3.2.4 The Fundamental Theorems of Asset Pricing

Basic Prerequisites: 02 · No-Arbitrage & Risk-Neutral and linear algebra (rank, null space).

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1. Intuition & Practical Objective

Everything so far has been constructive: build the hedge, read off the price. This page states the theorems that say when the construction is possible at all - the two results that turn derivative pricing from a collection of tricks into a theory.

• First Fundamental Theorem (FT1). No arbitrage $⟺$ there exists an equivalent martingale measure (EMM). Existence of a pricing measure is exactly the no-arbitrage condition - no more, no less.
• Second Fundamental Theorem (FT2). The market is complete - every claim is replicable, every price is unique - $⟺$ that measure is unique.

The practical objective: know the three cases a market can be in, and know which one you are in before quoting a price. Two of them do not have prices in the sense a trader means.

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2. Mathematical Ground Truth & Derivations

2.1 Definitions

Arbitrage (Shreve II Def 5.4.6; Björk Def 7.5). A self-financing strategy with $X(0)=0$, $X(T)\ge 0$ a.s. and $\mathbb{P}\{X(T)>0\}>0$ - equivalently, a way to beat the money-market account risklessly.

Equivalent martingale measure (Shreve II Def 5.4.3; Björk Def 3.7). A measure $\tilde{\mathbb{P}}\sim \mathbb{P}$ (same null sets) under which every discounted traded price $D(t)S_i(t)$ is a martingale. In the binomial, $\tilde{\mathbb{P}}$ is the $\tilde{p}/\tilde{q}$ pair.

2.2 The theorems

$$\boxed{\text{FT1:}\text{market is arbitrage-free}⟺\tilde{\mathbb{P}}\text{ exists}}$$ $$\boxed{\text{FT2:}\text{market is complete}⟺\tilde{\mathbb{P}}\text{ is unique}}$$

Shreve II proves FT1 as Thm 5.4.7 and FT2 as Thm 5.4.9; Björk states both for the finite-state one-period model in Ch 3 (no-arbitrage ⟺ existence of strictly positive state prices, via Farkas’ lemma; completeness ⟺ $\mathrm{Ker}[D]=0$).

The operator form. In Björk’s one-period $M$-state, $N$-asset market with normalized prices $Z$, a portfolio $h\in {\mathbb{R}}^N$ costs $h{Z}_0$ and pays $h{D}^Z$ (the $M\times N$ payoff matrix, first row all ones). No-arbitrage says: no $h$ with $h{Z}_0\le 0$ and $h{D}^Z>0$. Farkas’ lemma converts that statement about portfolios into the existence of a positive vector $q$ with

$$Z_0=D^{Z}q,q_j>0,{\sum }_j{q}_j=1.$$

Completeness is the dual statement $\mathrm{Im}[D^{\ast }]={\mathbb{R}}^M$ - enough independent assets to span every state. Two faces of one theorem: prices exist iff the payoff matrix is spanned; prices are unique iff it is spanned exactly.

2.3 Market price of risk (continuous formulation)

For $m$ stocks and $d$ Brownian drivers, $d{S}_i={\alpha }_i{S}_{i}dt+S_i{\sum }_j{\sigma }_{ij}d{W}_j$, the EMM is built from $\Theta$ solving (Shreve II eq. 5.4.18)

$${\alpha }_i(t)-R(t)=\sum _{j=1}^d{\sigma }_{ij}(t){\Theta }_j(t),i=1,\dots ,m.$$

These are $m$ equations in $d$ unknowns - one unknown per source of randomness, not per stock:

• no solution $\Rightarrow$ FT1 fails; the model has arbitrage (e.g. two stocks, one driver, inconsistent Sharpe ratios - Shreve Ex. 5.4.4).
• non-unique solution $\Rightarrow$ FT2 fails; the market is incomplete.
• unique solution ($m=d$, $\sigma$ invertible) $\Rightarrow$ complete market.

2.4 The three cases (Shreve II §5.7)

Case	EMM	Consequence
No solution	does not exist	model admits arbitrage - reject it
Many solutions	not unique	incomplete: non-replicable claims have a price interval, not a price (credit derivatives, jump risk, vol derivatives)
Exactly one	unique	complete: $V(t)=\frac{1}{D(t)}\tilde{\mathbb{E}}[D(T)V(T)\mid F(t)]$ is the price

And the stochastic-discount-factor view (Björk Prop 3.18) unifies them: $\Lambda =\frac{1}{1+R}\frac{dQ}{dP}$ has $\Pi (0;X)={\mathbb{E}}^{\mathbb{P}}[\Lambda X]$; distinct EMMs are distinct $\Lambda$‘s.

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3. Computational Implementation - completeness is a rank statement

A three-state, one-stock market. Two assets (bond, stock) and three states: the payoff matrix has rank $2<3$, so $\mathrm{Ker}[D]\ne \{0\}$, the EMM is not unique, and a non-replicable claim has a price interval. Stdlib only.

With two assets the state-price vector lives on a one-dimensional segment ($0<{\zeta }_3<0.357143$), and a claim whose payoff is not in the span of the traded assets has an entire interval of arbitrage-consistent prices, $[0.595238,0.666667]$. Add a third independent asset and the same claim prices uniquely at $0.628571$ - inside the interval, as FT2 guarantees it must be. Incompleteness is not a modelling detail; it is the difference between a price and a range.

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4. Failure Modes & First-Principles Breakdowns

1. Quoting a point price in an incomplete market. If the EMM is not unique, any single number is a choice of measure, not a no-arbitrage consequence. The honest output is the interval $[{\min }_{\zeta }\zeta \cdot X,{\max }_{\zeta }\zeta \cdot X]$ plus the economic argument that fixes a measure.
2. Confusing “calibrated” with “unique”. A model fitted to a rich set of liquid prices can still have a non-unique measure; calibration narrows the interval but FT2 is a rank statement about the payoff matrix, not a fit statistic.
3. Counting equations wrong. The market-price-of-risk system has $m$ equations and $d$ unknowns ($d$ = drivers, not assets). Adding correlated assets without adding drivers does not create a new measure dimension; it creates inconsistency → arbitrage (Shreve Ex. 5.4.4).
4. Assuming bounded payoffs are hedgeable. FT2 says every claim is replicable only if the tree/market spans all states. Jumps or a second untraded risk source break the span (05 · Failure Modes).
5. Treating $\Lambda$ as observable. The state-price density is implied by the market, not quoted in it; estimating it (or its $q$ in the tree) requires the market prices one is trying to price - a circularity that is exactly the calibration problem of the later pillar topics.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance I
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 03 · Trees & Convergence · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Siblings: BSM · 02 PDE & Derivation (completeness via the Martingale Representation Theorem) · Heston & SABR (an incomplete-market example)