One day of higher vol traded for extra maturity

Question

A call has $100$ days to expiry with implied volatility $20\%$ . For one day only, the volatility rises to $25\%$ , then returns to $20\%$ . By how much would you need to lengthen the maturity, keeping volatility at $20\%$ throughout, to get the same price impact?

Short answer

About $0.56$ of a day (roughly 13.5 hours). Price depends on total variance. The high-vol day contributes $0.25^2 = 0.0625$ instead of $0.2^2 = 0.04$ units of daily variance; the excess $0.0225$ equals $0.0225/0.04 = 0.5625$ extra days at $20\%$ .

Adapted from Crack, Heard on the Street, Q2.23, p.28; question and answer rewritten by SelfMadeQuant.com.

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