Worked Examples — Itô Calculus in Finance

The preceding lessons in this topic built the stochastic integral machinery: construction, isometry, martingale property, local martingales, Lévy's characterisation, Itô's formula and its extensions. This capstone assembles those tools on the four worked examples most central to derivatives pricing and fixed-income modelling:

1. GBM explicit solution — solving $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ in closed form via Itô's formula on $\ln S_t$.
2. Log-price dynamics and the $-\sigma^2/2$ correction — why realized log-returns underestimate $\mu$.
3. Vasicek interest rate model — solving $dr_t = \kappa(\theta - r_t) \, dt + \sigma \, dW_t$ via the integrating-factor method.
4. Black-Scholes PDE from Itô's formula — deriving $\partial_t V + rS\partial_S V + \frac{1}{2}\sigma^2 S^2 \partial_{SS}V - rV = 0$ from the self-financing condition.

Each example ends with a sanity check linking the stochastic calculus back to observable financial quantities.

• Solve the GBM SDE explicitly and compute the distribution of $S_T$.
• Compute $\mathbb{E}[S_T]$ and $\mathbb{E}[\ln(S_T/S_0)]$ and explain the role of the Itô correction.
• Derive the Vasicek bond price formula and identify the mean-reversion dynamics.
• Derive the Black-Scholes PDE from first principles using Itô's formula and the self-financing condition.
• Use Itô's formula to compute $dV$ for any smooth option price function $V(t, S_t)$.

1. Example 1: Geometric Brownian Motion — Explicit Solution
2. Example 2: The $-\sigma^2/2$ Correction — Why Log-Returns Underestimate $\mu$
3. Example 3: The Vasicek Interest Rate Model
4. Example 4: Deriving the Black-Scholes PDE