How does a digital (binary) option pay a fixed amount, and why does the discontinuous payoff create hedging challenges?

A digital (binary) option pays a fixed predetermined amount if the underlying finishes beyond the strike, and zero otherwise. This discontinuous payoff creates extreme delta and gamma near the strike at expiration, making hedging a significant challenge.\n\nTypes of Binary Options:\n\n| Type | Payoff (Call) | Payoff (Put) |\n|---|---|---|\n| Cash-or-Nothing | Q if S_T > K, else 0 | Q if S_T < K, else 0 |\n| Asset-or-Nothing | S_T if S_T > K, else 0 | S_T if S_T < K, else 0 |\n\nwhere Q is the fixed cash amount.\n\nPricing (Cash-or-Nothing Call):\n\nV = Q x e^{-rT} x N(d2)\n\nwhere d2 is the same d2 from Black-Scholes-Merton:\n\nd2 = [ln(S/K) + (r - q - sigma^2/2) x T] / (sigma x sqrt(T))\n\nThe price equals the present value of Q times the risk-neutral probability of finishing in the money.\n\nWorked Example:\nCedarpoint Trading sells a 30-day cash-or-nothing call on Avalon Semiconductors. Parameters:\n- Spot: $148, Strike: $150, Payout: $10,000, r = 5%, q = 0%, sigma = 32%\n\nd2 = [ln(148/150) + (0.05 - 0 - 0.0512) x 30/365] / (0.32 x sqrt(30/365))\nd2 = [-0.01342 + (-0.0001) x 0.0822] / (0.32 x 0.2867)\nd2 = -0.01343 / 0.09174 = -0.1464\n\nN(-0.1464) = 0.4418\n\nPrice = $10,000 x e^{-0.05 x 0.0822} x 0.4418 = $10,000 x 0.9959 x 0.4418 = $4,400\n\nThe Hedging Nightmare Near Expiration:\n\nWith 1 day remaining and the stock at $149.90 (just below the $150 strike):\n- A $0.20 move up triggers a $10,000 payout\n- Delta approaches infinity at the strike as expiration nears\n- The dealer must buy/sell enormous quantities of stock to delta-hedge tiny price changes\n- Gamma is extremely negative (for the seller) near the strike\n\nPractical dealers address this by:\n1. Spread replication: Approximating the binary with a tight bull call spread (buy K-epsilon call, sell K+epsilon call, scale the notional)\n2. Vega hedging: The skew around the strike significantly affects the binary's value\n3. Pin risk management: Accepting that perfect hedging is impossible near the strike and managing position limits\n4. Wider bid-ask spreads near expiration to compensate for hedging difficulty\n\nBinary-Vanilla Relationship:\nA cash-or-nothing call is the derivative of a vanilla call with respect to the strike:\n\nBinary Call = -dC/dK (scaled by Q)\n\nThis mathematical relationship is useful for constructing replicating portfolios.\n\nPractice exotic option pricing in our FRM question bank.