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## Option Valuation

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### Option Valuation CHAPTER 21 – PowerPoint PPT presentation

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Title: Option Valuation

1
CHAPTER 21
• Option Valuation

2
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

3
Figure 21.1 Call Option Value before Expiration
4
Table 21.1 Determinants of Call Option Values
5
Binomial Option Pricing Text Example
120
10
100
C
90
0
Call Option Value X 110
Stock Price
6
Binomial Option Pricing Text Example Continued
30
Alternative Portfolio Buy 1 share of stock at
100 Borrow 81.82 (10 Rate) Net outlay
18.18 Payoff Value of Stock 90 120 Repay
loan - 90 - 90 Net Payoff 0 30
18.18
0
Payoff Structure is exactly 3 times the Call
7
Binomial Option Pricing Text Example Continued
30
30
18.18
3C
0
0
3C 18.18 C 6.06
8
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 90/(1 rf) 90/(1.1)
81.82
• 3C 100 81.82
18.18
• Thus C 6.06

9
Hedge Ratio (H)
• Number of stocks per option (H)
• H (C - C-)/(S - S-)
• (10 0)/(120 90)
• 1/3
• Number of options per stock 1/H 3

10
Generalizing the Two-State Approach
• Assume that we can break the year into two
six-month 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)

11
Generalizing the Two-State Approach Continued
121
110
104.50
100
95
90.25
12
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

13
Expanding to Consider Three Intervals Continued
S
S
S -
S
S -
S
S - -
S -
S - -
S - - -
14
Possible Outcomes with Three Intervals
Event Probability Final 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
15
Figure 21.5 Probability Distributions
16
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 distribution will be less than d

17
Black-Scholes Option Valuation Continued
• 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

18
Figure 21.6 A Standard Normal Curve
19
Call Option Example
• So 100 X 95
• r .10 T .25 (quarter)
• ??? .50
• d1 ln(100/95)
• (.10(?5 2/2))(0.25)/(?5?.251/2)
• .43
• d2 .43 ((?5???.251/2)
• .18

20
Probabilities from Normal Distribution
• N (.43) .6664
• Table 21.2
• d N(d)
• .42 .6628
• .43 .6664 Interpolation
• .44 .6700

21
Probabilities from Normal Distribution Continued
• N (.18) .5714
• Table 21.2
• d N(d)
• .16 .5636
• .18 .5714
• .20 .5793

22
Table 21.2 Cumulative Normal Distribution
23
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?

24
Black-Scholes Option Values
25
Figure 21.7 Using Goal Seek to Find Implied
Volatility
26
Figure 21.8 Implied Volatility of the SP 500
(VIX Index)
27
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
• Replace S0 with S0 - PV (Dividends)

28
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

29
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

30
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 options value given a
1 change in the value of the underlying stock

31
Figure 21.9 Call Option Value and Hedge Ratio
32
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

33
Figure 21.10 Profit on a Protective Put Strategy
34
Figure 21.11 Hedge Ratios Change as the Stock
Price Fluctuates
35
Figure 21.12 SP 500 Cash-to-Futures Spread in
Points at 15 Minute Intervals
36
Hedging On Mispriced Options
• Option value is positively related to volatility
• If an investor believes that the volatility that
is implied in an options price is too low, a
• Profit must be hedged against a decline in the
value of the stock
• Performance depends on option price relative to
the implied volatility

37
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

Change in the value of the option Change of the
value of the stock
Delta
38
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
39
Table 21.3 Profit on a Hedged Put Portfolio
40
Table 21.4 Profits on Delta-Neutral Options
Portfolio
41
Figure 21.13 Implied Volatility of the SP 500
Index as a Function of Exercise Price