Part I: Portfolios and No-Arbitrage

These notebooks develop one-period pricing from restrictions on prices alone, before any model of investors’ preferences. They are available in HTML and PDF formats, and the sections build on each other progressively.

The first section shows how much follows from the law of one price, which only requires that a portfolio cost the sum of its parts. The course begins with the mathematics of portfolio selection: the geometry of the mean-variance frontier, beta pricing with respect to any frontier portfolio, and the simplification a risk-free asset brings. The section then restates the same law in the space of payoffs. With the geometry of that space in hand, any linear pricing rule is represented by a single random variable, the stochastic discount factor, and the SDF in the space of traded payoffs turns out to be a scaled frontier return. The mean-variance frontier and the SDF are two descriptions of the law of one price.

The second section asks what the law of one price leaves out. Its SDF can be negative in some states, which is harmless for traded assets but assigns negative prices to some options. The principle of no-arbitrage is exactly the condition for a strictly positive SDF to exist, and it is what lets beta pricing extend to options. Short-sale constraints, like positivity, go beyond the law of one price and turn pricing equalities into inequalities.

The Critical Line Algorithm in the numerical applications computes the exact mean-variance frontier when short sales are prohibited.

The story continues in Part II: Preferences and Equilibrium, which asks which portfolio prices assets and where the stochastic discount factor comes from.