Question

A stock, priced at $47.00, has 3-month call and put options with exercise prices of $45 and $50. The current market prices of these options are given by the following:

Exercise Price	Call	Put
45	$4.50	$2.20
50	$2.15	$4.80

Now, assume that you already hold a sizable block of the stock, currently priced at $47, and want to hedge your stock to lock in a minimum value of $45 per share at a very low up-front initial cost.

a) What hedge strategy from Chapter 7 would you recommend and what would you option transactions be to set up the holding (per 100 shares of stock that you already own) And, what would be the up-front cost to set up these option position?

b) What if the stock price falls appreciably over the next 3 months and ends up at $30. Relative to your starting point at time-zero when the stock was priced at $47, what is your dollar loss for the hedged position versus if you had not hedged and held the “long stock only” (again scaling by 100 shares of stock)? What would your percentage rate of return have been for your combined holdings (stock and options) from time-0 to time-T? What would your percentage rate of return have been for a comparable “long stock only” position over time-0 to time-T in this case? (Remember time-T is at option expiration).

c) Alternatively, what if the stock price had risen appreciably over the next 3 months and ends up at $65. Relative to your starting point at time-zero when the stock was priced at $47, what is your dollar gain for the hedged position versus if you had not hedged and held the “long stock only”? What would your percentage rate of return have been for your combined holdings (stock and options) from time-0 to time-T? What would your percentage rate of return have been for a comparable “long stock only” position over time-0 to time-T in this case?

Answer 1

Answer to Question a),

If you are in a long position in spot market ( have shares in your hand), you have to make situation to sell shares at your expected future date to hedge the value of your portfolio or investment. Here the put option gives you the right to sell shares. So you should enter into put by buying a put option at strike price of $45. you have to pay $2.2 per share for 100 shares which is given in question.so $2.20 X 100 = $220 is the upfront cost.

Answer to Question b),

if the stock price falls appreciably over the next 3 months and ends up at $30, the Dollar gain under the hedged position = value at time-T- Value at time-0 - Cost ) X number of shares, ($45-$47-$2.20) X 100,

Dollar gain = -$420.

Dollar gain under not hedged position (Long stock only position), Here value at time-0 is given in question $47 per share and Vlue at time-T = $30,

=($30 - $47) X 100,

Dollar gain = $1,700

Percentage return under different strategies are,

return = (((Value at time-T - Value at time-0) - Cast ) / Value at time-0) X 100.

Annualized return = (((Value at time-T - Value at time-0) - Cast ) / Value at time-0) X 100 X Annual period / holding period

Stock and option strategy:

here value at tme-T is $45, because you can exercise put option at $ 45, then you will get $45,

=(($45-$47)-$2.2) X 100 / ($47 X 100)) X 100,

= (-420/4700) X 100,

= -8.93617021276596 for 3 months period, Annualized return = (-$420 / ($47 X 100)) X 100 X 12 / 3,

=(-420/4700) X 100 X 12/3,

=-8.93617021276596 X 12/3,

= -35.7446808510638%.

Return Under Stock only Position

Here value at time-T = $30, here no cost

Return = ($30-$47) X 100 / ($47 X 100)) X 100 ,

= -1700/4700 X 100,

= -36.1702127659575,

Annualised Percentage = ($30-$47) X 100 / ($47 X 100)) X 100 X 12/3,

-1700/4700 X 100 X 12/3,

= -36.1702127659575 X 12/3,

-144.68085106383%. but loss willnot be more than investment. so it is better to say 100%.

Answer to c),

if the stock price falls appreciably over the next 3 months and ends up at $65, the Gain under the hedged position = value at time-T- Value at time-0 - Cost ) X number of shares,

=($65-$47-$2.20) X 100 = $1,580.

Gain under not hedged position (Long stock only position) = value at time-T- Value at time-0 - Cost ) X number of shares,

= ($65-$47) X 100 = $1,800

return = (((Value at time-T - Value at time-0) - Cast ) / Value at time-0) X 100.

Annualized return = (((Value at time-T - Value at time-0) - Cast ) / Value at time-0) X 100 X Annual period / holding period.

Stock and option strategy:

Value at timeT is $65 because you will lapse the option and sell shares in the market then you will get market price of the shares.

cost remains same as $2.2 per share for 100 shares.

Return = ((($65 - $47 - $2.2) X 100) / ($47 x 100)) X 100,

=33.6170212765957%,

Annualised Percentage = ((($65 - $47 - $2.2) X 100) / ($47 x 100)) X 100 X 12/3,

= 134.468085106383%.

Stock only Strategy

Here value at time-0 is given in question $47 per share and Vlue at time-T = $65, and no cost.

So return = (($65-$47) X 100) / ($47 X 100) X 100 X 12/3,

= 153.191489361702%,