QuantPrep | AI-Powered Learning for Quantitative Finance

This is it. This is the theoretical heart of all modern financial engineering. The "no-arbitrage" principle is the single most important assumption in asset pricing. Today, we will prove that this financial principle is, in fact, a deep statement about the geometry of vector spaces.

We will do this in three parts:

**Part 1: Setting the Stage:** We will model a financial market perfectly using the tools of linear algebra: matrices and vectors.
**Part 2: The Free Lunch:** We will define arbitrage with mathematical precision and find one with a concrete example.
**Part 3: The Law of the Universe:** We will introduce the Fundamental Theorem and show how it connects "no arbitrage" to the subspaces we have studied.

Part 1: The Financial Marketplace as a Vector Space

Building the World

First, we must build our world. Let's consider a simple, one-period market. There is "today" (time `t=0`) and "tomorrow" (time `t=1`).

1. States of the World

Tomorrow is uncertain. Let's assume there are `m` possible, distinct outcomes for the world. We'll call these **states**.

Example: For a farm, the states could be `s₁ = "Drought"`, `s₂ = "Normal Rain"`, `s₃ = "Flood"`.

2. Assets and Payoffs

In this world, there are `n` tradable assets. The value of each asset tomorrow depends on which state of the world occurs. The amount an asset pays in a given state is its **payoff**. We can represent each asset as a **payoff vector** in ℝᵐ.

Example: `p_umbrella = [15, 10, 5]` (Pays most in a flood, least in a drought).

3. The Payoff Matrix (`P`)

We combine all asset payoff vectors into a single `m x n` **Payoff Matrix, `P`**. Each column is an asset, and each row is a state.

P = \begin{bmatrix} 15 & 5 \\ 10 & 12 \\ 5 & 2 \end{bmatrix}

4. The Portfolio Vector (`h`)

An investor's portfolio is a vector `h` in ℝⁿ, where each component `hⱼ` is the number of shares held of asset `j`.

5. The Portfolio's Payoff

The total payoff of this portfolio in each state is a **linear combination** of the asset payoff vectors: `Portfolio Payoff = P * h`.

6. The Price Vector (`s`)

Each asset has a price today at `t=0`, represented in a price vector `s` in ℝⁿ. The initial cost of portfolio `h` is the dot product: `Cost = s · h = sᵀh`.

Part 2: The Free Lunch - Defining Arbitrage

The Mathematical Definition

An **arbitrage** is a "free lunch." It's a portfolio that costs nothing (or even pays you) today, but guarantees you can't lose money tomorrow, and you might even make some.

An arbitrage opportunity exists if you can find a portfolio vector `h` such that:

**Zero or Negative Cost:** `s · h ≤ 0`
**Non-Negative Payoff:** Every component of the payoff vector `Ph` is greater than or equal to zero (`Ph ≥ 0`).
**Strictly Positive Payoff:** At least one component of `Ph` is strictly greater than zero (`Ph ≠ 0`).

Part 3: The Fundamental Theorem of Asset Pricing (FTAP)

The Law of the Universe

The FTAP gives us a perfect mathematical test to see if a market, defined by `P` and `s`, has an arbitrage opportunity. It connects the financial concept to the geometry of our vector spaces.

The State Price Vector (`ψ`)

The theorem introduces a crucial vector called the **State Price Vector (`ψ`)**. Each component `ψᵢ` is the "price today" of receiving one single dollar if and only if state `i` occurs tomorrow. For this to make economic sense, all state prices must be strictly positive (`ψ &gt 0`).

The Law of One Price

In a no-arbitrage market, the price of any asset *must* equal the sum of its state-contingent payoffs, each weighted by its state price. `sⱼ = ψ · pⱼ`. For all assets, this gives the matrix equation:

s^T = \psi^T P \quad \text{or} \quad s = P^T \psi

This means the price vector `s` must lie in the **Row Space of `P`**.

The Theorem

A market (`P`, `s`) has **no arbitrage** if and only if there exists a **strictly positive state price vector `ψ`** such that `s = Pᵀψ`.

The Linear Algebra Connection

An arbitrage portfolio `h` requires `sᵀh = 0` (for zero cost) and `Ph ≥ 0` (with `Ph ≠ 0`). If the no-arbitrage condition holds (`s = Pᵀψ`), then the cost is:

\text{Cost} = s^T h = (P^T \psi)^T h = \psi^T P h = 0

This is the dot product of the strictly positive vector `ψ` and the non-negative vector `Ph`. The only way this sum can be zero is if `Ph` is the zero vector, which violates the definition of arbitrage.

The theorem ensures that any portfolio `h` that is "free" (orthogonal to the pricing vector `s` in the Row Space) cannot simultaneously generate a risk-free profit.

Arbitrage & The Fundamental Theorem of Asset Pricing