Chapter 20 · The Grand Unified Theory

The Two Fundamental Theorems of Asset Pricing

No arbitrage. Market completeness. The two questions every model must answer before it can be trusted.
Scaling up to reality

All previous chapters worked with one stock and one Brownian Motion. Real markets have hundreds of stocks, correlated to each other, driven by multiple sources of risk — interest rates, sector movements, macro shocks. The Two Fundamental Theorems of Asset Pricing are the grand unified answer to the two questions every model must answer: Is this model internally consistent? And: Can I hedge anything in it?

Two fundamental questions every financial model must answer Two boxes connected by a divider. Left shows First Theorem asking does a risk-neutral measure exist and answering the model is consistent. Right shows Second Theorem asking is that measure unique and answering the market is complete. First Fundamental Theorem Does a risk-neutral measure exist? → No Arbitrage Second Fundamental Theorem Is that measure unique? → Market Completeness &

Every model in quantitative finance must pass both tests. Failing the first means the model contains free money. Failing the second means some risks cannot be priced or hedged.

1 · The multidimensional market

In a market with m stocks and d independent Brownian Motions, every stock's price movement is a linear combination of all d random shocks. The matrix that links stocks to their Brownian drivers is the volatility matrix — and understanding its structure is the key to both fundamental theorems.

Multi-asset market structure — d Brownian drivers mapping through volatility matrix to m stock prices A diagram showing d Brownian Motion drivers on the left, connected through a volatility matrix sigma in the middle, to m stock prices on the right. Each stock has its own drift and is connected to all d drivers. d Brownian Drivers W₁ W₂ Wₐ ⋮ Volatility Matrix σ σᵢₖ = sensitivity of stock i to Brownian driver k m × d matrix Its rank determines everything m Stock Prices S₁ S₂ Sₘ ⋮

m stocks, d Brownian drivers, one volatility matrix σ (m×d). The structure of this matrix determines whether the market is arbitrage-free and complete.

The Market Price of Risk equations — a linear system
σ · θ = α − r · 1
σ (m×d)
The volatility matrix. Sensitivity of each stock to each Brownian driver.
θ (d×1)
The market price of risk vector. One entry per Brownian driver.
α − r·1 (m×1)
Excess return vector. How much each stock earns above the risk-free rate.

This linear system σθ = α − r·1 is the mathematical heart of the entire theory. If the system has a solution θ — a consistent price of risk — then we can build a risk-neutral measure and the First Fundamental Theorem holds. If it has no solution, the model contains an arbitrage. If it has a unique solution, the Second Fundamental Theorem holds and the market is complete.

2 · The first fundamental theorem — no arbitrage
FIRST FUNDAMENTAL THEOREM
No Arbitrage ↔ Risk-Neutral Measure
∃ risk-neutral measure ℙ̃  ⟺  No Arbitrage
A model is free of arbitrage if and only if there exists at least one probability measure ℙ̃ — equivalent to the real-world measure ℙ — under which all discounted stock prices are Martingales. If you cannot find any such measure, the model contains a "money machine."
No arbitrage check — solving the market price of risk system for a consistent theta A flow diagram showing the linear system sigma times theta equals alpha minus r. If a solution theta exists, the market has no arbitrage. If no solution exists, an arbitrage is found. Market model m stocks, d drivers volatility matrix σ returns α Solve for θ: σ · θ = α − r·1 linear algebra problem θ exists → ℙ̃ exists ✓ No arbitrage — model is valid No θ → No ℙ̃ ✗ Arbitrage exists — model fails

The no-arbitrage check is a linear algebra problem: does the system σθ = α − r·1 have a solution? If yes, the model is valid. If not, rebuild it.

3 · The arbitrage trap — when two stocks share one driver

The most illuminating example occurs when two stocks are driven by the same single Brownian Motion but have different reward-to-risk ratios. This is a direct violation of the First Fundamental Theorem — and the arbitrage is explicit and mechanically constructable.

⚡ The Arbitrage Trap: Example 5.4.4

Two stocks driven by the same W(t), with different Sharpe ratios:

(α₁ − r)/σ₁ ≠ (α₂ − r)/σ₂
1
Both stocks have the same noise source dW(t). Their only difference is how much they earn (α₁, α₂) per unit of risk (σ₁, σ₂).
2
Short the stock with the lower Sharpe ratio (cheap risk premium). Long the stock with the higher Sharpe ratio (expensive risk premium). Size the positions to make the dW terms cancel.
3
Result: zero net exposure to dW (zero risk) but guaranteed positive drift (free money). This is a textbook arbitrage — riskless profit from a structural inconsistency in the model.
✓
Prevention: any valid model must have one consistent market price of risk θ that applies equally to every stock exposed to the same Brownian driver. Different Sharpe ratios for the same driver is mathematically impossible in a no-arbitrage world.

