McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

Option Values
Intrinsic value - profit that could be made if the option was immediately exercised. (Alternatively, the value of the option, if today was its maturity date)
Call: Max(stock price - exercise price,0)
Put: max(exercise price - stock price,0)
Time value - the difference between the option price and the intrinsic value.
21-2

Option value
Time Value of Options: Call
Value of Call
Time value
X
Intrinsic Value
Stock Price
21-3

Factors Influencing Option Values: Calls
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Dividend Rate
Effect on value increases decreases increases increases increases decreases
21-4

Restrictions on Option Value: Call
Value cannot be negative
Value cannot exceed the stock value
Value of the call must be greater than the value of levered equity
C > S
0
- ( X + D ) / ( 1 + R f
) T
C > S
0
- PV ( X ) - PV ( D )
21-5

Call Value
Allowable Range for Call
PV (X) + PV (D)
Lower Bound
= S
0
- PV (X) - PV (D)
S
0
21-6

Arbitrage :
No possibility of a loss
A potential for a gain
No cash outlay
In finance, arbitrage is not allowed to persist.
“Absence of Arbitrage” = “No Free Lunch”
The “Absence of Arbitrage” rule is often used in finance to figure out prices of derivative securities.
Think about what would happen if arbitrage were allowed to persist. (Easy money for everybody)
21-7

The Upper Bound for a Call Option Price
Call option price must be less than the stock price .
Otherwise, arbitrage will be possible.
How?
Suppose you see a call option selling for $65, and the underlying stock is selling for $60.
The arbitrage: sell the call, and buy the stock.
Worst case? The option is exercised and you pocket $5.
Best case? The stock sells for less than $65 at option expiration, and you keep all of the $65.
There was zero cash outlay today, there was no possibility of loss, and there was a potential for gain.
21-8

The Upper Bound for a Put Option Price
Put option price must be less than the strike price. Otherwise, arbitrage will be possible.
How? Suppose there is a put option with a strike price of $50 and this put is selling for $60.
The Arbitrage : Sell the put, and invest the $60 in the bank. (Note you have zero cash outlay).
Worse case? Stock price goes to zero.
You must pay $50 for the stock (because you were the put writer).
But, you have $60 from the sale of the put (plus interest).
Best case? Stock price is at least $50 at expiration.
The put expires with zero value (and you are off the hook).
You keep the entire $60, plus interest.
21-9

The Lower Bound on Option Prices
Option prices must be at least zero.
By definition, an option can simply be discarded.
To derive a meaningful lower bound, we need to introduce a new term: intrinsic value .
The intrinsic value of an option is the payoff that an option holder receives if the underlying stock price does not change from its current value.
21-10

Option Intrinsic Values
Call option intrinsic value = max [ S – X ,0 ]
In words: The call option intrinsic value is the maximum of zero or the stock price minus the strike price.
Put option intrinsic value = max [X-S, 0 ]
In words: The put option intrinsic value is the maximum of zero or the strike price minus the stock price.
21-11

“In the Money” options have a positive intrinsic value.
For calls, the strike price is less than the stock price.
For puts, the strike price is greater than the stock price.
“Out of the Money” options have a zero intrinsic value.
For calls, the strike price is greater than the stock price.
For puts, the strike price is less than the stock price.
“At the Money” options is a term used for options when the stock price and the strike price are about the same.
21-12

Intrinsic Values and Arbitrage, Calls
Call options with American-style exercise must sell for at least their intrinsic value.
(Otherwise, there is arbitrage )
Suppose: S = $60; C = $5; X = $50.
Instant Arbitrage. How?
Buy the call for $5.
Immediately exercise the call, and buy the stock for
$50.
In the next instant, sell the stock at the market price of $60.
You made a profit with zero cash outlay.
21-13

Intrinsic Values and Arbitrage, Puts
Put options with American-style exercise must sell for at least their intrinsic value.
(Otherwise, there is arbitrage )
Suppose: S = $40; P = $5; K = $50.
Instant Arbitrage. How?
Buy the put for $5.
Buy the stock for $40.
Immediately exercise the put, and sell the stock for $50.
You made a profit with zero cash outlay.
21-14

Lower Bounds for Options (con’t)
As we have seen, to prevent arbitrage, option prices cannot be less than the option intrinsic value.
Otherwise, arbitrage will be possible.
Note that immediate exercise was needed.
Therefore, options needed to have American-style exercise.
Using equations: If S is the current stock price, and X is the strike price:
Call option price
 max [S-X,0 ]
Put option price
 max [X-S,0 ]
21-15

Pricing Bounds Summary
Calls:
Upper Bound: Stock Price, S
Lower Bound: Intrinsic Value : max [ S –X, 0]
Puts:
Upper Bound: Exercise Price, X
Lower Bound: Intrinsic Value max [X-S,0]
21-16

