Investments, tenth edition

 Figure 21.14 

 Call option elasticity as a function of stock price  

(Parameters:  s   5  .25;  T   5  .5,  r   5  .06;  X   5  100)  

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bod61671_ch21_722-769.indd   754

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7/27/13   1:45 AM

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  C H A P T E R  

2 1

 Option 

Valuation 

755

 This ratio tells us precisely how many shares of stock we must hold to offset our 

exposure to IBM. For example, if the delta is  2 .6, then the put will fall by $.60 in value 

for every one-point increase in IBM stock, and we need to hold .6 share of stock to hedge 

each put. If we purchase 10 option contracts, each for 100 shares, we would need to buy 

600 shares of stock. If the stock price rises by $1, each put option will decrease in value by 

$.60, resulting in a loss of $600. However, the loss on the puts will be offset by a gain on 

the stock holdings of $1 per share  3  600 shares. 

 To see how the profits on this strategy might develop, let’s use the following example.   

 

 Suppose option expiration  T  is 60 days; put price  P  is $4.495; exercise price  X  is $90; 

stock price  S  is $90; and the risk-free rate  r  is 4%. We assume that the stock will not pay 

a dividend in the next 60 days. Given these data, the implied volatility on the option is 

33%, as we posited. However, you believe the true volatility is 35%, implying that the 

fair put price is $4.785. Therefore, if the market assessment of volatility is revised to the 

value you believe is correct, your profit will be $.29 per put purchased. 

 Recall that the hedge ratio, or delta, of a put option equals  N ( d  

1

 )  2  1, where  N  (•) is 

the cumulative normal distribution function and 

  

ln(S/X ) 1 (r 1 s

2

/ 2)T

s√T

d

1

 5

 Using your estimate of  s   5  .35, you find that the hedge ratio  N ( d  

1

 )  2  1  5   2 .453. 

 Suppose, therefore, that you purchase 10 option contracts (1,000 puts) and purchase 

453 shares of stock. Once the market “catches up” to your presumably better volatil-

ity estimate, the put options purchased will increase in value. If the market assessment 

of volatility changes as soon as you purchase the options, your profits should equal 

1,000  3  $.29  5  $290. The option price will be affected as well by any change in the 

stock price, but this part of your exposure will be eliminated if the hedge ratio is chosen 

properly. Your profit should be based solely on the effect of the change in the implied 

volatility of the put, with the impact of the stock price hedged away. 

  Table 21.3  illustrates your profits as a function of the stock price assuming that the 

put price changes to reflect  your  estimate of volatility. Panel B shows that the put option 

alone can provide profits or losses depending on whether the stock price falls or rises. 

We see in panel C, however, that each  hedged  put option provides profits nearly equal 

to the original mispricing, regardless of the change in the stock price.