months Seller
7028
7029
7060
7060
7235
7245
2,204.60
lbs per metric tonne
$3.19 price per pound
$3.19
$3.20
$3.20
$3.28
$3.29
http://www.lme.com/metals/non-ferrous/copper/
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F520 – Futures
Copper Futures, price per pound, 25,000 pounds per contract
Daily Settlements for Copper Future Futures (FINAL) - Trade Date: 09/13/2013
Month
SEP 13
OCT 13
NOV 13
DEC 13
JAN 14
FEB 14
MAR 14
APR 14
MAY 14
JUN 14
JLY 14
AUG 14
SEP 14
Open
High
|
3.2010
3.2300
3.2060
3.2255
3.2015
3.2200
3.2000
3.2270
3.2030
3.2100
3.2090
3.2120
3.2350
3.2355
3.2115
3.2115
3.2255
3.2255
3.2200
3.2200
- 3.2505B
3.2405
3.2475
Low
3.1950
3.1905
3.1910
3.1905
3.1980
3.2020
3.2040
3.2115
3.2150
3.2200
3.2405
Last
3.2100
3.2040
3.2070
3.2040
-
Change
-.0055
-.0060
-.0065
-.0065
-.0065
-.0060
-.0070
-.0070
-.0065
-.0065
-.0065
-.0065
-.0065
Settle
3.2070
3.2030
3.2035
3.2035
3.2075
3.2110
3.2150
3.2190
3.2230
3.2275
3.2315
3.2360
3.2405
Estimate Prior Day
d
Open
Volume Interest
597
2,244
366
2,086
138
1,577
39,396 106,197
24
1,619
30
1,108
3,979
25,423
4
553
238
2,739
4
620
27
1,628
1
649
6
1,195
http://www.cmegroup.com/trading/metals/base/copper_quotes_settlements_futures.html
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Understanding Option Quotes
Copper Options on Futures (Call in Dec)
Strike Type
Open
High
Low
Chan
ge
Last
Settle
Estimated
Volume
Prior Day
Open
Interest
317
Call
-
-
.1370A
-
-.0055
.1335
0
0
318
Call
-
-
.1315A
-
-.0060
.1275
0
0
319
Call
-
-
.1260A
-
-.0055
.1220
0
0
320
Call
-
-
.1210A
-
-.0060
.1165
0
11
321
Call
-
-
.1105A
-
-.0060
.1110
0
0
322
Call
-
-
.1055A
-
-.0060
.1055
0
0
323
Call
-
-
.1010A
-
-.0065
.1005
0
0
324
Call
-
.1030B
.0960A
-
-.0065
.0955
0
0
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Understanding Option Quotes
Copper Options on Futures (Put in Dec)
Strike Type
Open
High
Low
Chan
ge
Last
Settle
Estimated
Volume
Prior Day
Open
Interest
317
Put
-
.1005B
.0955A
-
+.0010
.1000
0
0
318
Put
-
.1050B
.0990A
-
+.0005
.1040
0
0
319
Put
-
-
.1030A
-
+.0005
.1085
0
0
320
Put
-
.1180B
.1075A
-
+.0005
.1130
0
13
321
Put
-
-
.1115A
-
+.0005
.1175
0
0
322
Put
-
-
.1210A
-
+.0005
.1220
0
0
323
Put
-
-
.1255A
-
UNCH
.1270
0
0
324
Put
-
-
.1305A
-
UNCH
.1320
0
0
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Understanding December Quotes

How much does it cost to purchase:
»

one call of Copper Futures Options contract (exercise price of 317)?
Call = .1335 / lb * 25,000 lb per contract = $5,075.49 per contract
What is the intrinsic value of a call on Copper Futures Options (exercise price of 317)?
Call = max(0,F-X) = max(0,3.20 – 3.17) = 0.03 cents
Use the futures copper price (not the cash price)
What is the time value of money?
Option Price - Intrinsic Value = 0.1335 – 0.03 = 0.1035 / lb

What is the intrinsic value of a call on Copper Futures Options (exercise price of 324)?
Call = max(0,F-X) = max(0,3.30 – 3.24) = 0 cents
Use the futures copper price (not the cash price)
What is the time value of money?
Option Price - Intrinsic Value = 0.0955 – 0 = 0.0955 / lb

What is the intrinsic value of a call on Copper Futures Options (exercise price of 320)?
Option Price - Intrinsic Value = 0.1165 – 0 = 0.1165 / lb

