Part II: Continuous Time

These notebooks reformulate asset pricing using Brownian motions and Itô calculus. They build on the stochastic discount factor, preferences, and multiperiod results of Part I: Discrete Time, and several notebooks compare their results directly with the discrete-time versions.

The first section develops the SDF, optimal portfolio choice, and consumption-based pricing in continuous time, including time-varying investment opportunities and recursive utility.

This language also makes derivatives pricing natural: risk-neutral pricing, the Black-Scholes formula, and the pricing PDE all follow from a change of measure. The second section applies this framework to interest rate models, stochastic volatility, commodity markets, and foreign exchange options.

The final section steps back from theory to implementation. One notebook solves the exact recursive-utility value function numerically in the one-factor Gaussian model, making the nonlinear HJB concrete and showing how close the affine approximation is in practice. Another shows how to calibrate the Heston stochastic volatility model to real option data, using AAPL call prices from OptionMetrics to fit the four structural parameters and recover the implied volatility surface. I will be adding more numerical applications over time.