AppliedMultiple choiceFree solution

Which has more vega: a one-month at-the-money option or a one-year at-the-money option on the same underlying?

  1. AThe one-month option
  2. BThe one-year option
  3. CThey are the same at the money
  4. DIt depends on the level of implied volatility
Show the answer and worked solution

Answer: B – The one-year option

Vega for an at-the-money option is roughly ST ϕ(d1)S\sqrt{T}\,\phi(d_1), so it grows with the square root of time. A one-year option therefore has about 12≈3.5\sqrt{12} \approx 3.5 times the vega of a one-month one on the same underlying. The intuition is that volatility acts on the return over the life of the option, and there is simply more life over which it can act. Being at the money maximises vega for a given expiry but does nothing to equalise it across expiries, which is why long-dated options are how you take a view on the level of volatility and short-dated ones are how you take a view on movement.

Worked solution

  1. Formula
    VATM≈ST φ(d1)\mathcal{V}_{\text{ATM}} \approx S\sqrt{T}\,\varphi(d_1)
  2. Substitute
    T=112 against T=1T = \tfrac{1}{12} \text{ against } T = 1
  3. Solve
    V1yV1m=12\frac{\mathcal{V}_{1y}}{\mathcal{V}_{1m}} = \sqrt{12}
  4. ≈3.46\approx 3.46
  5. Answer
    The one-year option, by about 3.5×\text{The one-year option, by about } 3.5\times

Sanity check. Long-dated options express a view on the level of volatility; short-dated ones express a view on movement.

  • A. Short-dated options have little vega – there is not much time for volatility to matter.
  • B. Correct. Vega grows roughly with T\sqrt{T}, so the one-year option has about 12≈3.5\sqrt{12} \approx 3.5 times the vega.
  • C. Being at the money maximises vega for a given expiry, but does not equalise it across expiries.
  • D. The ordering by expiry holds across volatility levels.

Takeaway: Vega grows roughly with the square root of time to expiry.

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