Why there greater intrinsic value for options near the money.
F520 Options
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Understanding Option Prices
Option
Price
=
Intrinsic
Value
Security Price (S)
Exercise Price (X)
Time
Value
+
Volatility (s)
Interest rate (r)
Time to Expiration (T)
Call intrinsic value = max(0,S - X)
Put intrinsic value = max(0,X - S)
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Value of a Call Option
Call Price
Intrinsic value
max(0,S-X)
Time Value
X
Security Price
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Value of a Put Option
Put Price
Time Value
Intrinsic value
max(0,X-S)
X
Security Price
The Black-Scholes
Option Pricing Model
F520 Options

The B-S option pricing model for a call is:
C = S0 - Xe-rT + P
C = S0N(d1) - Xe-rTN(d2)
where
d1 = [ln(S/X)+(r+ ½s2)T]/sT
d2 = d1 - sT
N(d) = cumulative normal distribution
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Black-Scholes Put Price

Price of a European put is:
P = C - S0 + Xe-rT
= S0[N(d1)-1] - Xe-rT[N(d2)-1]
where d1, d2, and N(d) are defined as before.
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Black-Scholes Pricing Example

Assume:

Then:
C
» S0 = $100 d1
» X = $100 d1
» r = 5%
d1
» s = 22%
d2
» T = 1 year d2
» d1 = 0.34,
» d2 = 0.12
= S0N(d1) - Xe-rTN(d2)
= [ln(S/X)+(r+ ½s2)T]/sT
= [ln(100/100)+(.05+ ½(0.22)2)1]/(0)1
= 0 + .0742/.22 = .337274
= d1 - sT
= .33727 - 0.22/1 = .117273
N(d1) = 0.6331
N(d2) = 0.5478
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Call Option Example



Price of a call is then:
C = S0N(d1) - Xe-rTN(d2)
C= 100(0.6331) - 100(0.9512)(0.5478)
= $11.20
Price of a put is then:
P = S0[N(d1)-1] - Xe-rT[N(d2)-1]
P = 100[.6331 - 1] - 100(1/e(.05*1))(.5478-1)
P = 100(-0.3669) - 100(0.9512)(-0.4522)
= $6.32
Double check through Put-Call Parity:
P = C - S0 + Xe-rT
6.32 = 11.20 – 100 + 100(0.9512)
Relationship of Option and
Security Prices
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Put
Call
16
Option Price ($)
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8
4
0
80
85
90
95
100
105
110
115
120
Stock Price ($)
Parameters: X = $100, T = 3 months, r = 5%, and s = 25%
Changing S
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Relationship of Option Prices
to Interest Rates
F520 Options
Option Price ($)
7
Call
6
5
4
Put
3
2
0%
2%
4%
6%
8%
10%
12%
14%
16%
Interest Rate
Parameters: S=$100, X = $100, T = 3 months, and s = 25%
Changing r
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Relationship of Option Prices
to Volatility
12
Option Price ($)
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10
8
Call
Put
6
4
2
0
5%
15%
25%
35%
45%
Volatility
Parameters: S=$100, X = $100, T = 3 months, and r = 5%
Changing s
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Relationship of Option Prices
to Time to Expiration
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12
Call
10
8
6
Put
4
2
0
30
60
90
0
120 150 180 210 240 270 300
Days to Maturity
Parameters: S = $100, X = $100, r = 5%, and s = 25%
Changing t
Option Price
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
Parameters of the
Black-Scholes Model
Need to know:
» S, X, r, T, s.


All readily observable, except the last.
The interest rate should be a continuously
compounded rate
» To convert simple annualized rate to continuously
compounded rate:
r = ln(1+R)
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Volatility as a Parameter


In pricing options, analysts usually use some
measure of historical volatility of the
underlying security.
Volatility obtained from other than annualized
returns must be converted to annualized
volatility.
» e.g., Variance of weekly returns must be multiplied
by 52.
» e.g., Standard deviation of weekly returns must be
multiplied by  52.
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Implied Volatility

Alternatively, can use all the other inputs, and
infer a volatility estimate from the current
option price.
» Is called the implied volatility.

