Learning Objectives

1
Learning Objectives

• Use the Black-Scholes option pricing model
  (BS-OPM) to value call and put options on common
  stock

2
Options Terminology Summary / Review

• A call option gives the holder the right to buy
  one share of the underlying stock at a specified
  price within a stated time period.
• A put option gives the holder the right to sell
  one share of the underlying stock at a specified
  price within a stated time period.
• The fixed price is called the exercise or strike
  price.

3
Review of Properties of Options

• As stock price increases,
• value of call option increases
• value of put option increases
• As exercise price increases,
• value of call option decreases
• value of put option increases
• As the time to maturity increases,
• value of call option increases
• value of put option increases

4
Option Trading

• Exchange listed options
• Chicago Board of Options Exchange (CBOE)
• NYSE
• ASE
• Pacific Stock Exchange
• Philadelphia Stock Exchange
• Over-the-Counter options
• Non-standardized contracts

5
Warrants

• A warrant is a long-term call option issued by
  the firm.
• Entitles holder to buy a fixed number of shares
  from the firm, at a stated price, within a stated
  time period.
• When a warrant is exercised, the number of
  outstanding shares increases.

6
Option Pricing Models

• Binomial Option Pricing Model
• Black-Scholes Option Pricing Model
• Put-Call Parity Relationship

7
One-Period Binomial Option Pricing Model

• A stock has two possible prices at t1 80 with
  a risk adjusted probability of 0.60 and 30 with
  risk adjusted probability 0.40. A call option on
  this stock has an exercise price of 75. The risk
  free rate is 5 per period.
• Compute the current price of the stock and the
  value of the call option.

8
Stock Prices
80
p 0.60
P
30
p 0.40

• Expected price at t1 is
• (0.60)80 (0.40)(30) 60.00
• Present value at 5 is
• 60.00 / 1.05 57.14

9
Call Option Values at Maturity

• If stock price is 80, the call option is worth
  5 ( 80 - 75).
• If the stock price is 30, the call option is
  worthless.

10
Call Option Values
5
p 0.60
C
0
p 0.40

• Expected price at t1 is
• (0.60)5 (0.40)(0) 3.00
• Present value at 5 is
• 3.00 / 1.05 2.86

11
Two-Period Case - Stock Prices
130
104.76
81.63
80
57.14
30
12
Two-Period Case - Call Option Prices
55
33.33
20.14
5
2.86
0
13
Black-Scholes Option Pricing Model

• Assumptions
• The option and the underlying asset trade in
  perfect markets.
• The returns on the underlying assets are normally
  distributed with a constant ? over the life of
  the option.
• The riskless rate of interest is constant over
  the options life.
• Option contracts are European (cannot be
  exercised prior to maturity).
• Underlying asset does not provide any cash flows
  over the life of the option.

14
Black-Scholes Option Pricing Model

• S Strike price of the call option.
• P0 current value of the underlying asset.
• k riskless APR with continuous compounding.
• ?t time in years to option expiration.
• ? standard deviation of the (continuously
  compounded) returns on the asset.
• N(d) Cumulative distribution function for a
  standard normal random variable d.

15
Black-Scholes Option Pricing Model

• The value of the call option, C, is given by

16
Black-Scholes Option Pricing Model

• Find the value of a European call option on
  Hightone Records. The current stock price is 48,
  and the stocks volatility is 30. The risk free
  rate is 5 per year. The call option matures in 6
  months and has an exercise price of 50.

17
Black-Scholes Option Pricing Model
18
Black-Scholes Option Pricing Model
19
Black-Scholes Option Pricing Model

• N(d1) 0.51256
• N(d2) 0.42832

20
Black-Scholes Option Pricing Model

• N(d1) 0.51256
• N(d2) 0.42832

21
Put-Call Parity Relationship

• Consider a call and a put on the same underlying
  asset with the same strike price S0 and the same
  maturity date.
• Let Pu be the price of the put option and C be
  the price of the call option.

22
Put-Call Parity Relationship

• From Chapter 8, we know that at the maturity of
  the options,
• -P0 - Pu C - S0
• Taking present values, we get the put call parity
  relationship.

23
Put-Call Parity Relationship
24
Put-Call Parity Relationship

• Find the value of a put option on Hightone
  Records. The put option also has a strike price
  of 50 and expires 6 months from today.

25
Put-Call Parity Relationship

• Find the value of a put option on Hightone
  Records. The put option also has a strike price
  of 50 and expires 6 months from today.

26
Valuing Warrants

• Warrants are long term call options.
• When a warrant is exercised, the number of shares
  outstanding increases.
• Let ? be the proportionate increase in the number
  of outstanding shares after all warrants are
  exercised.

27
Valuing Warrants

• The value of the warrant before it is issued is
  simple C/(1?) where C is the value of a call
  option to buy one share.
• After the warrant is issued, its value is equal
  to that of the call option.