Problem 9.17 (from Textbook) Consider an exchange-traded call option contract to

Question

Problem 9.17 (from Textbook)

Consider an exchange-traded call option contract to buy 500 shares with a

strike price of $40 and maturity in four months. Explain how the terms of

the option contract change when there is

(a)

A 10% stock dividend

(b)

A 10% cash dividend

(c)

A 4-for-1 stock split

Problem 9.25 (from Textbook)

A trader writes five naked put option contracts, with each contract being on

100 shares.

The option price is $10, the time to maturity is six months, and the strike

price is $64.

(a) What is the margin requirement if the stock price is $58?

(c) How would the answer to (a) change if the stock price were $70?

(d) How would the answer to (a) change if the trader is buying instead of

selling the

options?

Explanation / Answer

Assumptions and Information: Let the price of the underlying stock be $100 at present. Each contract size is for 500 shares. Strike Price is $40 per share. Option Maturity =4 months

Therefore, market value of shares in each option= 500 x 100 = $50000

and Strike Price of shares in each option = 500 x 40 =$20000

It is assumed to be an European styled call option with no provision of early exit.

In every call option transaction there are two parties involved, the pruchaser of the call option and the seller of the call option. Let them be denoted by H (for holder of the call option) and S (seller of the call option respectively)

(a) Now if the stock issues a 10% stock dividend, it essentially means that the number of shares with a particular investor increases by 10%. In this case, the investor is S as he/she being the call option seller implies that he/she holds the underlying asset ( the stock) for the call option sold.

If S had only those 500 shares (which constitute an option) then his/her share holding increases by 50 (10% of 500). So total share holding by S =(500+50)=550.

Now the thing with stock dividends is that although the number of shares increase the total value of the firm remains constant, as dividends do not create any extra value for investors and is only a income returning instrument.  

Assuming the underlying stock had a value $ 10 m with 100000 shares outstanding ( with each share priced at $100 a piece), the stock dividend would increase the number of shares by 10% of 100000 i.e 10000.

Therefore, price per share of the stock = $ 10 m / 110000 = $90.91

This implies that option seller S losses in terms of value of the underlying stock. However, since he would have had to sell stocks worth $100 for the strike price of $40 upon option maturity, entailing a loss of $60, a reduction in underlying asset (the stocks) price to $90.91 from $100 actually reduces his loss to $50.91.

Therefore, the option holder S loses less money upon option exercise. However, this lesser loss is wiped off by the loss in the value of underlying asset.

For the option holder H, his earlier gains of $60 is reduced to $50.91.

ASSUMPTION: Underlying stock price stay constant at $90.91 from ex-dividend date to option maturity date.

(b) A 10% cash dividend means that S (the option seller and underlying asset holder) receives $10 (10% of market price of $100) per share. However, with cash dividends, the amount of the dividend per share is immediately reduced from the existing share price of the stock. Therefore, on the ex-dividend date the stock price becomes (100-10) =$90.

Option Seller reduces loss in option position, but increases loss in the stock possition.

Option buyer H experiences a lesser gain of $50 instead of $ 60 upon exercising the option.

(c) A 4 - 1 stock split implies that the holder of one share receives four shares in exchange for the share already being held.This essentially means that an investor would receive 3 additional shares for each share he/she originally had. Stock splits only increase the number of outstanding shares (somewhat like stock dividends) without adding any value to the firm.

Hence, firm value =$ 10 m (assumed in (b)) and no. of outstanding shares = 100000

After 4-1 stock split no. of outstanding shares = 4 x 100000 = 400000

Therefore, stock price per share, $ 10 m / 400000 = $25.

Also the option which initially constituted of 500 shares after the stock split comprises of 4 x 500 =2000 shares.

Additionally, the strike price would be adjusted accordingly to one-fourth of the orginial value of $40 (because the no. of shares in each contract increases by a factor of 4, the strike price for each share has to be reduced by a factor of 4 to keep the overall contract value unchanged).

Therefore, new strike price =40 /4 =$10.

Therefore, the option holder H gains = (25-10) =$15 per share instead of the original $60 per share on maturity of the option.

The option seller S however, still suffers the same loss on an overall basis but lesser loss on a per share basis.

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