Investments, tenth edition

Payoff to a Strip

S

T

 " X

S

T

 + X

2 Puts

2(X 2 S

T

)

0

1 Call

0

S

T

 2 X

Payoff to a Strap

S

T

 " X

S

T

 + X

1 Put

X 2 S

T

0

2 Calls

0

2(S

T

 2 X)

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720

P A R T   V I

  Options, Futures, and Other Derivatives

   5.  The payoff table on a per-share basis is as follows: 

Slope = −1

Slope = 2

Payoff and

Profit

Payoff

Profit

− P − 2C

 X − P − 2C

X

S

T

X

S

T

 " 90

90 " S

T

 " 110

S

T

 + 110

Buy put (X 5 90)

90 2 S

T

0

0

Share

S

T

S

T

S

T

Write call (X 5 110)

0

0

2

(S

T

 2 110)

    TOTAL

90

S

T

110

 The graph of the payoff is as follows. If you multiply the per-share values by 2,000, you will see that 

the collar provides a minimum payoff of $180,000 (representing a maximum loss of $20,000) and a 

maximum payoff of $220,000 (which is the cost of the house).      

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  C H A P T E R  

2 0

  Options Markets: Introduction

721

   6.  The covered call strategy would consist of a straight bond with a call written on the bond. The 

value of the strategy at option expiration as a function of the value of the straight bond is given by 

the solid colored payoff line in the following figure, which is virtually identical to  Figure 20.11 .      

$90

$110

$90

$110

Collar

Payoff

S

T

   7.  The call option is worth less as call protection is expanded. Therefore, the coupon rate need not 

be as high.  

   8.  Lower. Investors will accept a lower coupon rate in return for the conversion option.  

   9.  The depositor’s implicit cost per dollar invested is now only ($.03  2  $.005)/1.03  5  $.02427  per 

6-month period. Calls cost 50/1,000  5  $.05 per dollar invested in the index. The multiplier falls 

to .02427/.05  5  .4854.         

Value of Straight Bond

Call Written

Value of Straight Bond

Payoff of Covered Call

X

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21 

  IN THE PREVIOUS 

chapter we examined 

option markets and strategies. We noted that 

many securities contain embedded options 

that affect both their values and their risk–

return characteristics. In this chapter, we turn 

our attention to option-valuation issues. To 

understand most option-valuation models 

requires considerable mathematical and sta-

tistical background. Still, many of the ideas 

and insights of these models can be demon-

strated in simple examples, and we will con-

centrate on these. 

 We start with a discussion of the factors 

that ought to affect option prices. After 

this discussion, we present several bounds 

within which option prices must lie. Next we 

turn to quantitative models, starting with a 

simple “two-state” option-valuation model, 

and then showing how this approach can be 

generalized into a useful and accurate pric-

ing tool. We then move on to one particular 

valuation formula, the famous Black-Scholes 

model, one of the most significant break-

throughs in finance theory in several decades. 

Finally, we look at some of the more impor-

tant applications of option-pricing theory in 

portfolio management and control. 

 Option-pricing models allow us to “back 

out” market estimates of stock-price volatil-

ity, and we will examine these measures of 

implied volatility. Next we turn to some of the 

more important applications of option-pricing 

theory in risk management. Finally, we take a 

brief look at some of the empirical evidence 

on option pricing, and the implications of 

that evidence concerning the limitations of 

the Black-Scholes model.  

 CHAPTER TWENTY-ONE