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.

Please consider the following calculations

Explanation:

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

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