AppliedNumeric answerFree solution

An at-the-money one-year call on a $100\$100 stock trades at $8\$8, with rates at zero. Estimate the implied volatility, as a percentage.

Answer with a number in %. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

Show the answer and worked solution

Answer

20%20\%

Invert the at-the-money approximation C≈0.4 SσTC \approx 0.4\,S\sigma\sqrt{T}, which gives σ≈C/(0.4ST)=8/40=20%\sigma \approx C/(0.4 S\sqrt{T}) = 8/40 = 20\%. Near the money the price is very nearly linear in volatility, so this is accurate to a fraction of a point and is exactly the first guess a Newton solver wants. Away from the money the linearity fails and the approximation degrades, which is the same conditioning problem that makes a pure Newton solver unsafe in the wings.

Worked solution

  1. Formula
    σ≈C0.4 ST\sigma \approx \frac{C}{0.4\,S\sqrt{T}}
  2. Substitute
    =80.4×100×1= \frac{8}{0.4 \times 100 \times 1}
  3. Solve
    =840= \frac{8}{40}
  4. Answer
    σ≈20%\sigma \approx 20\%

Sanity check. The exact root is 20.09%20.09\%, so the approximation is within a tenth of a point.

Takeaway: Invert the at-the-money approximation for a first guess at implied volatility.

Answer it in practice – your answer is marked and recorded.

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An at-the-money one-year call on a $100 stock trades at $8, with rates at zero. · QuantMax