The one-month option implies 60% and the three-month implies 40% on the same name. What volatility is the market pricing for the two months between the two expiries, and what does that tell you about the front expiry?

The one-month option implies 60% and the three-month implies 40% on the same name. What volatility is the market pricing for the two months between the two expiries, and what does that tell you about the front expiry?

Approach: Convert each implied volatility to total variance, take the difference across the two maturities, and divide by the length of the gap before taking a square root.

24.5%. Total variance to one month is 0.60^2*(1/12) = 0.03 and to three months is 0.40^2*0.25 = 0.04, so the forward variance across the gap is (0.04 - 0.03)/(0.25 - 1/12) = 0.01/0.16667 = 0.06 and the forward volatility is sqrt(0.06) = 24.5%. An inverted term structure like this almost always means a dated event sits inside the front expiry, and variance additivity says the market expects the name to be quiet at 24.5% once the event has passed. Two consequences follow. Buying the front expiry is buying the event at 60% rather than buying volatility, and a calendar spread that sells the front and buys the back is short the event with a residual long vega position that survives it. If the forward variance had come out negative the surface would carry a calendar arbitrage rather than an event.

Follow-up: The event is confirmed to move to the week after the front expiry. What should the two implied volatilities become?

Key concepts: forward variance, term structure, variance additivity, event volatility.