Chapter 1: Discrete Models and the Fundamental Theorem of Asset Pricing¶

This chapter develops the theory of arbitrage-free pricing from first principles in discrete-time, finite-state markets. It begins with static pricing using portfolios and state prices, builds toward dynamic hedging and replication in the binomial model, and culminates in the Fundamental Theorem of Asset Pricing, which provides the general mathematical foundation for all modern pricing theory.

Conceptual Roadmap¶

The chapter follows a single guiding idea:

Prices are determined by the absence of arbitrage.

This idea unfolds in stages:

flowchart LR
A[Static Payoffs]
--> B[State Prices]
--> C[Replication]
--> D[Risk-Neutral Pricing]
--> E[Multi-Period Models]
--> F[Continuous Limit]
--> G[FTAP]

Structure of the Chapter¶

1. Discrete-Time Foundations¶

We begin with a one-period model, where all future uncertainty is captured by a finite set of states.

• Assets are described by a payoff matrix
• Portfolios generate state-dependent payoffs
• Arbitrage is defined as a free lunch across states
• Arrow–Debreu securities provide a basis for all payoffs

The key result is:

where \(\psi_s\) are state prices. They exist if and only if the market is arbitrage-free, and are unique if and only if the market is complete.

2. The Binomial Model¶

We introduce dynamics through the binomial model, where prices evolve over time.

Three seemingly different approaches are actually equivalent:

• Replication: construct a portfolio matching the payoff
• Delta hedging: eliminate risk dynamically
• Risk-neutral pricing: compute discounted expectations

All lead to:

where \(\mathbb{Q}\) is the risk-neutral measure. This is the first appearance of the central idea:

pricing = expectation under a special probability measure

3. Multi-Period Models and American Options¶

Extending to multiple periods introduces:

• Recombining trees
• Backward induction
• Dynamic hedging strategies

American options introduce optimal stopping, where pricing becomes:

The trinomial model reveals an important limitation: when markets are incomplete, prices are no longer unique.

4. Binomial to Black–Scholes Limit¶

As the time step shrinks:

• the binomial model converges to Brownian motion
• prices converge to the Black–Scholes model
• the discrete recursion becomes the Black–Scholes PDE

This shows that continuous-time finance is not new theory, but a limit of discrete models.

5. Fundamental Theorem of Asset Pricing¶

The chapter culminates in the FTAP, which generalizes everything developed so far:

The proof relies on convex geometry (separating hyperplanes), revealing that:

• arbitrage is a geometric inconsistency
• pricing measures arise as separating functionals

This unifies state prices, risk-neutral probabilities, and replication-based pricing into a single framework.

6. Numéraire and Change of Measure¶

Finally, the theory is extended by changing the unit of account (numéraire).

Prices are invariant under the choice of numéraire.

Different numéraires produce different martingale measures, but the same prices. This leads to the general pricing formula:

and provides powerful tools for simplifying complex pricing problems.

What This Chapter Achieves¶

By the end of the chapter, the reader understands that:

• arbitrage-free pricing is fundamentally a linear pricing rule
• dynamic hedging and risk-neutral expectation are equivalent
• continuous-time models arise as limits of discrete ones
• all pricing can be expressed as expectation under a martingale measure