AdvancedMultiple choiceFree solution

Why will no desk hold a naked short digital into expiry with the spot sitting on the strike?

  1. AThe pricing model stops working that close to expiry
  2. BThe delta is unbounded and reverses on a one-tick move
  3. CThe vega becomes very large close to expiry
  4. DExchanges prohibit holding digitals through expiry
Show the answer and worked solution

Answer: B – The delta is unbounded and reverses on a one-tick move

The payoff is discontinuous at the strike, so just before expiry the delta is enormous, of order the payoff divided by the width of the region where the price is still uncertain, and it collapses to zero on either side. Sitting on the strike, a one-tick move flips the required hedge from nearly everything to nearly nothing, so any static hedge is wrong the instant the price ticks back and trading it dynamically simply pays the spread again and again. Vega is not the problem, since it goes to zero at expiry; it is delta and gamma that blow up. The practical response is to hedge with a finite call spread and accept the residual.

Worked solution

  1. Formula
    Δ≈payoffwidth of the uncertain region→∞\Delta \approx \frac{\text{payoff}}{\text{width of the uncertain region}} \to \infty
  2. Substitute
    S→K, T→0S \to K, \ T \to 0
  3. Solve
    hedge is near-total just below K\text{hedge is near-total just below } K
  4. and near-zero just above it\text{and near-zero just above it}
  5. Answer
    Unbounded delta, flipping on a tick\text{Unbounded delta, flipping on a tick}

Sanity check. Vega goes to zero at expiry, so it is delta and gamma that blow up, and the response is to hedge with a finite call spread.

  • A. The price is fine – it is the hedge that is impossible.
  • B. Correct. Any hedge is wrong the moment the price ticks back, and trading it dynamically just pays the spread repeatedly.
  • C. Vega goes to zero at expiry; it is delta and gamma that blow up.
  • D. No such rule; the constraint is risk, not regulation.

Takeaway: A discontinuous payoff has unbounded delta at the strike near expiry.

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

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