F520 Options1 Options. F520 Options2 Financial options contracts l An option is a right (rather than...
Transcript of F520 Options1 Options. F520 Options2 Financial options contracts l An option is a right (rather than...
F520 Options 2
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.
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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 pre-specified 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 pre-specified price» t=0 receive C t=1 pay Max(0,PR-X)
C = Call Premium
PR = Price of underlying security
X = Exercise Price
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Net Payoff of a Call Option(includes call premium)
A
Net Payoff
X
Pa
yoff
of C
all (
$)
Security Price
0
-C
Buyer of a Call
X A
Writer of a Call
Pa
yoff
of C
all (
$)
0
+C
Security Price
Net Payoff
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Value of a Call Option
Intrinsic valuemax(0,S-X)
X
Call Price
Time Value
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 pre-specified price.» t=0 receive P t=1 pay Max(0,X-PR)
P = Put Premium
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Net Payoff of a Put Option
X
Pa
yoff
of P
ut (
$)
Security Price
0
-P
Buyer of a Put
XB
Writer of a Put
Pa
yoff
of P
ut (
$)
0
+P
Security Price
Net Payoff
B
-(S0-P)
+(S0-P)
Net Payoff
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Value of a Put Option
Intrinsic valuemax(0,X-S)
X
Put Price
Time Value
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
OptionPrice
IntrinsicValue
TimeValue
= +
Security Price (S)Exercise Price (X)
Volatility ()Interest rate (r)
Time to Expiration (T)
Call intrinsic value = max(0,S - X)Put intrinsic value = max(0,X - S)
F520 – Futures
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COPPER 2,204.60 lbs per metric tonne
Cash Buyer 7028 $3.19price per pound
Cash Seller & Settlement
7029$3.19
3-months Buyer 7060 $3.203-months Seller 7060 $3.2015-months Buyer 7235 $3.2815-months Seller 7245 $3.29
Copper Cash and Forward PricesLME Official Prices (US$/tonne) for 13 September 2013
http://www.lme.com/metals/non-ferrous/copper/
F520 – Futures
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|
http://www.cmegroup.com/trading/metals/base/copper_quotes_settlements_futures.html
Copper Futures, price per pound, 25,000 pounds per contractDaily Settlements for Copper Future Futures (FINAL) - Trade Date: 09/13/2013
Month Open High Low Last Change SettleEstimatedVolume
Prior DayOpen Interest
SEP 13 3.2010 3.2300 3.1950 3.2100 -.0055 3.2070 597 2,244
OCT 13 3.2060 3.2255 3.1905 3.2040 -.0060 3.2030 366 2,086
NOV 13 3.2015 3.2200 3.1910 - -.0065 3.2035 138 1,577
DEC 13 3.2000 3.2270 3.1905 3.2070 -.0065 3.2035 39,396 106,197
JAN 14 3.2030 3.2100 3.1980 - -.0065 3.2075 24 1,619
FEB 14 3.2090 3.2120 3.2020 - -.0060 3.2110 30 1,108
MAR 14 3.2350 3.2355 3.2040 3.2040 -.0070 3.2150 3,979 25,423
APR 14 3.2115 3.2115 3.2115 - -.0070 3.2190 4 553
MAY 14 3.2255 3.2255 3.2150 - -.0065 3.2230 238 2,739
JUN 14 3.2200 3.2200 3.2200 - -.0065 3.2275 4 620
JLY 14 - 3.2505B - - -.0065 3.2315 27 1,628
AUG 14 - - - - -.0065 3.2360 1 649
SEP 14 3.2405 3.2475 3.2405 - -.0065 3.2405 6 1,195
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Understanding Option QuotesCopper Options on Futures (Call in
Dec)Strike Type Open High Low Last
Change
SettleEstimatedVolume
Prior DayOpen 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 QuotesCopper Options on Futures (Put in
Dec)Strike Type Open High Low Last
Change
SettleEstimatedVolume
Prior DayOpen 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
F520 Options 15
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 centsUse 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 centsUse 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.
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Understanding Option Prices
OptionPrice
IntrinsicValue
TimeValue
= +
Security Price (S)Exercise Price (X)
Volatility ()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
Intrinsic valuemax(0,S-X)
X
Call Price
Time Value
Security Price
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Value of a Put Option
Intrinsic valuemax(0,X-S)
X
Put Price
Time Value
Security Price
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The Black-Scholes Option Pricing Model
The B-S option pricing model for a call is:
C S0 - Xe-rT + P
C = S0N(d1) - Xe-rTN(d2)
whered1 = [ln(S/X)+(r+ ½2)T]/T
d2 = d1 - 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:» S0 = $100
» X = $100» r = 5%» = 22%» T = 1 year
Then:» d1 = 0.34, N(d1) = 0.6331
» d2 = 0.12 N(d2) = 0.5478
C = S0N(d1) - Xe-rTN(d2)
d1 = [ln(S/X)+(r+ ½2)T]/T
d1 = [ln(100/100)+(.05+ ½(0.222)1]/(01
d1 = 0 + .0742/.22 = .337274
d2 = d1 - T
d2 = .33727 - 0.22/1 = .117273
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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)= $6.32
Double check through Put-Call Parity:
P = C - S0 + Xe-rT
6.32 = 11.20 – 100 + 100(0.9512)
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Relationship of Option and Security Prices
0
4
8
12
16
20
80 85 90 95 100 105 110 115 120
Stock Price ($)
Op
tio
n P
rice
($)
CallPut
Parameters: X = $100, T = 3 months, r = 5%, and = 25%Changing S
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Relationship of Option Prices to Interest Rates
2
3
4
5
6
7
0% 2% 4% 6% 8% 10% 12% 14% 16%
Interest Rate
Op
tio
n P
rice
($)
Call
Put
Parameters: S=$100, X = $100, T = 3 months, and = 25%Changing r
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Relationship of Option Prices to Volatility
0
2
4
6
8
10
12
5% 15% 25% 35% 45%
Volatility
Op
tio
n P
rice
($)
Put
Call
Parameters: S=$100, X = $100, T = 3 months, and r = 5%Changing
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Relationship of Option Prices to Time to Expiration
0
2
4
6
8
10
12
0 30 60 90 120 150 180 210 240 270 300
Days to Maturity
Op
tio
n P
rice
Call
Put
Parameters: S = $100, X = $100, r = 5%, and = 25%Changing t
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Parameters of the Black-Scholes Model
Need to know:» S, X, r, T, .
