Deriving the Black–Scholes Equation with Delta Hedging

Summary

This document outlines the risk-neutral argument behind the Black–Scholes partial differential equation for a European option. It starts with a stochastic model of a stock price and applies Itô’s lemma to describe how an option’s value changes with the underlying price and time. A portfolio combining the option and the underlying asset is then adjusted using delta hedging to remove its instantaneous exposure to stock-price movements.

Under the model’s assumptions, the hedged portfolio must earn the risk-free rate; otherwise, an arbitrage opportunity would exist. This condition yields the Black–Scholes equation. The text then notes that the equation alone does not specify a unique option value: it also needs a payoff or other boundary condition, illustrated by a European call at expiration. The derivation is only sketched, with key equations omitted from the supplied text, and it does not discuss practical complications such as transaction costs, discrete rebalancing, or model misspecification.

Key ideas

• Itô’s lemma translates the stock-price model into a dynamic equation for an option’s value.
• Delta hedging combines an option with the underlying asset to eliminate instantaneous price risk.
• No-arbitrage reasoning requires the riskless hedged portfolio to grow at the risk-free rate.
• The resulting pricing equation needs an expiration payoff or another boundary condition to determine a solution.
• The text sketches the derivation but omits equations and practical hedging frictions.

Tags

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