Itô's Lemma and Stochastic Calculus › Summary

This chapter introduced the mathematical framework of stochastic calculus, essential for modeling and pricing derivatives in continuous time.

Core concepts covered:

• Stochastic differential equations (SDEs) describe how random processes evolve, with a drift component $\mu \, dt$ representing the expected trend and a diffusion component $\sigma \, dW_t$ capturing random fluctuations.

• The quadratic variation property $(dW_t)^2 = dt$ is the key insight distinguishing stochastic from ordinary calculus. This means second-order terms in Brownian motion contribute to first-order changes in functions of the process.

• Itô's Lemma extends the chain rule to stochastic processes:

$$df = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma \frac{\partial f}{\partial x} \, dW_t$$

where:

• $df$: differential of the function $f$

• $\mu, \sigma$: drift and diffusion coefficients of the underlying process

• partial derivatives: sensitivity of $f$ to time and state

Key applications demonstrated:

• The log of GBM follows simple Brownian motion: $d(\ln S_t) = (\mu - \frac{1}{2}\sigma^2) dt + \sigma \, dW_t$

• The Itô correction $-\frac{1}{2}\sigma^2$ explains why expected log-returns differ from expected simple returns

• The exponential martingale $e^{\sigma W_t - \frac{1}{2}\sigma^2 t}$ has zero drift, foreshadowing risk-neutral pricing

With Itô's Lemma in hand, we're now equipped to tackle the central problems of derivatives pricing. In the next chapter, we'll introduce the no-arbitrage principle and risk-neutral valuation, which together with Itô's Lemma form the foundation for deriving the Black-Scholes-Merton equation.