Question 1: What, to the nearest cent, is the lower bound for the price of a two-year European call option on a stock
when the stock price is $20, the strike price is $15, and the risk-free interest rate with continuous compounding is
5% and there are no dividends?
Solution:
We have S0 = 20, X = 15, r = 0.05, t = 2
Lower bound = S0 – X e-rt
= 20 – 15 e-0.05 x 2
= 20 -15 (0.9048)
= $6.43
Question 2: A three-month call with a strike price of $25 costs $2. A three-month put with a strike price of $20 and
costs $3. A trader uses the options to create a strangle. For what two values of the stock price in three months does
the trader breakeven with a profit of zero?
Solution:
Its payoff is given by:-
Payoff strangle
i) -$3 - $2 + ($20 – ST if ST ≤ 20) or $15 – ST if ST ≤ 20
ii) -$3 - $2 + (0 if 20 < ST ≤ 25) or -$5 if 20 < ST ≤ 25
iii) -$3 - $2 + (ST - $25 if ST > $25) or ST - $30 if ST > $25
Hence, the breakeven prices are $15 and $30.
Question 3: A portfolio of derivatives on a stock has a delta of 2400 and a gamma of –100. An option on the stock
with a delta of 0.6 and a gamma of 0.04 can be traded.
1. What position in the option creates a portfolio that is gamma neutral? Give size of position and state whether it is
long or short
2. After this position has been taken what position in the stock is then necessary for delta neutrality? Give size of
position and state whether it is long or short
Solution:
1) Value = 2,400 – (-100) = $2,500
Therefore, long 2,500 stocks to obtain portfolio that is gamma-neutral.
when the stock price is $20, the strike price is $15, and the risk-free interest rate with continuous compounding is
5% and there are no dividends?
Solution:
We have S0 = 20, X = 15, r = 0.05, t = 2
Lower bound = S0 – X e-rt
= 20 – 15 e-0.05 x 2
= 20 -15 (0.9048)
= $6.43
Question 2: A three-month call with a strike price of $25 costs $2. A three-month put with a strike price of $20 and
costs $3. A trader uses the options to create a strangle. For what two values of the stock price in three months does
the trader breakeven with a profit of zero?
Solution:
Its payoff is given by:-
Payoff strangle
i) -$3 - $2 + ($20 – ST if ST ≤ 20) or $15 – ST if ST ≤ 20
ii) -$3 - $2 + (0 if 20 < ST ≤ 25) or -$5 if 20 < ST ≤ 25
iii) -$3 - $2 + (ST - $25 if ST > $25) or ST - $30 if ST > $25
Hence, the breakeven prices are $15 and $30.
Question 3: A portfolio of derivatives on a stock has a delta of 2400 and a gamma of –100. An option on the stock
with a delta of 0.6 and a gamma of 0.04 can be traded.
1. What position in the option creates a portfolio that is gamma neutral? Give size of position and state whether it is
long or short
2. After this position has been taken what position in the stock is then necessary for delta neutrality? Give size of
position and state whether it is long or short
Solution:
1) Value = 2,400 – (-100) = $2,500
Therefore, long 2,500 stocks to obtain portfolio that is gamma-neutral.
