Stochastic Calculus, Brownian Motion, and Black–Scholes Pricing

Summary

This introduction explains why ordinary differential calculus is inadequate for many random price processes: Brownian paths are continuous but generally not differentiable. In quantitative finance, Ito calculus provides a way to work with these processes through stochastic integral equations and stochastic differential equations.

The article connects geometric Brownian motion to modeling a nonnegative asset price and describes Ito’s Lemma as the stochastic counterpart of the chain rule. It outlines how a portfolio combining an asset and a contingent claim can be constructed to cancel random components, after which a no-arbitrage argument leads to the Black–Scholes equation and European call pricing. The discussion is conceptual and previews a derivation rather than supplying its full steps or empirical validation. Its modeling setup relies on assumptions such as nonnegative prices and geometric Brownian motion, so it is an introduction to a pricing framework rather than evidence that real asset returns follow that process.

Key ideas

• Ito calculus handles stochastic processes whose paths are not differentiable.
• Brownian motion and geometric Brownian motion provide models for random asset-price movement.
• Ito’s Lemma extends the chain rule to functions of stochastic variables.
• A hedging portfolio can cancel stochastic exposure and support no-arbitrage option pricing.
• The framework relies on model assumptions and does not establish that market prices follow geometric Brownian motion.

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