How would you price an option?
Step 1. Project the distribution of ending stock price (based on the stock price volatility)
Step 2. Compute the payoff for that distribution
Step 3. Using the correct risk adjusted discount rate, discount the expected payoffs
Nice idea – but although Step 1 and 2 are “easy” to do, what is the RADR for an option? Option problem took more than 75 years to solve
21-17

Binomial Option Pricing
Call options pay off when the stock prices rises above the eXercise price. Thus, to price options, we need a model that has variable stock prices
The simplest way to represent an uncertain stock price is to say it can either go up or down a given amount
21-18

Binomial Option Pricing Example
Today
Expiration
200
Today Expiration
75
100
C
50
0
Stock Price
Call Option Value
X = 125
21-19

Hedge ratio approach
If the stock goes up in value, so will the call option (as seen in the previous slide).
If you go long in the stock, and short in the call, its easy to set up a risk free future, by carefully choosing the fraction of stock to hold long for each option you write.
Buy d stock for every option you write, so that the payoff in the up state equals the payoff in the down state
21-20

Create Equal Expiration Cash Flows
Upstate: d
Su – Cu = d
200 – 75
Downstate: d
Sd – Cd = d
50 – 0
Set equal, and solve for d
(called a d hedge, or hedge ratio)
200 d – 75 = 50 d which gives: d
= 0.5
By inspection we can see that the hedge ratio is: d 
C
S u u


C
S d d
21-21

Create Equal Expiration Cash Flows
CF in up = .5*200 – 75 = 25 = CF down
What’s the PV of receiving $25 for sure? = 25/(1+r f
) for r f
= 8% = 23.15
So, the value of 0.5 stocks, minus the value of one Call option is $23.15 today
0.5(100) – C = 23.15
Solve to C, and you have = 26.85
21-22

Binomial Option Pricing: Alternate Portfolio
Alternative Portfolio
Buy 1 share of stock at $100
Borrow enough money to get a zero payout in down state
(Owe $50 at expiration, i.e. borrow $46.30 (8% Rate)
Net outlay $53.70
Payoff
Value of Stock 50 200
Repay loan - 50 -50
Net Payoff 0 150
53.70
Expiration
150
0
Payoff Structure is exactly 2 times the Call
21-23

53.70
Binomial Option Pricing
150
C
0
2C = $53.70
C = $26.85
75
0
21-24

Implied Probability
If the stock has a beta = 1.5, and the expected return on the market is 15%
(risk free as noted is 8%), then what is the probability the stock will go up to the higher value?
CAPM –
Hpr = {[ p
Su + (1p
)Sd] – So}/So
Solve for p
21-25

RADR for the Call
What is the discount rate implied on the call (ie RADR)
What is the Beta of the Call?
21-26

Price of the put
Use Put-Call parity to price the put
21-27

RADR for the Put
What is the RADR for the put?
What is the Beta for the put?
21-28

Option value if volatility is smaller
The size of the up or down movement must be set according to the volatility of the stock.
Lets say the upstate would yield a price of 160 and the downstate a price of
62.5. What happens to the value of the put and the call (with this lower volatility)? What happens to the RADR and the implied beta of the call and put
21-29

What if we change the eXercise Price?
If the strike price is $100, what happens to the value of the call and the put, using the revised pricing values?
21-30

What if the risk free interest rate decreases
For the previous example, what happens when we use a 2% risk free rate, rather than an 8% rate?
21-31

Generalizing the Two-State Approach
Assume that we can break the year into two sixmonth segments.
In each six-month segment the stock could increase by 10% or decrease by 5%.
Assume the stock is initially selling at 100.
Possible outcomes:
Increase by 10% twice
Decrease by 5% twice
Increase once and decrease once (2 paths).
21-32

Generalizing the Two-State Approach
Note: The size of the up and down jump is determined by the volatility of the stock price and the length of the period
121
110
104.50
100
95
90.25
21-33

Delta hedging
Assume that the eXercise price is $100
How many stock should you own at the outset? How many stock after the first period
How do I do that? Habit 2 – begin with the end in mind (solve like a dynamic program)
First solve for Su position, then for Sd position, then work back to origination
21-34

Compute hedge ratio and option values
At Su, d
= (21 – 4.5)/(121-104.5)=1
Risk free portfolio = 121-21 = 100
PV of ptf = 100/(1.04) = 96.1538
1*110 – Cu = 96.1538, so Cu = 13.8462
Hedge ratio for down state is 0.3103
Cd = 0.316
What do you notice about hedge ratios in relationship to the stock price versus exercise price?
21-35

Wind back to the beginning
Hedge ratio at beginning:
(13.85 – 2.60)/(110-95) = .75
Risk free ptf =
Present value ptf =
Value of call =
If I program a computer to buy or sell stock to keep my hedge in place (pursuing a
“delta hedging” strategy), when do I buy, and when do I sell?
21-36

