Question

the cash price of a one year treasury bill is 95 per 100 of face value. a 2 year bond with a face value of 100usd that pays annual coupons of

8. The table below gives today's prices of six-month European put and call options written on a share of ABC stock at different strike prices. The stock does not pay a dividend and the risk-free interest rate is 0% per annum. Put Price (S) Call Price (S) 13.1 9.7 7.9 8.2 Strike Price (S) 105 110 115 9.7 12.9 a) Using call options with strike prices of 105 and 110, create a bear spread and show in a table the profit of the spread at maturity as a function of the then stock price. Give the range of stock prices at maturity for which this portfolio of options is profitable. b) Using put options with strike prices of 110 and 115, create a bull spread and show in a table the profit of the spread at maturity as a function of the then stock price. Give the range of stock prices at maturity for which this portfolio of options is profitable.

Answer 1

Part a.

Using the strike prices of 105 & 110, a bear spread can be created by selling (going short) a 105 call option & buying (going long) a 110 strike call option. The profit of the spread will simply be the sum of profit/loss from these 2 individual option positions as shown in the table below. The formulas used are:

Payoff of long call option= MAX((stock price at expiry - strike price),0) - call price ($)

Payoff of short call option= call price ($) - MAX((stock price at expiry-strike price ),0)

Total P/L= sum of above two

Bear Spread (Sell (short) call at 105 strike price, buy (long) call at 110 strike price) stock price at expiry Payoff of long call (strike payoff of price 110) short call Total P/L of Bear spread -9.7 -9. 13.1 13.1 3.4 13.1 1O| 104 105 106 107 108 109 2.4 12.1 11.1 10.1| 9.7 1.4 0.4 -0.6 -1.6 -1.6 110 | 8.1 -9.7 -8.7 111 112 7.7 -6.7 113 114 -1.6 -1.6 -57 -4.7 -3.7 115 116 -1.6 -1.6

As we can see in the table above, this spread is profitable if the stock price at expiry is at 108 or below that till 0. The maximum possible profit is $3.4 which occurs if stock price goes to 105 or less.

Part b.

Using similar logic as given above, the bull spread payoff table will be as following:

Bull spread (Sell (short) put at 115 strike, Buy (long) put at 110 strike) stock payoff of Payoff of short price at long put @ put (strike expiry 110 strike price=115) Total P/L of Bull spread -1.8 100 104 105 -2.11 1.9 2.9 -1.8 -1.8 3.9 106/ 107 108 -1.8 -1.8 -1.8 -1.8 5.9 6.91 1091 110 111 112 1139 114 115 116 0.3 3.71 -4.71 -5.7 -6.71 -7.7 -8.71 -9.7 -9.71 9.7 .7 9.7 -9.7 -9.7 8.91 9.9 10.91 11.9 12.9 12.9 -0.8 0.2 1.2 2.2 3.2 3.2

The formulas used are:

Payoff of long put= -9.7(i.e. price paid for put) + MAX(110 (i.e. strike price) - stock price at expiry,0)

Payoff of short put (at 115 strike price) = 12.9 (i.e. price received for put ) - MAX(115 (i.e. strike price) - stock price at expiry,0) .

Total P/L= sum of above two

As clear from the above table, the range of stock price at expiry at which it is profitable is from 112 to infinity. The maximum possible profit is $3.2 (the difference between put prices).