Implied vol from a straddle quote

A stock trades at $100. Its 3-month at-the-money straddle (one call plus one put, same strike, T=0.25T=0.25 years) is quoted at $8.00. Using the standard small-vol approximation that an at-the-money call is worth about 0.4SσT0.4\,S\sigma\sqrt{T}, back out the implied annualized volatility σ\sigma that the market is pricing in. Give it as a decimal (e.g. 0.200.20 for 20%20\%), to three places.

Show hints (2)+
  1. A straddle is a call plus a put — at the money that's ~twice the call, so straddle 0.8SσT\approx 0.8\,S\sigma\sqrt{T}.
  2. Solve σ=straddle/(0.8ST)\sigma = \text{straddle}/(0.8\,S\sqrt{T}) with T=0.5\sqrt{T}=0.5.

Answer

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0.2 (± 0.005)

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