Options

F520 Options 1

Options

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.

F520 Options 3

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

F520 Options 4

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

F520 Options 5

Value of a Call Option

Intrinsic valuemax(0,S-X)

X

Call Price

Time Value

Security Price

F520 Options 6

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

F520 Options 7

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

F520 Options 8

Value of a Put Option

Intrinsic valuemax(0,X-S)

X

Put Price

Time Value

Security Price

F520 Options 9

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.

F520 Options 10

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

11

COPPER 2,204.60 lbs per metric tonne

Cash Buyer 4814 $2.18 price per pound

Cash Seller & Settlement 4815$2.18

3-months Buyer 4838 $2.193-months Seller 4840 $2.20Dec 3 2017 Buyer 4930 $2.24Dec 3 2017 Seller 4940 $2.24

http://www.lme.com/metals/non-ferrous/copper/

F520 – Futures

12

|

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: 09/23/2016

Month Open High Low Last Change SettleEstimatedVolume

Prior DayOpen Interest

SEP 16 2.1885 2.1960 2.1885 2.1915 +.0050 2.1925 723 1,439OCT 16 2.1910 2.1995 2.1810 2.1950 +.0030 2.1940 960 3,344NOV 16 2.1920 2.2035 2.1850 2.1975 +.0050 2.1975 134 1,094DEC 16 2.1885 2.2065 2.1855 2.2000 +.0065 2.2010 58,725 136,614JAN 17 2.1945 2.2095 2.1945 2.2090 +.0070 2.2055 25 699FEB 17 - - - - +.0070 2.2080 0 562MAR 17 2.2005 2.2155B 2.1950 2.2090 +.0075 2.2105 5,618 32,826APR 17 - - - - +.0080 2.2150 0 328MAY 17 2.2145 2.2215B 2.2010A 2.2165 +.0080 2.2170 1,345 5,612JUN 17 2.2215 2.2215 2.2215 2.2215 +.0085 2.2215 1 297JLY 17 2.2110 2.2225 2.2100A 2.2220 +.0090 2.2225 103 2,318AUG 17 2.2275 2.2275 2.2275 2.2275 +.0095 2.2270 1 297SEP 17 2.2280 2.2280 2.2280 2.2280 +.0100 2.2280 28 1,027

F520 Options 13

Understanding Option QuotesCopper Options on Futures (Call in

Dec)Strike Type Open High Low Last

Change

SettleEstimatedVolume


217 Call - - - - +.0045 .0795 0 20

218 Call - - - - +.0045 .0735 0 4

219 Call - - - - +.0050 .0680 0 1

220 Call .0625 .0625 .0625 - +.0050 .0630 10 47

221 Call - - - - +.0040 .0570 0 0

222 Call - - - - +.0035 .0520 0 0

223 Call - - - - +.0025 .0465 0 0

224 Call - - - - +.0015 .0415 0 0

F520 Options 14

Understanding December Quotes

Put in Dec How much does it cost to purchase:

» one call of Copper Futures Options contract (exercise price of 217)?Call = .0795 / lb * 25,000 lb per contract = $1,987.50 per contract

What is the intrinsic value of a call on Copper Futures Options (exercise price of 317)?Call = max(0,F-X) = max(0,2.20 – 2.17) = 0.03 centsUse the futures copper price (not the cash price)

What is the time value of money?Option Price - Intrinsic Value = 0.0795 – 0.03 = 0.0495 / lb

What is the intrinsic value of a call on Copper Futures Options (exercise price of 224)?Call = max(0,F-X) = max(0,2.20 – 2.24) = 0 centsUse the futures copper price (not the cash price)

What is the time value of money?Option Price - Intrinsic Value = 0.0415 – 0 = 0.0415 / lb

What is the intrinsic value of a call on Copper Futures Options (exercise price of 220)? Option Price - Intrinsic Value = 0.0630 – 0 = 0.0630 / lb

Why there greater intrinsic value for options near the money.

F520 Options 15

Understanding Option QuotesCopper Options on Futures (Put in

Dec)Strike Type Open High Low Last

Change

SettleEstimatedVolume


217 Put - .0510B - - -.0020 .0485 0 13

218 Put - .0555B - - -.0020 .0525 0 2

219 Put - .0600B - - -.0015 .0570 0 3

220 Put - .0650B - - -.0015 .0620 0 24

221 Put - .0710B .0680A - -.0025 .0660 0 0

222 Put - .0765B .0735A - -.0035 .0705 0 0

223 Put - - .0790A - -.0040 .0755 0 0

224 Put - - .0840A - -.0050 .0805 0 0

F520 Options 16

Understanding December Quotes

Put in Dec How much does it cost to purchase:

» one put of Copper Futures Options contract (exercise price of 217)?Call = .0485 / lb * 25,000 lb per contract = $1,212.50 per contract

What is the intrinsic value of a put on Copper Futures Options (exercise price of 317)?Call = max(0,X-F) = max(0,2.17 – 2.20) = 0.0 centsUse the futures copper price (not the cash price)


What is the intrinsic value of a put on Copper Futures Options (exercise price of 224)?Call = max(0,X-F) = max(0,2.24 – 2.20) = 0.04 centsUse the futures copper price (not the cash price)


What is the intrinsic value of a put on Copper Futures Options (exercise price of 220)? Option Price - Intrinsic Value = 0.0620 – 0 = 0.0620 / lb

Why there greater intrinsic value for options near the money.

F520 Options 17

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)

F520 Options 18

Value of a Call Option

Intrinsic valuemax(0,S-X)

X

Call Price

Time Value

Security Price

F520 Options 19

Value of a Put Option

Intrinsic valuemax(0,X-S)

X

Put Price

Time Value

Security Price

F520 Options 20

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

F520 Options 21

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.

F520 Options 22

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

F520 Options 23

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)

F520 Options 24

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

F520 Options 25

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

F520 Options 26

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

F520 Options 27

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

F520 Options 28

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)

F520 Options 29

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.

F520 Options 30

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 31

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

F520 Options 32

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).

F520 Options 33

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

F520 Options 34

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?

F520 Options 35

Hedged


Firm in Period 1

Price of oil high 0.5 600Price of oil low 0.5 600



t=1


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.

F520 Options 36

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 37Will the Unhedged firm add a risk-free project when new capital must be added by

equityholders


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?



t=1


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 38Would New Bondholders add the new capital?Bondholders generally enter as subordinate to the old

bonds.


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?



t=1


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.

F520 Options 39

Will the hedged firm add take a risk-free project?


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



t=1


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).

F520 Options 40

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

F520 Options 41

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 42

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 43

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?

F520 Options 44

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 45

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 46

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 47

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

F520 Options 48

Contracts

Options and Futures http://www.cmegroup.com/education/getting-started.html

Future and Option contractshttp://www.cmegroup.com/globex/

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