Butterfly spread

Question

Three call options are available with the following strikes and prices:

Strike 1000, price 4
Strike 1010, price 3.5
Strike 1020, price 2.75

An arbitrage exists. Using one contract at each of the outer strikes and two contracts at the middle strike, what is the guaranteed profit (in dollars)?

Accepted Answer

The three calls have equally spaced strikes ( $\Delta K = 10$ ) and prices $C_1=4$ , $C_2=3.5$ , $C_3=2.75$ . A no-arbitrage condition for call options is that the price as a function of strike must be convex: $C_1 + C_3 \ge 2C_2$ . Here $4 + 2.75 = 6.75 < 7 = 2 \times 3.5$ , so convexity is violated, creating an arbitrage.

Construct a butterfly spread: buy one call at strike 1000, buy one call at strike 1020, and sell two calls at strike 1010. The net premium is

C_1 + C_3 - 2C_2 = 4 + 2.75 - 2(3.5) = 6.75 - 7 = -0.25,

so the position generates an upfront credit of $0.25$ .

The payoff at expiration $S_T$ is

V(S_T) = \max(S_T-1000,0) + \max(S_T-1020,0) - 2\max(S_T-1010,0).

Evaluating piecewise:

$S_T \le 1000$ : all options expire worthless, $V=0$ .
$1000 < S_T \le 1010$ : $V = (S_T-1000) \ge 0$ .
$1010 < S_T \le 1020$ : $V = (S_T-1000) - 2(S_T-1010) = 1020 - S_T \ge 0$ .
$S_T > 1020$ : $V = (S_T-1000)+(S_T-1020)-2(S_T-1010)=0$ .

The payoff is non-negative for all $S_T$ and strictly positive for $1000 < S_T < 1020$ . Since the position was entered at a net credit of $0.25$ , the guaranteed profit is $0.25$ dollars (25 cents).

Butterfly spread

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