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Financial Options and Applications in Corporate Finance

CHAPTER 8

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Topics in Chapter
Financial Options Terminology
Option Price Relationships
Black-Scholes Option Pricing Model
Put-Call Parity

The Big Picture: The Value of a Stock Option

What is a financial option?
An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.

What is the single most important
characteristic of an option?
It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset.

Option Terminology (1 of 8)
Call option: An option to buy a specified number of shares of a security within some future period.
Put option: An option to sell a specified number of shares of a security within some future period.

Option Terminology (2 of 8)
Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold.
Expiration date: The last date the option can be exercised.

Option Terminology (3 of 8)
Exercise value: The value of a call option if it were exercised today =
Max[0, Current stock price – Strike price]
Note: The exercise value is zero if the stock price is less than the strike price.
Option price: The market price of the option contract.

Option Terminology (4 of 8)
Time value: Option price minus the exercise value. It is the additional value because the option has remaining time until it expires.

Option Terminology (5 of 8)
Writing a call option: For every new option, there is an investor who “writes” the option.
A writer creates the contract, sells it to another investor, and must fulfill the option contract if it is exercised.
For example, the writer of a call must be prepared to sell a share of stock to the investor who owns the call.

Option Terminology (6 of 8)
Covered option: A call option written against stock held in an investor’s portfolio.
Naked (uncovered) option: An option written without the stock to back it up.

Option Terminology (7 of 8)
In-the-money call: A call whose strike price is less than the current price of the underlying stock.
Out-of-the-money call: A call option whose strike price exceeds the current stock price.

Option Terminology (8 of 8)
LEAPS: Long-term Equity AnticiPation Securities that are similar to conventional options except that they are long-term options with maturities of up to 2 ½ years.

Consider the following data:
Strike price = $25.
Stock Price Call Option Price
$25 $3.00
30 7.50
35 12.00
40 16.50
45 21.00
50 25.50

Exercise Value of Option
Price of
stock (a) Strike
price (b) Exercise value
of option (a)–(b)
$25.00 $25.00 $0.00
30.00 25.00 5.00
35.00 25.00 10.00
40.00 25.00 15.00
45.00 25.00 20.00
50.00 25.00 25.00

Market Price of Option
Price of
stock (a) Strike
price (b) Exer.
val. (c) Mkt. Price
of opt. (d)
$25.00 $25.00 $0.00 $3.00
30.00 25.00 5.00 7.50
35.00 25.00 10.00 12.00
40.00 25.00 15.00 16.50
45.00 25.00 20.00 21.00
50.00 25.00 25.00 25.50

Time Value of Option
Price of
stock (a) Strike
price (b) Exer.
Val. (c) Mkt. P of
opt. (d) Time value
(d) – (c)
$25.00 $25.00 $0.00 $3.00 $3.00
30.00 25.00 5.00 7.50 2.50
35.00 25.00 10.00 12.00 2.00
40.00 25.00 15.00 16.50 1.50
45.00 25.00 20.00 21.00 1.00
50.00 25.00 25.00 25.50 0.50

Call Time Value Diagram

Option Time Value Versus Exercise Value
The time value, which is the option price less its exercise value, declines as the stock price increases.
This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.

The Binomial Model
Stock assumptions:
Current price: P = $27
In next 6 months, stock can either
Go up by factor of 1.41
Go down by factor of 0.71
Call option assumptions
Expires in t = 6 months = 0.5 years
Exercise price: X = $25
Risk-free rate: rRF = 6%

Binomial Payoffs at Call’s Expiration

Create portfolio by writing 1 option and buying Ns shares of stock.
Portfolio payoffs:
Stock is up: Ns(P)(u) − Cu
Stock is down: Ns(P)(d) − Cd

The Hedge Portfolio with a Riskless Payoff
Set payoffs for up and down equal, solve for number of shares:
Ns= (Cu − Cd) / P(u − d)
In our example:
Ns= ($13.07 − $0) / $27(1.41 − 0.71)
Ns=0.6915

Riskless Portfolio’s Payoffs at Call’s Expiration: $13.26

Riskless payoffs earn the risk-free rate of return.
Find PV of riskless payoff. Discount at risk-free rate compounded daily.
N = 0.5(365)
I/YR = 6/365
PMT = 0
FV = −$13.26 (because we want to know how much we would want now to give up the FV)
PV = $12.87

Alternatively, use the PV formula
(daily compounding).
PV = $13.26 / (1 + 0.06/365)365*0.5
= $12.87

The Value of the Call Option
Because the portfolio is riskless:
VPortfolio = PV of riskless payoff
By definition, the value of the portfolio is:
VPortfolio = Ns(P) − VC
Equating these and rearranging, we get the value of the call:
VC = Ns(P) − PV of riskless payoff

Value of Call
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
VC = 0.6915($27)
− $13.26 / (1 + 0.06/365)365*0.5
= $18.67 − $12.87
= $5.80
(VC = $5.81 if no rounding in any intermediate steps.)

