chapters 14/15 – part 1 options: basic concepts

1

Chapters 14/15 – Part 1Options: Basic Concepts

Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option‑Pricing Formula Investment in Real Projects and Options Summary and Conclusions

2

Options Contracts: Preliminaries

Option Definition. Calls versus Puts Call options Put options. Exercising the Option Strike Price or Exercise Price Expiration Date European versus American options

3

Options Contracts: Preliminaries

Intrinsic Value Speculative Value

Option Premium =

Intrinsic Value

Speculative Value

+

4

Value of an Option at Expiration

Impact of leverage…

Stock price is $50. Buy 100 shares

Call strike is $50, price is $10. Buy 1 contract.

Put strike is $50, price is $10. Buy 1 contract.

=====================

C = S – E

P = E - S

5

Call Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Write a call

Buy a call

6

Put Option Payoffs

Write a put

Buy a put

-20

0

-40

20

0

-60

40

60

Op

tio

n p

ayo

ffs

($)

Stock price ($)

1009080706010 20 30 40 50

7

Call Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a call

Exercise price = $50

8

Call Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Write a call

Exercise price = $50

9

Call Option Profits

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Write a call

Buy a call

Exercise price = $50; option premium = $10

10

Put Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a put

Exercise price = $50

11

Put Option Payoffs

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Op

tio

n p

ayo

ffs

($)

write a put

Exercise price = $50

Stock price ($)

12

Put Option Profits

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Buy a put

Write a put

Exercise price = $50; option premium = $10

10

-10

13

Selling Options – Writing Options

The seller (or writer) of an option has an obligation.

The purchaser of an option has an option.

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Buy a put

Write a put

10

-10

-20

100908070600 10 20 30 40 50

-40

20

0

-60

40

60

Stock price ($)

Op

tio

n p

rofi

ts (

$)

Write a call

Buy a call

14

Call Option Payoffs at Expiration (Δ exercise)

Stock price ($)

100908070600 10 20 30 40 50

Buy a call

20

10

40

30

0

50

60

Op

tio

n p

ayo

ffs

($)

E=50E=0

15

Option Pricing Bounds at Expiration

Upper bounds Call Options Put Options

Lower Bounds Call option intrinsic value = max [0, S - E] Put option intrinsic value = max [0, E - S]

In-the-money / Out-of-the-money Time premium/time decay At expiration, an American call option is worth

the same as a European option with the same characteristics.

16

Reading The Wall Street Journal

Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

17

Valuing Options

The last section concerned itself with the value of an option at expiration.

This section considers the value of an option prior to the expiration date.

18

Option Value Determinants

Call Put1. Exercise price2. Stock price 3. Interest rate 4. Volatility in the stock price 5. Expiration date

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on these factors.

19

Varying Option Input Values

Stock price: Call: as stock price increases call option price

increases Put: as stock price increases put option price

decreases

Strike price: Call: as strike price increases call option price

decreases Put: as strike price increases put option price

increases

20

Varying Option Input Values

Time until expiration: Call & Put: as time to expiration increases call and put

option price increase

Volatility: Call & Put: as volatility increases call & put value

increase

Risk-free rate: Call: as the risk-free rate increases call option price

increases Put: as the risk-free rate increases put option price

decreases

21

Figure 15.1. Put and Call Option Prices

0

5

10

15

20

25

Stock Price ($)

Op

tio

n P

rice

($) Call PricePut Price

22

Figure 15.2. Option Prices and Time to Expiration

0

5

10

15

20

25

30

35

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Time to Expiration (months)

Op

tio

n P

rice

($)

Call Price

Put Price

23

Figure 15.3. Option Prices and Sigma

0

5

10

15

20

25

Sigma (%)

Op

tio

n P

rice

($)

Call Price

Put Price

24

Figure 15.4. Options Prices and Interest Rates

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Interest Rate (%)

Op

tio

n P

rice

($)

Call Price

Put Price

25

Option Value Determinants

Call Put1. Exercise price – +2. Stock price + –3. Interest rate + –4. Volatility in the stock price + +5. Expiration date + +

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on these factors.

26

Market Value, Time Value and Intrinsic Value for an American Call

CaT > Max[ST - E, 0]Profit

lossE ST

Market Value

Intrinsic value

S T - E

Time value

Out-of-the-money In-the-money

S T

The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.

27

Combinations of Options

Puts and calls can serve as the building blocks for more complex option contracts.

If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

28

Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiration

Buy a put with an exercise price of $50

Buy the stock

Protective Put strategy has downside protection and upside potential

$50

$0

$50

Value at expiration

Value of stock at

expiration

29

Protective Put Strategy Profits

Buy a put with exercise price of $50 for $10

Buy the stock at $40

$40

Protective Put strategy has

downside protection and upside potential

$40

$0

-$40

$50

Value at expiration

Value of stock at

expiration

30

Covered Call Strategy

Sell a call with exercise price of $50 for $10

Buy the stock at $40

$40

Covered call

$40

$0

-$40

$10

-$30

$30 $50

Value of stock at expiration

Value at expiration

31

Long Straddle: Buy a Call and a Put

Buy a put with an exercise price of

$50 for $10$40

A Long Straddle only makes money if the stock price moves $20 away from $50.

