AppliedMultiple choiceFree solution

How do you replicate a cash-or-nothing digital call?

  1. AA single call struck at the digital’s strike
  2. BA tight call spread around the strike, scaled up
  3. CA straddle struck at the digital’s strike
  4. DA position in the underlying plus a cash borrowing
Show the answer and worked solution

Answer: B – A tight call spread around the strike, scaled up

A digital pays a fixed amount above the strike, which is the derivative of the vanilla payoff with respect to strike, so its price is −∂C/∂K-\partial C/\partial K. Construct it as the limit of a call spread: buy the K−εK-\varepsilon call, sell the K+εK+\varepsilon call, and scale the position by 1/2ε1/2\varepsilon so the ramp between the strikes has slope one. As ε→0\varepsilon \to 0 the ramp becomes a step. In practice desks keep ε\varepsilon finite and sell the slightly over-hedged spread, because the limit itself is unhedgeable near expiry.

Worked solution

  1. Formula
    D=−∂C∂K=lim⁡ε→0C(K−ε)−C(K+ε)2εD = -\frac{\partial C}{\partial K} = \lim_{\varepsilon\to0}\frac{C(K-\varepsilon) - C(K+\varepsilon)}{2\varepsilon}
  2. Substitute
    buy K−ε, sell K+ε, scale 1/2ε\text{buy } K-\varepsilon, \text{ sell } K+\varepsilon, \text{ scale } 1/2\varepsilon
  3. Solve
    the ramp between strikes has slope 1\text{the ramp between strikes has slope } 1
  4. ε→0⇒a step\varepsilon \to 0 \Rightarrow \text{a step}
  5. Answer
    A tight call spread, scaled up\text{A tight call spread, scaled up}

Sanity check. Desks keep ε\varepsilon finite and sell the slightly over-hedged spread, because the limit itself is unhedgeable near expiry.

  • A. A call’s payoff ramps; a digital’s jumps.
  • B. Correct: buy at K−εK-\varepsilon, sell at K+εK+\varepsilon, scale by 1/2ε1/2\varepsilon, and take the limit.
  • C. A straddle is symmetric and unbounded; a digital is one-sided and capped.
  • D. That replicates a forward, which has a linear payoff.

Takeaway: A digital is the limit of a tight call spread.

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

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