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.
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Implied Volatility Using the Black-Scholes Model
VolatilityAssumptions Put Price Call Price
15% $1.41 $2.0420% 1.98 2.6125% 2.55 3.1830% 3.11 3.7435% 3.68 4.31
Volatility implied by option prices
Given InformationS0 = $100, X = 100r = 8%, T = 30 days,
P = $3.10, and C = $3.73
http://www.numa.com/derivs/ref/calculat/option/calc-opa.htm
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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
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Unhedged
Outcome ProbabilityValue of the
Firm in Period 1
Price of oil high 0.5 1000Price of oil low 0.5 200
Capital Structure Book Values
Price of Oil High Market Value at
t=1
Price of Oil Low Market Value at
t=1Market Value
Debt 500 500 200 350Equity 500 500 0 250
1000 200 600
Does hedging this company's risk increase value?
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Hedged
Outcome ProbabilityValue of the
Firm in Period 1
Price of oil high 0.5 600Price of oil low 0.5 600
Capital Structure Book Values
Price of Oil High Market Value at
t=1
Price of Oil Low Market Value at
t=1Market Value
Debt 500 500 500 500Equity 500 100 100 100
600 600 600
The total market value is not affected (both are $600); however the distributionis affected. The Stockholder value was decreased from $250 to $100 withhedging, showing that there is a transfer of wealth to bondholders. This is dueto the fact that the firm is on the brink of insolvency.
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Note the similarities between the payoff on stock and a call option.
Net Payoff
X = Debt Amount
Pa
yoff
on
Fir
m (
$)
Market Value of Assets
0
-C
Buyer of a Call / Stock
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.
F520 Options 36Will the Unhedged firm add a risk-free project when new capital must be added by
equityholders
Outcome ProbabilityValue of the
Firm in Period 1
Value of the Firm in Period 1
w/Investment
Price of oil high 0.5 1000 1300Price of oil low 0.5 200 500New Investment 200Cash Flow at t=1 300 Should the investment be taken?
Capital Structure Book Values
Price of Oil High Market Value at
t=1
Price of Oil Low Market Value at
t=1Market Value
Debt 500 500 500 500Equity 700 800 0 400
1300 500 900
Equityholders have a value of $400, compared to a value of $250 if no projectis taken. But remember, that the equityholders added $200 to make theinvestment. So they gained $150 but it cost them $200 to obtain this gain. Onlythe bondholders have benefited.
F520 Options 37Would New Bondholders add the new capital?Bondholders generally enter as subordinate to the old
bonds.
Outcome ProbabilityValue of the
Firm in Period 1
Value of the Firm in Period 1
w/Investment
Price of oil high 0.5 1000 1300Price of oil low 0.5 200 500New Investment 200Cash Flow at t=1 300 Should the investment be taken?
Capital Structure Book Values
Price of Oil High Market Value at
t=1
Price of Oil Low Market Value at
t=1Market Value
Senior Debt 500 500 500 500Sub. Debt 200 200 0 100Equity 500 600 0 300
1300 500 900
New debtholders will not enter into this transaction, it has a guaranteed lossfor the new debtholders.
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Will the hedged firm add take a risk-free project?
Outcome ProbabilityValue of the
Firm in Period 1
Value of the Firm in Period 1
w/Investment
Price of oil high 0.5 600 900Price of oil low 0.5 600 900New Investment 200Cash Flow at t=1 300
Capital Structure Book Values
Price of Oil High Market Value at
t=1
Price of Oil Low Market Value at
t=1Market Value
Debt 500 500 500 500Equity 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 theNPV of the project was greater than $150 (the amount of value lost from thedecision 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 100,000 lbs 100,000 lbs 100,000 lbs
Receive $62,300 $63,050 $63,750 $64,600
3 6 9 12 mo
F520 Options 41
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 PMTPMT = $63,406.19
A swap will have four equal payments of $63,158.72 at the end of each quarter.
Deliver 100,000 lbs 100,000 lbs 100,000 lbs 100,000 lbs
Receive $63,406 $63,406 $63,406 $63,406
3 6 9 12 mo
F520 Options 42
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?
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Should you take a long or a short position?
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?
F520 Options 44
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
F520 Options 45
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 contract40 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
F520 Options 46
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