Black-Scholes Formula and European Option Pricing › Black-Scholes Formula and European Option Pricing
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In the previous chapter, we derived the Black-Scholes-Merton partial differential equation (PDE), which describes how the price of a derivative must evolve to prevent arbitrage opportunities. That PDE, while elegant, doesn't immediately tell us what an option is worth. To price an option, we need to solve the PDE subject to the appropriate boundary conditions.
This chapter presents the solution: the famous Black-Scholes formula for European options. Published by Fischer Black and Myron Scholes in 1973, with an alternative derivation and extensions published by Robert Merton that same year, this formula transformed finance by providing a closed-form European option price grounded in dynamic replication and no-arbitrage reasoning. The formula gave market participants a systematic way to connect option prices with the cost of dynamically hedging their payoffs. Cboe reports growth from 5.6 million contracts in 1974 to roughly 1.3 billion across its exchanges in 2014, including futures. The 1997 Nobel committee credited the pricing methodology with advances in valuation and risk management.
The formula expresses an option's price in terms of five observable or estimable quantities: the current stock price, the strike price, time to expiration, the risk-free interest rate, and the volatility of the underlying asset. The expected return of the stock does not appear in the formula. As we learned when studying risk-neutral valuation, this non-dividend-paying stock has expected growth rate under the pricing measure. More generally, the martingale condition applies to appropriately discounted total gains, including dividends, rather than to every ex-dividend asset price.
We'll work through the solution to the Black-Scholes PDE, interpret each component of the resulting formula, implement it in Python, and verify that it satisfies fundamental relationships like put-call parity. We'll close with an introduction to the Greeks, the sensitivity measures that quantify how option prices respond to changes in the underlying parameters.