Lesson 5
Option
Investments
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Part 2: Option
Valuation
(Textbook chapter 21)
Thanh Trúc – TCNH – UEL
5.2-1
Outline
 Option valuation: Introduction
 Restriction on option values
 Binomial option pricing
 Black-Scholes option valuation
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5.2- 2
Option valuation: Introduction
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5.2- 3
Option Values
Intrinsic value - profit that could be
made if the option was immediately
exercised.
– Call: stock price - exercise price
– Put: exercise price - stock price
Time value - the difference between the
option price and the intrinsic value.
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5.2- 4
Figure 21.1 Call Option Value before Expiration
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5.2- 5
Table 21.1 Determinants of Call Option Values
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5.2- 6
Restrictions on Option Value
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5.2- 7
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 )
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5.2- 8
Figure 21.2 Range of Possible Call Option
Values
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5.2- 9
Figure 21.3 Call Option Value as a Function of
the Current Stock Price
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5.2- 10
Figure 21.4 Put Option Values as a Function of
the Current Stock Price
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5.2- 11
Binomial Option Pricing
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5.2- 12
Binomial Option Pricing: Text Example
Page 735/1041
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5.2- 13
Binomial Option Pricing: Text Example
120
100
C
90
Stock Price
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10
0
Call Option Value
X = 110
5.2- 14
Binomial Option Pricing: Text Example
Alternative Portfolio
Buy 1 share of stock at $100
Borrow $81.82 (10% Rate) 18.18
Net outlay $18.18
Payoff
Value of Stock 90 120
Repay loan
- 90 - 90
Net Payoff
0 30
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30
0
Payoff Structure
is exactly 3 times
the Call
5.2- 15
Binomial Option Pricing: Text Example
30
30
18.18
C
0
0
3C = $18.18
C = $6.06
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5.2- 16
Replication of Payoffs and Option Values
Alternative Portfolio - one share of stock
and 3 calls written (X = 110)
Portfolio is perfectly hedged
Stock Value
90
120
Call Obligation
0
-30
Net payoff
90
90
Hence 100 - 3C = 81.82 or C = 6.06
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5.2- 17
Why three call option?-The hedge ratio
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5.2- 18
Why three call option?-The hedge ratio
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5.2- 19
Arbitrage if the option is mispriced
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What if the option is underpriced?
Reverse the arbitrage strategy
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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).
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5.2- 21
Generalizing the Two-State Approach
121
110
104.50
100
95
90.25
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5.2- 22
Generalizing the Two-State Approach
Example: page 738/1041
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5.2- 23
Generalizing the Two-State Approach
Example: page 738/1041
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5.2- 24
Generalizing the Two-State Approach
Example: page 738/1041
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5.2- 25
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.
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5.2- 26
Expanding to Consider Three Intervals
S+++
S++
S++-
S+
S+-
S
S+-SS-S---
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5.2- 27
Possible Outcomes with Three Intervals
Event
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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
5.2- 28
Valuation of Put option
Excersice 9,10: page 766,767/1041
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5.2- 29
Valuation of Put option
Range of stock price: 80 – 130; range of Put option value 0 – 30
The hedge ratio = (30 – 0)/(130 – 80) = 3/5
The strategy: Buy 3 stocks at price of $100 and buy 5 put options
The payoffs:
Initial CFs
S = 80
S = 130
Three stocks
-300
240
390
5 put options
-5P
150
0
300 + 5P
390
390
The value of the Portfolio = 390/1.1 = 354.545 = 300 + 5P
P = (354.545 – 300)/5 = $10.91
Put – Call Parity: S0 + P = PV(X) + C
C = S0 + P – PV(X) = 100 + 10.91 – 110/1.1 = $10.91
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5.2- 30
Figure 21.5 Probability Distributions
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Black-Scholes Option Valuation
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5.2- 32
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.
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5.2- 33
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
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5.2- 34
Figure 21.6 A Standard Normal Curve
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5.2- 35
Call Option Example
So = 100
r = .10
X = 95
T = .25 (quarter)
= .50
d1 = [ln(100/95) + (.10+(5 2/2))] /
(5 .251/2)
= .43
d2 = .43 + ((5.251/2)
= .18
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5.2- 36
Probabilities from Normal Dist
N (.43) = .6664
Table 21.2
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d
.42
N(d)
.6628
.43
.44
.6664 Interpolation
.6700
5.2- 37
Probabilities from Normal Dist.
N (.18) = .5714
Table 21.2
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d
.16
N(d)
.5636
.18
.20
.5714
.5793
5.2- 38
Table 21.2 Cumulative Normal Distribution
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5.2- 39
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?
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5.2- 40
Spreadsheet 21.1 Spreadsheet to Calculate
Black-Scholes Option Values
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5.2- 41
Figure 21.7 Using Goal Seek to Find Implied
Volatility
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5.2- 42
Figure 21.8 Implied Volatility of the S&P 500
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Black-Scholes Model with Dividends
The call option formula applies to stocks
that pay dividends.
One approach is to replace the stock
price with a dividend adjusted stock
price.
Replace S0 with S0 - PV (Dividends)
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5.2- 44
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
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5.2- 45
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
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5.2- 46
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.
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5.2- 47
Figure 21.9 Call Option Value and Hedge Ratio
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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.
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5.2- 49
Figure 21.10 Profit on a Protective Put Strategy
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Figure 21.11 Hedge Ratios Change as the
Stock Price Fluctuates
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Figure 21.12 S&P 500 Cash-to-Futures Spread
in Points at 15 Minute Intervals
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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.
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5.2- 53
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
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5.2- 54
Mispriced Option: Text Example
Implied volatility
= 33%
Investor believes volatility should = 35%
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Option maturity
= 60 days
Put price P
= $4.495
Exercise price and stock price
= $90
Risk-free rate r
= 4%
Delta
= -.453
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Table 21.3 Profit on a Hedged Put Portfolio
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Table 21.4 Profits on Delta-Neutral Options
Portfolio
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Figure 21.13 Implied Volatility of the S&P 500
Index as a Function of Exercise Price
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5.2- 58