Can then compare implied volatility with
recent historical volatility.
» Higher implied than historical may indicate the
option is expensive.
» Lower implied than historical may indicate the
option is cheap.
F520 Options
Implied Volatility Using the
Black-Scholes Model
http://www.numa.com/derivs/ref/calculat/option/calc-opa.htm
Volatility
Assumptions
15%
20%
25%
30%
35%
Put Price
$1.41
1.98
2.55
3.11
3.68
Volatility implied by option prices
Call Price
$2.04
2.61
3.18
3.74
4.31
Given Information
S0 = $100, X = 100
r = 8%, T = 30 days,
P = $3.10, and C = $3.73
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





Assumptions In Original
Option Pricing Model
Underlying returns log normally distributed.
Variance is constant over time.
The interest rate is constant over time.
No sudden jumps in underlying price.
No dividends.
No early exercise (i.e., European option).
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


Enhancing Firm Value through
Hedging
Reducing Volatility of cash flows does not
guarantee increased value.
Hedging has transaction costs, so hedging is
not free.
Hedging can add value if
» Taxes are reduced
» Transaction costs (like default risk) is reduced
» When it aligns incentives to take positive NPV
projects
32
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Unhedged
Outcome
Price of oil high
Price of oil low
Probability
0.5
0.5
Value of the
Firm in Period 1
1000
200
Price of Oil High Price of Oil Low
Market Value at Market Value at Market
Value
t=1
t=1
Capital Structure Book Values
350
200
500
500
Debt
250
0
500
500
Equity
600
200
1000
Does hedging this company's risk increase value?
F520 Options
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Hedged
Outcome
Price of oil high
Price of oil low
Probability
0.5
0.5
Value of the
Firm in Period 1
600
600
Price of Oil High Price of Oil Low
Market Value at Market Value at Market
Capital Structure Book Values
t=1
t=1
Value
Debt
500
500
500
500
Equity
500
100
100
100
600
600
600
The total market value is not affected (both are $600); however the distribution
is affected. The Stockholder value was decreased from $250 to $100 with
hedging, showing that there is a transfer of wealth to bondholders. This is due
to the fact that the firm is on the brink of insolvency.
Payoff on Firm ($)
F520 Options
Note the similarities between the
payoff on stock and a call option.
Buyer of a Call / Stock
Net Payoff
0
-C
X = Debt
Amount
Market Value of Assets
In our prior example, stockholders only get paid after the debtholders receive their
value. Therefore, the value of the debt is like the exercise price on a call option.
If the value of the firm is less than the value of the debt, stockholders will walk
away and leave the firm to the debtholders. If the value of the firm is greater than
the value of the debt, the stockholders remain in control of the firm.
This also shows why reducing volatility (through hedging) does not guarantee an
increase in the value of the firm. In fact, as shown in the Black Scholes formula,
decreasing volatility can reduce the value of the firm to equity holders (see the
hedging example several slides earlier.
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Will the Unhedged firm add a risk-free project when
new capital must be added by equityholders
Outcome
Price of oil high
Price of oil low
New Investment
Cash Flow at t=1
Value of the
Value of the
Firm in Period 1
Firm in Period 1 w/Investment
1000
1300
200
500
Probability
0.5
0.5
200
300
Should the investment be taken?
Price of Oil High Price of Oil Low
Market Value at Market Value at Market
Capital Structure Book Values
t=1
t=1
Value
Debt
500
500
500
500
Equity
700
800
0
400
1300
500
900
Equityholders have a value of $400, compared to a value of $250 if no project
is taken. But remember, that the equityholders added $200 to make the
investment. So they gained $150 but it cost them $200 to obtain this gain. Only
the bondholders have benefited.
F520 Options
Would New Bondholders add the new capital?
Bondholders generally enter as subordinate to the old bonds.
Outcome
Price of oil high
Price of oil low
New Investment
Cash Flow at t=1
Value of the
Value of the
Firm in Period 1
Firm in Period 1 w/Investment
1000
1300
200
500
Probability
0.5
0.5
200
300
Should the investment be taken?
Price of Oil High Price of Oil Low
Market Value at Market Value at Market
Capital Structure Book Values
t=1
t=1
Value
Senior Debt
500
500
500
500
Sub. Debt
200
200
0
100
Equity
500
600
0
300
1300
500
900
New debtholders will not enter into this transaction, it has a guaranteed loss
for the new debtholders.
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Will the hedged firm add take a risk-free project?
Outcome
Price of oil high
Price of oil low
New Investment
Cash Flow at t=1
Probability
0.5
0.5
200
300
Value of the
Value of the
Firm in Period 1
Firm in Period 1 w/Investment
600
900
600
900
Price of Oil High Price of Oil Low
Market Value at Market Value at Market
Capital Structure Book Values
t=1
t=1
Value
Debt
500
500
500
500
Equity
700
400
400
400
900
900
900
When the firm does not have concerns about market value falling below the
debt outstanding, then the firm will take any positive NPV projects.
Note: From our original example, we would only choose to hedge the firm if the
NPV of the project was greater than $150 (the amount of value lost from the
decision to hedge in the prior slide).
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Swaps



