Investments, tenth edition

 Table 21.3 

 Profit on hedged 

put portfolio 

A. Cost to establish hedged position

   1,000 put options @ $4.495/option

$  4,495

   453 shares @ $90/share

  40,770

      TOTAL outlay

$45,265

B. Value of put option as a function of the stock price at implied volatility of 35%

   Stock Price:

89

90

91

   Put price

$  5.254

$  4.785

$  4.347

   Profit (loss) on each put

0.759

0.290

(0.148)

C. Value of and profit on hedged put portfolio

   Stock Price:

89

90

91

   Value of 1,000 put options

$  5,254

$  4,785

$  4,347

   Value of 453 shares

 40,317

  40,770

  41,223

      TOTAL

$45,571

$45,555

$45,570

   Profit (5 Value 2 Cost from panel A)

306

290

305

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  C H A P T E R  

2 1

 Option 

Valuation

757

 Suppose the true annual volatility of the stock is midway between the two implied vola-

tilities, so  s   5  30%. We know that the delta of a call option is  N ( d  

1

 ). Therefore, the deltas 

of the two options and the hedge ratio are computed as follows:

    Option with strike price 90: 

   

 d

1

5

ln(90/90) 1 (.04 1 .30

2

/2) 3 45/365

.30

"45/365

5 .0995

 

 N(d

1

) 5 .5396

       

Option with strike price 95:  

  

 d

1

5

ln(90/95) 1 (.04 1 .30

2

/2) 3 45/365

.30

"45/365

5 2.4138

 

 N(d

1

) 5 .3395

       

Hedge ratio:  

  

.5396

.3395

5 1.589    

Therefore, for every 1,000 call options purchased with strike price 90, we need to write 

1,589 call options with strike price 95. Following this strategy enables us to bet on the rela-

tive mispricing of the two options without taking a position on IBM. Panel A of  Table 21.4  

shows that the position will result in a cash inflow of $151.30. The premium income on the 

calls written exceeds the cost of the calls purchased. 

 When you establish a position in stocks and options that is hedged with respect to fluc-

tuations in the price of the underlying asset, your portfolio is said to be    delta  neutral    ,  

meaning that the portfolio has no tendency to either increase or decrease in value when the 

stock price fluctuates. 

 Let’s check that our options position is in fact delta neutral. Suppose that the implied 

volatilities of the two options come back into alignment just after you establish your 

 Table 21.4 

 Profits on delta-

neutral options 

portfolio