Binary Call Replication via Call Spread (Part 1)

Replicating binary options using vanilla call spreads

This medium-difficulty question tests your ability to bridge exotic option theory and static replication—a core skill in derivatives pricing and risk management. It asks you to construct a hedging portfolio using only standard instruments, which is essential for practitioners who need to price or risk-manage payoffs that are not directly traded.

The problem probes whether you can think about an option's payoff as a limit of simpler payoff structures, and how to express that relationship mathematically. You'll need to reason about the geometry of call payoffs at different strikes, understand how a spread converges to a binary payoff as the strikes narrow, and write down the explicit hedge ratio as a function of the spread width.

• Call spread payoff diagrams and their limiting behavior
• Static replication and completion of markets
• Asymptotic scaling of hedge ratios
• Link between vanilla and exotic derivatives