Download presentation
Presentation is loading. Please wait.
Published byAugust Snow Modified over 7 years ago
1
CHAPTER 21 Option Valuation Investments Cover image Slides by
Richard D. Johnson McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
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
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 )
6
Figure 21.2 Range of Possible Call Option Values
7
Figure 21.3 Call Option Value as a Function of the Current Stock Price
8
Figure 21.4 Put Option Values as a Function of the Current Stock Price
9
Binomial Option Pricing: Text Example
120 10 100 C 90 Call Option Value X = 110 Stock Price
10
Binomial Option Pricing: Text Example
30 Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock Repay loan Net Payoff 18.18 Payoff Structure is exactly 3 times the Call
11
Binomial Option Pricing: Text Example
30 30 18.18 C 3C = $18.18 C = $6.06
12
Replication of Payoffs and Option Values
Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged Stock Value Call Obligation Net payoff Hence C = or C = 6.06
13
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).
14
Generalizing the Two-State Approach
121 110 104.50 100 95 90.25
15
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.
16
Expanding to Consider Three Intervals
17
Possible Outcomes with Three Intervals
Event Probability Stock Price 3 up 1/ (1.05)3 =115.76 2 up 1 down 3/ (1.05)2 (.97) =106.94 1 up 2 down 3/ (1.05) (.97)2 = 98.79 3 down 1/ (.97)3 = 91.27
18
Figure 21.5 Probability Distributions
19
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.
20
Black-Scholes Option Valuation
X = Exercise price e = , 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
Figure 21.6 A Standard Normal Curve
22
Call Option Example So = 100 X = 95 r = T = .25 (quarter) = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = ((5.251/2) = .18
23
Probabilities from Normal Dist
Table 21.2 d N(d) Interpolation
24
Probabilities from Normal Dist.
Table 21.2 d N(d)
25
Table 21.2 Cumulative Normal Distribution
26
Call Option Value Co = SoN(d1) - Xe-rTN(d2) Co = 100 X 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?
27
Spreadsheet 21.1 Spreadsheet to Calculate Black-Scholes Option Values
28
Figure 21.7 Using Goal Seek to Find Implied Volatility
29
Figure 21.8 Implied Volatility of the S&P 500
30
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)
31
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( )-100( ) = 6.35
32
Put Option Valuation: Using Put-Call Parity
P = C + PV (X) - So = C + Xe-rT - So Using the example data C = X = 95 S = 100 r = .10 T = .25 P = e -.10 X P =
33
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.
34
Figure 21.9 Call Option Value and Hedge Ratio
35
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.
36
Figure 21.10 Profit on a Protective Put Strategy
37
Figure 21.11 Hedge Ratios Change as the Stock Price Fluctuates
38
Figure 21.12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals
39
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.
40
The appropriate hedge will depend on the delta.
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 =
41
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
42
Table 21.3 Profit on a Hedged Put Portfolio
43
Table 21.4 Profits on Delta-Neutral Options Portfolio
44
Figure 21.13 Implied Volatility of the S&P 500 Index as a Function of Exercise Price
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.