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A butterfly through the spread

Market makingHard

Calls on one stock with one expiry are quoted bid and ask: strike $90$ at $13.5$ and $14$ , strike $100$ at $9$ and $9.5$ , strike $110$ at $3$ and $3.5$ . Rates are zero. You buy $1$ call at strike $90$ and $1$ at strike $110$ , and sell $2$ at strike $100$ , trading at the quoted prices you can actually hit. What is the smallest profit you can end with at expiry? Give an exact decimal.

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Hint

A long butterfly never pays less than zero at expiry, so look at the cash today.

Solution

Buying pays the ask on the wings: $14 + 3.5 = 17.5$ .
Selling receives the bid on the body: $2 \times 9 = 18$ .
You collect $18 - 17.5 = 0.5$ today, and the payoff at expiry is at least $0$ , reached when the stock ends at or below $90$ or at or above $110$ , so the smallest profit is $0.5 + 0 = 0.5$ .

Answer: $0.5$

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