Straddle to Vol, Ten Seconds

Question

Desk drill: the 3-month at-the-money straddle on an 80 stock trades at 6.40 (rates zero). Back out the implied volatility in your head, and quote the implied daily move in dollars.

Accepted Answer

Invert the rule of thumb. ATM straddle approx 0.8,S,sigmasqrtT, so sigma approx fractextstraddle0.8, S sqrtT = frac6.400.8 times 80 times 0.5 = frac6.4032 = mathbf20%. (Exact Black–Scholes implied for a 6.40 straddle here: 20.06%, the 0.8 rule is good to a few basis points of vol at these maturities. The 0.8 is sqrt2/pi approx 0.7979.) Implied daily move: per-day vol approx fracsigma_textannualsqrt252 approx fracsigma16 = frac20%16 = 1.25%, i.e. 80 times 1.25% = mathbf1.00 text per day , the size of daily change at which long-gamma-at-this-vol breaks even against theta. The cross-checks a desk person runs instantly: straddle over spot is 6.40/80 = 8%, the implied expected absolute move over the quarter; and 0.8sigmasqrtT = 0.8 times 20% times 0.5 = 8% ✓ consistent. Memorize the three conversion constants and the whole loop is mental arithmetic: 0.8 (that is, sqrt2/pi) between straddle and total vol; 16 (approx sqrt252) between annual and daily vol; straddle/spot as the expected-move percentage. Why interviewers drill it: market-makers translate between premium-space and vol-space dozens of times an hour, a quoted straddle IS a vol quote, and someone who needs a pricer to see that 6

Straddle to Vol, Ten Seconds

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