Derivation of the Black-Scholes-Merton PDE › Summary

This chapter derived the Black-Scholes-Merton partial differential equation, the fundamental equation governing derivative prices under certain idealized conditions. The key insights are:

• Stock price dynamics: The underlying follows geometric Brownian motion with constant volatility, $dS = \mu S \, dt + \sigma S \, dW$.

• Delta hedging eliminates risk: By holding $\Delta = \frac{\partial V}{\partial S}$ shares of stock against one derivative, we create a portfolio with no exposure to the random $dW$ term.

• No-arbitrage implies the PDE: A riskless portfolio must earn the risk-free rate, leading directly to the Black-Scholes PDE.

• The drift disappears: The expected return $\mu$ does not appear in the final equation; only the volatility $\sigma$ and risk-free rate $r$ matter for pricing.

• Boundary conditions specify the derivative: The same PDE applies to all derivatives; what distinguishes calls from puts is the terminal payoff condition.

• The PDE is linear and backward parabolic: This mathematical structure enables both analytical solutions and efficient numerical methods.

In the next chapter, we'll solve this PDE subject to call and put option boundary conditions, deriving the famous Black-Scholes formula that gives explicit closed-form prices for European options.