The 3-month 100-strike implies 22% and the 6-month 100-strike implies 15% on the same non-dividend stock with zero rates. Is there an arbitrage, what is the trade, and what is the constraint being broken?

The 3-month 100-strike implies 22% and the 6-month 100-strike implies 15% on the same non-dividend stock with zero rates. Is there an arbitrage, what is the trade, and what is the constraint being broken?

Approach: Compare total variance rather than volatility across the two maturities at the same strike, and remember what a calendar spread is worth at the near expiry.

Yes, buy the 6-month and sell the 3-month at the 100 strike. Total variance must be non-decreasing in maturity at a fixed strike, and here the 3-month carries 0.22^2*0.25, or 0.01210, while the 6-month carries only 0.15^2*0.50, or 0.01125, so the calendar arbitrage condition is broken. With zero rates and no dividends the longer-dated call is worth at least as much as the shorter one at the same strike, because at the near expiry the surviving option is worth at least its intrinsic value, which is exactly what the expiring short leg pays. The calendar spread therefore has a non-negative payoff for a negative premium. In practice the first thing to check is whether the two expiries face different dividends or a hard borrow, which can legitimately push the forwards apart and dissolve the apparent free money.

Follow-up: How would you restate the constraint in terms of forward variance, and what does negative forward variance imply about the implied density?

Key concepts: calendar arbitrage, total variance, calendar spread, no-arbitrage bound.