With spot at 100 the 90-strike put trades at 2.50 on 22% implied and the 80-strike put at 0.90 on 26% implied. What is the payoff ratio on the 90/80 put spread, and how does the skew change it against a flat surface where the 80 strike would be 0.55?

With spot at 100 the 90-strike put trades at 2.50 on 22% implied and the 80-strike put at 0.90 on 26% implied. What is the payoff ratio on the 90/80 put spread, and how does the skew change it against a flat surface where the 80 strike would be 0.55?

Approach: Compute the maximum payoff over the net premium in both cases, then attribute the difference to what the skew does to the leg you are selling.

6.25 to 1. The spread costs 2.50 - 0.90 = 1.60 and pays at most 90 - 80 = 10, giving 10/1.60 = 6.25 to 1. On a flat implied volatility surface the lower strike would cost 0.55, the spread would cost 1.95 and the ratio would fall to 5.13 to 1, so the skew improves the payoff ratio by 22%. The reason is that skew makes the option you sell relatively more expensive than the one you buy, so any structure that is short a lower strike against a higher one is helped by it. That is why a hedger who wants protection against a 10% fall but not against a crash buys the put spread rather than the outright put, and why the outright put is the poor value leg in an index with a steep skew. What the ratio hides is that the spread caps out at 80, so it gives no protection at all in the scenario the outright put exists for.

Follow-up: You need protection for a $50m portfolio against a fall of more than 10%. How would you compare a put spread against a collar on the same budget?

Key concepts: put spread, skew, payoff ratio, implied volatility.