A swap is an agreement whereby two parties (called
counterparties) agree to exchange periodic
payments. The dollar amount of the payments
exchanged is based on some predetermined dollar
principal (or commodity quantity), which is called the
notional amount.
It can be considered the same as entering a series of
forward contracts, since it is an agreement to make
the exchange at several points in the future.
Types of swaps include
»
»
»
»
»
Interest rate swaps
Interest rate-equity swaps
Equity swaps
Currency swaps
Commodity swaps
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Comparing Forwards and swaps


Assume the following forward prices for commodity X
» 3 months
$0.6230 per pound
» 6 months
$0.6305 per pound
» 9 months
$0.6375 per pound
» 12 months
$0.6460 per pound
A company enters 4 forward contracts (one in each month) with
a promise to deliver 100,000 pounds of copper each month for
the prices set above.
Deliver
100,000 lbs
3
Receive
$62,300
100,000 lbs
6
$63,050
100,000 lbs 100,000 lbs
9
12 mo
$63,750
$64,600
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Calculating swap payment



Find the Present Value of the Cash Flows (assume 2% per quarter)
PV = $62,300/(1.02)1 + $63,050/(1.02)2 + $63,750/(1.02)3 +
$64,600/ /(1.02)4 = $241,433.59
Now spread this value over 4 equal payments at the end of each
period (4 period annuity).
PV = $241,433.59, I = 2, N = 4, FV = 0, compute PMT
PMT = $63,406.19
A swap will have four equal payments of $63,158.72 at the end of
each quarter.
Deliver
100,000 lbs
3
Receive
$63,406
100,000 lbs
6
$63,406
100,000 lbs 100,000 lbs
9
12 mo
$63,406
$63,406
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Using Duration in Hedging




Hedge the future issuance of 90-day commercial
paper. Assume today is August 10, 200X. Our
projected date of cash flow needs is November 25,
200X.
 The amount of commercial paper that will be
issued is $50 million.
 The Euro-dollar Futures contract has a face
value of $1 million is if for a 90-day maturity
Eurodollar issue to be made 107 days from today.
Should you take a long or a short position?
Should you take a long or a short
position?
F520 Options

You want to protect against rising interest
rates that results in falling prices. Therefore,
you want the futures contract to make money
when prices fall (a short position). These
profits from the futures contract will offset the
lower set of funds your will be able to bring in
if interest rates increase and you issue the
commercial paper at a larger discount.

How many contracts do we need?
43
F520 Options
How many contracts do we
need?
Formula from Hedging notes:
$ amt. of security
duration of asset
# of Contracts = --------------------------- X -----------------------$ amt. of fut. contract
duration of future sec
$50,000,000
90-days
# of Contracts = ----------------------- X ------------------------ = 50
$1,000,000
90-days
contracts
44
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If commercial paper rates go up by 40 basis points (from 3.53% to 3.93%) and Eurodollar
future rates also go up by 40 basis points (from 3.565% to 3.965), how much money will get
from the futures contract and how much money will we get from our commercial paper
issuance?
Futures contract
40 bp * $25 per basis point *50 contracts = $50,000 profit
Commercial paper issuance:
1,000,000 * (1-.0393*(90/360)) = $990,175
x 50 contracts
$49,508,750
This is exactly $50,000 less than what we had anticipated raising if rates had remained at 3.53%.
Between the profits from the futures contract and the expected commercial paper issuance proceeds,
we have locked in our expected cash flow. Now let’s just hope that our basis risk (difference
between spot and futures prices) remains the same over this time period.
Notes:

The change in $1 million for a 1 bp interest rate change is equal to
$1,000,000*(.0001*(90/360)) = $25

1,000,000 * (1-.0353*(90/360)) = $990,175
x 50 contracts
$49,558,750
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How many contracts do we need?
Situation 2
If we had a desire to issue commercial paper 107 days from
now with 120-days to maturity, how many contracts would
we need?
$50,000,000
120-days
# of Contracts = --------------------- X --------------------- = 66.67
$1,000,000
90-days
contracts
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Contracts

Options and Futures
http://www.cmegroup.com/education/getting-started.html

Future and Option contracts
http://www.cmegroup.com/globex/

www.cmegroup.com