Question

Current price of Stock A is $53. One investor is making a volatility bet: profits will be highest when volatility is low, such that if the stock price ends up in the interval between $50 and $60. Devise a portfolio using only call options and shares of stock with the following payoff at the option expiration date.

Buy stock, short call at 50, short call at 60, long call at 110

Short stock, long call at 50, long call at 60, short call at 110

Buy stock, short 2 calls at 50, long call at 60, short call at 110

Short stock, long 2 calls at 50, short call at 60, long call at 110

Answer 1

Before we get into the question, let's understand the payoff of following situations: Let S0 be the current stock price, S be the stock price on date of expiration and K is the strike price of call option.

• Long stock = S - S0 = S - 53
• Short stock = S0 - S = 53 - S
• Long call option = max (S - K, 0)
• Short call option = - max (S - K, 0)

Buy stock, short call at 50, short call at 60, long call at 110

Payoff = S - 53 - max (S - K1, 0) - max (S - K2, 0) + max (S - K3, 0)

When S = 50, Payoff = 50 - 53 - max (50 - 50, 0) - max (50 - 60, 0) + max (50 - 110, 0) = -3

When S = 60, Payoff = 60 - 53 - max (60 - 50, 0) - max (60 - 60, 0) + max (60 - 110, 0) = -17

Short stock, long call at 50, long call at 60, short call at 110

Payoff = 53 - S + max (S - K1, 0) + max (S - K2, 0) - max (S - K3, 0)

When S = 50, Payoff = 53 - 50 + max (50 - 50, 0) + max (50 - 60, 0) - max (50 - 110, 0) = 3

When S = 60, Payoff = 53 - 60 + max (60 - 50, 0) + max (60 - 60, 0) - max (60 - 110, 0) = 17

Buy stock, short 2 calls at 50, long call at 60, short call at 110

Payoff = S - 53 - 2 x max (S - K1, 0) + max (S - K2, 0) - max (S - K3, 0)

When S = 50, Payoff = 50 - 53 - 2 x max (50 - 50, 0) + max (50 - 60, 0) - max (50 - 110, 0) = -3

When S = 60, Payoff = 60 - 53 - 2 x max (60 - 50, 0) + max (60 - 60, 0) - max (60 - 110, 0) = -13

Short stock, long 2 calls at 50, short call at 60, long call at 110

Payoff = 53 - S + 2 x max (S - K1, 0) - max (S - K2, 0) + max (S - K3, 0)

When S = 50, Payoff = 50 - 53 + 2 x max (50 - 50, 0) - max (50 - 60, 0) + max (50 - 110, 0) = -3

When S = 60, Payoff = 60 - 53 + 2 x max (60 - 50, 0) - max (60 - 60, 0) + max (60 - 110, 0) = 27

Thus, option 2 is the only option where there is always a profit irrespective of where stock price lands up in the range of $ 50 to $ 60. Hence, please choose the second option position:

Short stock, long call at 50, long call at 60, short call at 110