Black-Scholes Option Pricing from No-Arbitrage Principles

Summary

The article introduces European call and put options and defines their payoffs at maturity. It models a market containing a risk-free asset, an underlying stock following geometric Brownian motion, and an option whose value is assumed to be a smooth function of time and the stock price. Fair pricing is framed through the absence of arbitrage: there should be no self-financed portfolio that starts with zero wealth, cannot lose, and has a chance of gaining.

Under those assumptions, the option value must satisfy the Black-Scholes partial differential equation with the relevant call or put payoff as its terminal condition. The discussion establishes a mathematical pricing framework rather than providing a numerical valuation example or empirical test. It notes that the model has significant limitations and should be treated carefully, leaving their detailed treatment to later articles. The result depends on the specified market assumptions and is not presented as a universal description of observed option prices.

Key ideas

• European calls and puts grant rights to trade the underlying at a fixed strike on a specified maturity date.
• The model represents the stock with geometric Brownian motion and includes a risk-free asset.
• No-arbitrage assumptions lead to a partial differential equation for a smooth option price function.
• The terminal condition for the pricing equation is the option payoff at maturity.
• The framework relies on simplifying assumptions and has limitations in describing real markets.

Tags

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