Portfolio Replicating the Call Option
From the previous slide we have:
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
The right side of the equation is the same as creating a portfolio by buying Ns shares of stock and borrowing an amount equal to the present value of the hedge portfolio’s riskless payoff (which must be repaid).
The payoffs of the replicating portfolio are the same as the option’s payoffs.

Replicating Portfolio Payoffs: Amount Borrowed and Repaid
Amount borrowed:
PV of payoff = $12.87
Repayment due to borrowing this amount:
Repayment = $12.87 (1 + rRF/365)365*t
Repayment = $13.26
Notice that this is the same as the payoff of the hedge portfolio.

Replicating Portfolio Net Payoffs
Stock up:
Value of stock = 0.6915($38.07) =$26.33
Repayment of borrowing = $13.26
Net portfolio payoff = $13.07
Stock down:
Value of stock = 0.6915($19.17) =$13.26
Repayment of borrowing = $13.26
Net portfolio payoff = $0
Notice that the replicating portfolio’s payoffs exactly equal those of the option.

Replicating Portfolios and Arbitrage
The payoffs of the replicating portfolio exactly equal those of the call option.
Cost of replicating portfolio
= Ns(P) − Amount borrowed
= 0.6915($27) − $12.87
= $18.67 − $12.87
= $5.80
If the call option’s price is not the same as the cost of the replicating portfolio, then there will be an opportunity for arbitrage.

Arbitrage Example
Suppose the option sells for $6.
You can write option, receiving $6.
Create replicating portfolio for $5.80, netting $6.00 −$5.80 = $0.20.
Arbitrage:
You invested none of your own money.
You have no risk (the replicating portfolio’s payoffs exactly equal the payoffs you will owe because you wrote the option.
You have cash ($0.20) in your pocket.

Arbitrage and Equilibrium Prices
If you could make a sure arbitrage profit, you would want to repeat it (and so would other investors).
With so many trying to write (sell) options, the extra “supply” would drive the option’s price down until it reached $5.80 and there were no more arbitrage profits available.
The opposite would occur if the option sold for less than $5.80.

Multi-Period Binomial Pricing
If you divided time into smaller periods and allowed the stock price to go up or down each period, you would have a more reasonable outcome of possible stock prices when the option expires.
This type of problem can be solved with a binomial lattice.
As time periods get smaller, the binomial option price converges to the Black-Scholes price, which we discuss in later slides.

Assumptions of the
Black-Scholes Option Pricing Model
The stock underlying the call option provides no dividends during the call option’s life.
There are no transactions costs for the sale/purchase of either the stock or the option.
Risk-free rate, rRF, is known and constant during the option’s life.

Assumptions
Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate.
No penalty for short selling and sellers receive immediately full cash proceeds at today’s price.
Call option can be exercised only on its expiration date.
Security trading takes place in continuous time, and stock prices move randomly in continuous time.

What are the three equations that make up the OPM?

What is the value of the following call option according to the OPM?
Assume:
P = $27
X = $25
rRF = 6%
t = 0.5 years
σ = 0.49

First, find d1.

Second, find d2.
d2 = d1
d2 = 0.4819
d2 = 0.1355

Third, find N(d1) and N(d2)
N(d1) = N(0.4819) = 0.6851
N(d2) = N(0.1355) = 0.5539
Note: Values obtained from Excel using NORMSDIST function. For example:
N(d1) = NORMSDIST(0.4819)

Fourth, find value of option.
VC = $27(0.6851) – $25 e-(0.06)(0.5) (0.5539)
= $18.4977 – $13.4383
= $5.06

What impact do the following parameters have on a call option’s value?
Current stock price: Call option value increases as the current stock price increases.
Strike price: As the exercise price increases, a call option’s value decreases.

Impact on Call Value (1 of 2)
Option period: As the expiration date is lengthened, a call option’s value increases.
Longer time to expiration increases probability of very high stock price, which has big payoff.
Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price.
Risk-free rate: Call option’s value tends to increase as rRF increases (reduces the PV of the exercise price).

Impact on Call Value (2 of 2)
Stock return variance: Option value increases with variance of the underlying stock.
Higher variance increases probability of very high stock price, which has big payoff.
Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price.

Put Options
A put option gives its holder the right to sell a share of stock at a specified stock on or before a particular date.

Put-Call Parity
Portfolio 1:
Put option,
Share of stock, P
Portfolio 2:
Call option, VC
PV of exercise price, X

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