In real markets, this arbitrage is exactly what statistical arbitrage strategies look for: pairs or baskets of assets that share common factors but have temporarily diverged in their risk-adjusted returns. The First Fundamental Theorem says these divergences must close — and traders who spot them first collect the convergence profit.

4 · The second fundamental theorem — completeness
SECOND FUNDAMENTAL THEOREM
Unique Risk-Neutral Measure ↔ Complete Market
ℙ̃ is unique  ⟺  Market is complete
If the risk-neutral measure is unique, then every derivative security — no matter how exotic — can be perfectly replicated by trading in the available stocks and bank account. If multiple risk-neutral measures exist, some payoffs cannot be perfectly hedged and pricing becomes ambiguous.

Uniqueness of θ (and therefore ℙ̃) depends entirely on the structure of the volatility matrix σ. This is the Goldilocks problem: the relationship between m (stocks) and d (Brownian drivers) must be just right.

📉
Too many risks
d > m
More Brownian drivers than stocks. Cannot pin down all risk prices. Multiple θ solutions → multiple ℙ̃ measures.
Incomplete ✗
✅
Just right
m = d, σ invertible
Equal stocks and drivers, full-rank matrix. Unique solution θ = σ⁻¹(α − r·1) → unique ℙ̃.
Complete ✓
📊
Too many stocks
m > d
More stocks than drivers. Redundant assets exist. Still complete — can hedge everything — but some stocks are combinations of others.
Complete but redundant
Completeness and the rank of the volatility matrix — three scenarios showing d greater m, d equals m, and d less m Three matrix diagrams. Left shows a tall matrix with more rows than columns representing too many risks. Middle shows a square invertible matrix representing the complete case. Right shows a wide matrix with more columns than rows representing redundant stocks. d > m (too many risks) d = m (just right) m > d (redundant stocks) σ₁₁ σ₁₂ σ₂₁ σ₂₂ σ₃₁ σ₃₂ σ₄₁ σ₄₂ Multiple θ → Incomplete σ₁₁ σ₁₂ σ₂₁ σ₂₂ Unique θ = σ⁻¹(α−r) → Complete σ₁₁ σ₁₂ σ₁₃ σ₂₁ σ₂₂ σ₂₃ Unique θ, redundant stocks

The shape of the volatility matrix determines completeness. A square invertible matrix (m = d, full rank) is the Goldilocks condition for a unique risk-neutral measure.

5 · The two theorems — the diagnostic toolkit
TheoremFocusConditionLinear algebraWhat it tells you
First Theorem Existence ∃ a risk-neutral measure ℙ̃ System σθ = α − r·1 has at least one solution Model is internally consistent — no free money.
Second Theorem Uniqueness Only ONE risk-neutral measure System has a unique solution (σ is square and invertible) Model is perfectly hedgeable — every payoff can be replicated.

When building a two-factor NSE model: check the First Theorem first — solve σθ = α − r·1 and confirm a solution exists. If it does, check uniqueness — is σ square and invertible? If both pass, the model is both arbitrage-free and complete. If uniqueness fails (d > m), you must accept that some risks cannot be perfectly hedged and build a minimum-variance hedge instead of a perfect one.

Try it — the arbitrage detector and completeness checker

Set parameters for two stocks driven by one or two Brownian Motions. The simulator checks whether the market price of risk system has a solution (First Theorem) and whether it is unique (Second Theorem). It also shows the arbitrage portfolio if one exists.

12%
20%
18%
20%
5%
1 (same driver for both)
Sharpe₁ (α₁−r)/σ₁
—
Sharpe₂ (α₂−r)/σ₂
—
1st Theorem
—
2nd Theorem
—

When d=1 and Sharpe₁ ≠ Sharpe₂: First Theorem fails — the arbitrage path (gold) grows without risk. When d=2: both theorems can hold with the right parameters.

Chapter 20 — the two fundamental theorems:

Your diagnostic toolkit: Before trusting any multi-asset model, run both checks. First: does σθ = α − r·1 have a solution? (No arbitrage.) Second: is that solution unique? (Complete market.) These two theorems separate models that are mathematically valid from those that appear to work in backtests only because they secretly contain impossible free profits.