FoundationMultiple choiceFree solution

What is the sign of gamma on a long put?

  1. ANegative, because delta is negative
  2. BPositive
  3. CZero at the money
  4. DIt depends on whether the put is in the money
Show the answer and worked solution

Answer: B – Positive

Gamma is the second derivative of value with respect to spot, and any long option position is long convexity, so gamma is positive for a long put just as for a long call. The delta of a put is negative, but that is the first derivative and says nothing about the curvature. Moneyness changes the size of gamma, peaking at the money, but never its sign for a long position. The one thing that flips the sign is being short the option.

Worked solution

  1. Formula
    Γ=∂2V∂S2=φ(d1)SσT>0\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\varphi(d_1)}{S\sigma\sqrt{T}} > 0
  2. Substitute
    same expression for a call and a put\text{same expression for a call and a put}
  3. Solve
    Δput=Δcall−1\Delta_{\text{put}} = \Delta_{\text{call}} - 1
  4. Γput=Γcall>0\Gamma_{\text{put}} = \Gamma_{\text{call}} > 0
    Differentiating the constant gives zero, so the curvature is shared.
  5. Answer
    Γ>0\Gamma > 0

Sanity check. Delta is the first derivative and its sign says nothing about the second; only being short the option flips gamma.

  • A. Gamma is the rate of change of delta, not its sign. The two are unrelated.
  • B. Correct. Long any option is long convexity – you gain from movement in either direction.
  • C. Gamma is largest at the money, not zero.
  • D. Moneyness changes the size of gamma, never its sign for a long position.

Takeaway: Buying any option buys convexity: long calls and long puts are both long gamma.

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

Reported in interviews at

Optiver, SIG (Susquehanna), IMC Trading

More the greeks and hedging questions

What is the sign of gamma on a long put? · QuantMax