Deriving the Black-Scholes PDE Through Delta Hedging

Summary

The document derives the Black-Scholes partial differential equation for a European contingent claim whose underlying asset follows geometric Brownian motion. It applies Ito’s lemma to express the option price change in terms of time, asset price, and randomness. A portfolio combining the option with a position in the underlying is then formed, and the position is chosen to cancel the stochastic term. With the resulting portfolio treated as risk-free, the no-arbitrage argument requires its growth to match the continuously compounded risk-free rate. Rearranging yields the pricing equation.

The derivation assumes constant asset drift, volatility, and risk-free rate, and it treats the option price as a sufficiently well-defined function of asset price and time. It does not by itself produce a unique option value: the PDE needs boundary conditions, such as a payoff at maturity. A European call payoff is offered as an example of such a condition, with solving the resulting equation left as the next step. The argument explains the model’s hedging logic, but its conclusions depend on the stated assumptions and idealized ability to form the hedge.

Key ideas

• Ito’s lemma expresses the change in an option price as the underlying asset moves stochastically.
• A position in the underlying can be selected to cancel the option portfolio’s instantaneous randomness.
• The no-arbitrage argument sets the return of the hedged portfolio equal to the risk-free rate.
• The resulting Black-Scholes PDE needs payoff or other boundary conditions to determine a unique solution.
• The derivation assumes constant drift, volatility, and risk-free interest rate.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.