Transcript intrinsic value

Option Valuation
Chapter 21
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
Time Value of Options: Call
Option
value
Value of Call
Intrinsic Value
Time value
X
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 > S0 - ( X + D ) / ( 1 + Rf )T
C > S0 - PV ( X ) - PV ( D )
21-5
Allowable Range for Call
Call Value
Lower Bound
= S0 - PV (X) - PV (D)
S0
PV (X) + PV (D)
21-6
Arbitrage
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
Option “Moneyness”
“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
Today
200
100
Expiration
75
C
50
Stock Price
0
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: dSu – Cu = d200 – 75
Downstate: dSd – Cd = d50 – 0
Set equal, and solve for d (called a d hedge, or
hedge ratio)
200d – 75 = 50d which gives: d = 0.5
By inspection we can see that the hedge ratio
is:
Cu  Cd
d
Su  S 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+rf) for rf = 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
Expiration
150
53.70
0
Payoff Structure
is exactly 2 times
the Call
21-23
Binomial Option Pricing
150
53.70
75
C
0
0
2C = $53.70
C = $26.85
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 = {[pSu + (1-p)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
Probability
Stock Price
3 up
1/8
100 (1.05)3
=115.76
2 up 1 down
3/8
100 (1.05)2 (.97)
=106.94
1 up 2 down
3/8
100 (1.05) (.97)2
= 98.79
3 down
1/8
100 (.97)3
= 91.27
21-40
Black-Scholes Option Valuation
Co = SoN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r + 2/2)T] / ( T1/2)
d2 = d1 + ( T1/2)
where
Co = Current call option value.
So = 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
So = 100
X = 95
r = .10
T = .25 (quarter)
 = .50
d1 = [ln(100/95) + (.10+(5 2/2))] / (5 .251/2)
= .43
d2 = .43 + ((5 .251/2)
= .18
21-43
Probabilities from Normal Dist
N (.43) = .6664
Table 17.2
d
N(d)
.42
.6628
.43
.6664 Interpolation
.44
.6700
21-44
Probabilities from Normal Dist.
N (.18) = .5714
Table 17.2
d
N(d)
.16
.5636
.18
.5714
.20
.5793
21-45
Call Option Value
Co = SoN(d1) - Xe-rTN(d2)
Co = 100 X .6664 - 95 e- .10 X .25 X .5714
Co = 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(d2)] - S0 [1-N(d1)]
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) - So
= C + Xe-rT - So
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 S0 with S0 - 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 (d1)
Put = N (d1) - 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
= 60 days
Put price P
= $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
40,770
45,265
21-55
Profit Position on Hedged Put Portfolio
Value of put option: implied vol. = 35%
Stock Price
Put Price
Profit (loss) for each put
89
90
91
$5.254
$4.785
$4.347
.759
.290
(.148)
Value of and profit on hedged portfolio
Stock Price
89
90
91
Value of 1,000 puts
$ 5,254
$ 4,785
$ 4,347
Value of 453 shares
40,317
40,770
41,223
Total
45,571
45,555
5,570
Profit
306
290
305
21-56