Binary Call Pricing via Differentiation (Part 2)

How binary option pricing relates to vanilla call derivatives

This is a hard option-theory problem that tests whether you can connect the discrete payoff structure of a digital option to the continuous framework of vanilla European calls. It requires fluency in risk-neutral valuation, replication arguments, and calculus-based reasoning about limits.

The question probes your ability to set up a replicating portfolio in continuous time and then take a mathematical limit to uncover a deep relationship between option prices. Specifically, you'll need to recognize how the payoff of a binary call can be expressed in terms of the derivative of vanilla call prices with respect to the strike, and justify this connection via arbitrage-free pricing principles. Traders and quant researchers rely on this kind of reasoning to hedge exotic payoffs using liquid vanilla instruments.

• Risk-neutral valuation and Girsanov's theorem
• Replication and self-financing portfolios
• Sensitivity of option price to strike (vega-like properties)
• Limits of spreads and their relationship to digital payoffs