21-37

Expanding to Consider Three Intervals
Assume that we can break the year into three intervals.
For each interval the stock could increase by 5% or decrease by 3%.
Assume the stock is initially selling at
100.
21-38

Expanding to Consider Three Intervals
S
S +
S -
S + +
S + -
S - -
S + + +
S + + -
S + - -
S - - -
21-39

Possible Outcomes with Three Intervals
Event
3 up
Probability
1/8
2 up 1 down 3/8
1 up 2 down 3/8
3 down 1/8
Stock Price
100 (1.05) 3
100 (1.05) 2 (.97)
100 (1.05) (.97) 2
100 (.97) 3
=115.76
=106.94
= 98.79
= 91.27
21-40

Black-Scholes Option Valuation
C o
= S o
N(d
1
) - Xe -rT N(d
2
) d d
1
2
= [ln(S o
/X) + (r +

2 /2)T] / (

T 1/2 )
= d
1
+ (

T 1/2 ) where
C o
= Current call option value.
S o
= Current stock price
N(d) = probability that a random draw from a normal dist. will be less than d.
21-41

Black-Scholes Option Valuation
X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option)
T = time to maturity of the option in years ln = Natural log function
 
Standard deviation of annualized cont. compounded rate of return on the stock
21-42

Call Option Example
S o
= 100 X = 95 r = .10
T = .25 (quarter)

= .50
d
1
= [ln(100/95) + (.10+(

5 2 /2))] / (

5 .25
1/2 ) d
2
= .43
= .43 + ((

5

.25
1/2 )
= .18
21-43

Probabilities from Normal Dist
N (.43) = .6664
Table 17.2
d
.42
N(d)
.6628
.43
.44
.6664 Interpolation
.6700
21-44

Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d
.16
N(d)
.5636
.18
.20
.5714
.5793
21-45

Call Option Value
C o
= S o
N(d
1
) - Xe -rT N(d
2
)
C o
= 100 X .6664 - 95 e - .10 X .25
X .5714
C o
= 13.70
Implied Volatility
Using Black-Scholes and the actual price of the option, solve for volatility.
Is the implied volatility consistent with the stock?
21-46

Put Value Using Black-Scholes
P = Xe -rT [1-N(d
2
)] - S
0
[1-N(d
1
)]
Using the sample call data
S = 100 r = .10 X = 95 g = .5 T = .25
95e -10x.25
(1-.5714)-100(1-.6664) = 6.35
21-47

Put Option Valuation: Using Put-Call Parity
P = C + PV (X) - S o
= C + Xe -rT - S o
Using the example data
C = 13.70
X = 95 S = 100 r = .10
T = .25
P = 13.70 + 95 e -.10 X .25
- 100
P = 6.35
21-48

Black-Scholes Model with Dividends
The call option formula applies to stocks that do not pay dividends.
One approach is to replace the stock price with a dividend adjusted stock price.
Replace S
0 with S
0
- PV (Dividends)
21-49

Using the Black-Scholes Formula
Hedging: Hedge ratio or delta
The number of stocks required to hedge against the price risk of holding one option.
Call = N (d
1
)
Put = N (d
1
) - 1
Option Elasticity
Percentage change in the option’s value given a 1% change in the value of the underlying stock.
21-50

Portfolio Insurance
Buying Puts - results in downside protection with unlimited upside potential.
Limitations
Tracking errors if indexes are used for the puts.
Maturity of puts may be too short.
Hedge ratios or deltas change as stock values change.
21-51

Hedging On Mispriced Options
Option value is positively related to volatility:
If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible.
Profit must be hedged against a decline in the value of the stock.
Performance depends on option price relative to the implied volatility.
21-52

Hedging and Delta
The appropriate hedge will depend on the delta.
Recall the delta is the change in the value of the option relative to the change in the value of the stock.
Delta =
Change in the value of the option
Change of the value of the stock
21-53

Mispriced Option: Text Example
Implied volatility = 33%
Investor believes volatility should = 35%
Option maturity
Put price P
= 60 days
= $4.495
Exercise price and stock price = $90
Risk-free rate r = 4%
Delta = -.453
21-54

Hedged Put Portfolio
Cost to establish the hedged position
1000 put options at $4.495 / option
453 shares at $90 / share
Total outlay
$ 4,495
45,265
40,770
21-55

Profit Position on Hedged Put Portfolio
Value of put option: implied vol. = 35%
Stock Price 89 90 91
Put Price $5.254
Profit (loss) for each put .759
$4.785
.290
$4.347
(.148)
Value of and profit on hedged portfolio
Stock Price
Value of 1,000 puts
89
$ 5,254
90 91
$ 4,785 $ 4,347
Value of 453 shares
Total
Profit
40,317 40,770 41,223
45,571 45,555 5,570
306 290 305
21-56