Investments, tenth edition

        Excel Questions 

    1.  Find the value of the call and put options using the param-

eters given in this box but changing the standard deviation 

to .25. What happens to the value of each option?  

   2.  What is implied volatility if the call option is selling for $9?    

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Chapter 21- Black-Scholes Option Pricing

Call Valuation & Call Time Premiums

Standard deviation (

σ)

Variance (annual, 

σ

2

)

Time to expiration (years, T)

Risk-free rate (annual, r)

Current stock price (S

0

)

Exercise price (X)

Dividend yield (annual, 

δ)

d

1

d

2

N(d

1

)

N(d

2

)

Black-Scholes call value

Black-Scholes put value

Intrinsic value of call

Time value of call

Intrinsic value of put

Time value of put

0.27830

0.07745

0.50

6.00%

$100.00

$105.00

0.00%

0.0029095

−0.193878

0.50116

0.42314

$6.99992

$8.89670

$0.00000

6.99992

$5.00000

3.89670

A

B

C

D

Standard

Deviation

Call

Option

Value

0.15

0.18

0.20

0.23

0.25

0.28

0.30

0.33

0.35

0.38

0.40

0.43

0.45

0.48

0.50

E

7.000

3.388

4.089

4.792

5.497

6.202

6.907

7.612

8.317

9.022

9.726

10.429

11.132

11.834

12.536

13.236

F

G

Standard

Deviation

Call

Time

Value

H

I

J

Stock

Price

Call

Option

Value

$60

$65

$70

$75

$80

$85

$90

$95

$100

$105

$110

$115

$120

$125

$130

$135.00

K

7.000

0.017

0.061

0.179

0.440

0.935

1.763

3.014

4.750

7.000

9.754

12.974

16.602

20.572

24.817

29.275

33.893

L

M

Stock

Price

Call

Time

Value

$60

$65

$70

$75

$80

$85

$90

$95

$100

$105

$110

$115

$120

$125

$130

$135

N

7.000

0.017

0.061

0.179

0.440

0.935

1.763

3.014

4.750

7.000

9.754

7.974

6.602

5.572

4.817

4.275

3.893

0.150

0.175

0.200

0.225

0.250

0.275

0.300

0.325

0.350

0.375

0.400

0.425

0.450

0.475

0.500

7.000

3.388

4.089

4.792

5.497

6.202

6.907

7.612

8.317

9.022

9.726

10.429

11.132

11.834

12.536

13.236

LEGEND:

Enter data

Value calculated

See comment

bod61671_ch21_722-769.indd   748

bod61671_ch21_722-769.indd   748

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

  C H A P T E R  

2 1

 Option 

Valuation

749

While the protective put is 

a simple and convenient way 

to achieve    portfolio  insurance    ,  

that is, to limit the worst-case 

portfolio rate of return, there are 

practical difficulties in trying 

to insure a portfolio of stocks. 

First, unless the investor’s port-

folio corresponds to a standard 

market index for which puts are 

traded, a put option on the port-

folio will not be available for 

purchase. And if index puts are 

used to protect a non-indexed 

portfolio, tracking error can 

result. For example, if the port-

folio falls in value while the 

market index rises, the put will 

fail to provide the intended pro-

tection. Moreover, the maturi-

ties of traded options may not match the investor’s horizon. Therefore, rather than using 

option strategies, investors may use trading strategies that mimic the payoff to a protective 

put option. 

 Here is the general idea. Even if a put option on the desired portfolio does not exist, a 

theoretical option-pricing model (such as the Black-Scholes model) can be used to deter-

mine how that option’s price would respond to the portfolio’s value if it did trade. For 

example, if stock prices were to fall, the put option would increase in value. The option 

model could quantify this relationship. The net exposure of the (hypothetical) protective 

put portfolio to swings in stock prices is the sum of the exposures of the two components 

of the portfolio, the stock and the put. The net exposure of the portfolio equals the equity 

exposure less the (offsetting) put option exposure. 

 We can create “synthetic” protective put positions by holding a quantity of stocks with 

the same net exposure to market swings as the hypothetical protective put position. The 

key to this strategy is the option delta, or hedge ratio, that is, the change in the price of the 

protective put option per change in the value of the underlying stock portfolio.  

Change in Value

of Protected Position

Change in Value

of Underlying Asset

2P

0

0

Cost of Put

 Figure 21.10 

Profit on a protective put strategy  

 Suppose a portfolio is currently valued at $100 million. An at-the-money put option 

on the portfolio might have a hedge ratio or delta of  2 .6, meaning the option’s value 

swings $.60 for every dollar change in portfolio value, but in an opposite direction. Sup-

pose the stock portfolio falls in value by 2%. The profit on a hypothetical protective put 

position (if the put existed) would be as follows (in millions of dollars): 

Loss on stocks:

2% of $100 5 $2.00

Gain on put:

.6 3 $2.00 5   1.20

 Net 

loss

     5 $  .80

 Example  21.7 

Synthetic Protective Put Options 

bod61671_ch21_722-769.indd   749

bod61671_ch21_722-769.indd   749

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

750

P A R T   V I

  Options, Futures, and Other Derivatives

 

The challenge with this procedure is that deltas constantly change.  

