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Expected Profit and Risk of an At-the-Money Straddle Near Expiry

Article Quant Q&A · Author: JesperHansen
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Summary

The document asks how to derive approximations for the expected profit and standard deviation of profit from buying an at-the-money straddle close to expiration. It reports formulas attributed to a quantitative finance textbook: expected profit is proportional to the difference between actual and implied volatility, the underlying price, and the square root of time remaining. The stated profit standard deviation depends on actual volatility, the underlying price, and time to expiry.

These expressions describe a near-expiry approximation, not a general valuation or risk formula for straddles. The document provides no derivation, worked example, or empirical evidence, and its author is seeking an explanation. The results should therefore be understood within the stated assumptions and timing context; the text does not spell out additional conditions such as transaction costs, volatility dynamics, or how the position is managed before expiry.

Key ideas

  • The document concerns buying an at-the-money straddle close to expiration.
  • Expected profit is described as depending on the gap between actual and implied volatility.
  • The reported profit standard deviation scales with actual volatility and time remaining.
  • The formulas are presented as approximations, and no derivation or worked evidence is supplied.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Expected profit from straddle and its standard deviation


# 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.

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