$40

$0

-$20$50

Buy a call with an exercise price of $50 for $10

-$10

$30

$60$30 $70

Value of stock at

expiration

Value at expiration

32

Short Straddle: Sell a Call and a Put

Sell a put with exercise price of$50 for $10

$40

A Short Straddle only loses money if the stock price moves $20 away from $50.

-$40

$0

-$30$50

Sell a call with an exercise price of $50 for $10

$10

$20

$60$30 $70

Value of stock at expiration

Value at expiration

33

Put-Call Parity

TreECPS

TreECPS

Buy the stock, buy a put, and write a call; the sum ofwhich equals the strike price discounted at the risk-free rate

C = Call option price P = Put option priceS = Current stock price E = Option strike pricer = Risk-free rate T = Time until option

expiration

34

Put-Call ParityBuy Stock & Buy Put

Share Price

Pos

itio

n V

alue

Combination: Long Stock & Long Put

Long Put

Long Stock

35

Put-Call ParityBuy Call & Buy Zero Coupon Risk-Free Bond @ Exercise Price

Long Bond

Share Price

Pos

itio

n V

alue

Combination: Long Stock & Long Bond

Long Call

36

Put-Call Parity

Share Price

Pos

itio

n V

alue Combination:

Long Stock & Long Put

Long Put

Long Stock

Share Price

Pos

itio

n V

alue

Combination: Long Stock & Long Bond

Long Call

Long Bond

TreECPS In market equilibrium, it must be the case that option prices are set such that:

Otherwise, riskless portfolios with positive payoffs exist.

37

The Black-Scholes Model

Value of a stock option is a function of 6 input factors:1. Current price of underlying stock.2. Strike price specified in the option contract.3. Risk-free interest rate over the life of the contract.4. Time remaining until the option contract expires.5. Price volatility of the underlying stock.

The price of a call option equals:

)()( 21 dNeEdNSC Tr

38

Black-Scholes Model

Where the inputs are:S = Current stock priceE = Option strike pricer = Risk-free interest rateT = Time remaining until option expiration = Sigma, representing stock price volatility,

standard deviation

)()( 21 dNeEdNSC Tr

39

Black-Scholes Model

)()( 21 dNeEdNSC Tr

Where d1 and d2 equal:

T

TrES

d2

2

1

2ln

Tdd 212

40

Black-Scholes Models

ESC

SEP

Also, remember at expiration:

Remembering put-call parity, the value of a put,given the value of a call equals:

TreECPS TreESCP

41

The Black-Scholes Model

Find the value of a six-month call option on the Microsoft with an exercise price of $150

The current value of a share of Microsoft is $160The interest rate available in the U.S. is r = 5%.The option maturity is 6 months (half of a year).The standard deviation of the underlying asset is

30% per annum.Before we start, note that the intrinsic value of the

option is $10—our answer must be at least that amount.

42

The Black-Scholes Model

Then d2,

T

TσrESd

)5.()/ln( 2

1

First calculate d1 and d2

Tdd 12

5282.05.30.0

5).)30.0(5.05(.)150/160ln( 2

1

d

31602.05.30.052815.02 d

Assume S = $160, X = $150, T = 6 months, r = 5%, and = 30%, calculate the value of a call.

43

The Black-Scholes Model

N(d1) = N(0.52815) = 0.7013

N(d2) = N(0.31602) = 0.62401

5282.01 d

31602.02 d

)N()N( 210 dEedSC rT

92.20$

62401.01507013.0160$

0

5.05.0

C

eC

44

Assume S = $50, X = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put.

12514550328 500100 .$$$.$ ).(. eP

328754045812050 500100 .$).().( ).(. eC

8840

500280

5002280

1004550

2

1 ...

..

.ln

d

686050028088402 .... d

From a standard normal probability table, look up N(d1) = 0.812 and N(d2) = 0.754 (or use Excel’s “normsdist” function)

Another Black-Scholes Example

45

Real Options

Real estate developer buys 70 acres in a rural area. He plans on building a subdivision when the population from the city expands this direction. If growth is less than anticipated, the developer thinks he can sell the land to a country club to build a golf course on the property.

The development option is a ______ option. The golf course option is a _______ option. How would these real options change the standard

NPV analysis?

46

Collar: Buy a Put, Buy the Stock, Sell the Call

Buy a put with exercise price of $50 for $0.67

Buy the stock at $80

$80

$49.33

$0

-$80

$50

Value at expiration

Value of stock at

expiration

$120

Sell a call with exercise price of $120 for $2.76

$2.76

-$27.91

$42.11

Collar

$0.67

NTS