Figure  21.11 

 

shows that as the stock price falls, the magnitude of the appropriate hedge ratio increases. 

Therefore, market declines require extra hedging, that is, additional conversion of equity 

into cash. This constant updating of the hedge ratio is called    dynamic  hedging     (alterna-

tively, delta hedging).   

Dynamic hedging is one reason portfolio insurance has been said to contribute to mar-

ket volatility. Market declines trigger additional sales of stock as portfolio insurers strive to 

increase their hedging. These additional sales 

are seen as reinforcing or exaggerating mar-

ket downturns. 

 

In practice, portfolio insurers often do 

not actually buy or sell stocks directly when 

they update their hedge positions. Instead, 

they minimize trading costs by buying or 

selling stock index futures as a substitute 

for sale of the stocks themselves. As you 

will see in the next chapter, stock prices and 

index futures prices usually are very tightly 

linked by cross-market arbitrageurs so that 

futures transactions can be used as reliable 

proxies for stock transactions. Instead of 

selling equities based on the put option’s 

delta, insurers will sell an equivalent number 

of futures contracts.  

17

     

 Several portfolio insurers suffered great set-

backs during the market crash of October 19, 

1987, when the market suffered an unprece-

dented 1-day loss of about 20%. A description 

 We create the synthetic option position by selling a proportion of shares equal to the 

put option’s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free 

T-bills. The rationale is that the hypothetical put option would have offset 60% of any 

change in the stock portfolio’s value, so one must reduce portfolio risk directly by sell-

ing 60% of the equity and putting the proceeds into a risk-free asset. Total return on a 

synthetic protective put position with $60 million in risk-free investments such as T-bills 

and $40 million in equity is 

   Loss on stocks: 

 2% of $40  5  $.80 

1

  Loss on bills: 

  5       0

   Net 

loss 

  5  $.80 

 The synthetic and actual protective put positions have equal returns. We conclude 

that if you sell a proportion of shares equal to the put option’s delta and place the 

proceeds in cash equivalents, your exposure to the stock market will equal that of the 

desired protective put position. 

0

Value of a Put (P)

S

0

 

Low Slope 

5

Low Hedge Ratio

Higher Slope 

5

High Hedge Ratio

 Figure 21.11 

Hedge ratios change as the stock price 

fluctuates  

17

 Notice, however, that the use of index futures reintroduces the problem of tracking error between the portfolio 

and the market index. 

bod61671_ch21_722-769.indd   750

bod61671_ch21_722-769.indd   750

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

  C H A P T E R  

2 1

 Option 

Valuation

751

of what happened then should let you appreciate the complexities of applying a seemingly 

straightforward hedging concept. 

    1.  Market volatility at the crash was much greater than ever encountered before. Put 

option deltas based on historical experience were too low; insurers underhedged, 

held too much equity, and suffered excessive losses.  

   2.  Prices moved so fast that insurers could not keep up with the necessary rebalancing. 

They were “chasing deltas” that kept getting away from them. The futures market 

also saw a “gap” opening, where the opening price was nearly 10% below the previ-

ous day’s close. The price dropped before insurers could update their hedge ratios.  

   3.  Execution problems were severe. First, current market prices were unavailable, with 

trade execution and the price quotation system hours behind, which made computa-

tion of correct hedge ratios impossible. Moreover, trading in stocks and stock futures 

ceased during some periods. The continuous rebalancing capability that is essential 

for a viable insurance program vanished during the precipitous market collapse.  

   4.  Futures prices traded at steep discounts to their proper levels compared to reported 

stock prices, thereby making the sale of futures (as a proxy for equity sales) seem 

expensive. Although you will see in the next chapter that stock index futures prices 

normally exceed the value of the stock index,  Figure 21.12  shows that on October 19, 

futures sold far below the stock index level. When some insurers gambled that the 

futures price would recover to its usual premium over the stock index, and chose to 

defer sales, they remained underhedged. As the market fell farther, their portfolios 

experienced substantial losses.    

 Although most observers at the time believed that the portfolio insurance industry 

would never recover from the market crash, delta hedging is still alive and well on Wall 

Street. Dynamic hedges are widely used by large firms to hedge potential losses from 

options positions. For example, the nearby box notes that when Microsoft ended its 

employee stock option program and J. P. Morgan purchased many already-issued options 

0

10

210

220

230

240

10

11

12

1

2

3

4 10

11

12

1

2

3

4

October 19

October 20

 Figure 21.12 

S&P 500 cash-to-futures spread in points at 15-minute intervals    

Note: Trading in futures contracts halted between 12:15 and 1:05.  

 Source:  The Wall Street Journal.  Reprinted by permission of  The Wall Street Journal,  © 1987 Dow Jones & 

Company, Inc. All rights reserved worldwide. 

bod61671_ch21_722-769.indd   751

bod61671_ch21_722-769.indd   751

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

752

of Microsoft employees, it was widely expected that Morgan would protect its options 

position by selling shares in Microsoft in accord with a delta hedging strategy.