# Expected profit from straddle and its standard deviation

I was reading "Paul Wilmott introduces quantitative finance". In chapter 10 page 227 he states that:

If you buy an at-the-money straddle close to expiry the profit you expect to make from this strategy is approx. $$\sqrt{2(T-t)/\pi}(\sigma - \sigma_{\text{implied}})S$$ and its standard deviation of the profit (the risk) is approx. $$\sqrt{1-(2/\pi)}\sigma S\sqrt{T-t},$$ where $\sigma$ is the actual volatility, $\sigma_{\text{implied}}$ is the BSM implied volatility, $t$ is current time and $T$ is the maturity time.

I can't figure out how to derive these results. Any help would be greatly appreciated.

• That does not look right. Isn't it $(\sigma^2-\sigma_{implied}^2)$ ? – noob2 Apr 18 '17 at 15:59
• Take a look at this answer. – msitt Apr 18 '17 at 17:29
• @msitt thank you so much for your answer! That sure helped a lot. – JesperHansen Apr 18 '17 at 21:41