The Fundamental Theorems of Asset Pricing
No arbitrage \(\Leftrightarrow\) an equivalent martingale measure exists; completeness \(\Leftrightarrow\) it is unique.
Learning Objectives
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- State the First Fundamental Theorem precisely and explain the equivalence of measures.
- State the Second Fundamental Theorem and connect uniqueness of the EMM to market completeness.
- Distinguish complete from incomplete markets and give an example of each.
- Explain why incomplete markets yield a price interval rather than a single price.
- Compute no-arbitrage price bounds in an incomplete (trinomial) one-period market.
Key Vocabulary
- Equivalent martingale measure (EMM)
- A probability \(\mathbb{Q}\) equivalent to \(\Prob\) (same null sets) under which discounted asset prices are martingales; also called a risk-neutral measure.
- Equivalent measures
- Two measures with the same events of probability zero; they agree on what is possible, disagree on how likely.
- First Fundamental Theorem (FTAP1)
- No arbitrage \(\Leftrightarrow\) at least one EMM exists.
- Second Fundamental Theorem (FTAP2)
- An arbitrage-free market is complete \(\Leftrightarrow\) the EMM is unique.
- Complete market
- One in which every contingent claim is replicable by a self-financing strategy; then prices are unique.
- Incomplete market
- Some claims cannot be replicated; multiple EMMs exist and no-arbitrage pins a claim only to a price interval.
Intuition & Motivation
Equivalence of measures
Two probability measures \(\Prob\) and \(\mathbb{Q}\) are equivalent (written \(\mathbb{Q}\sim\Prob\)) if they have the same null sets: \(\Prob(A)=0\iff\mathbb{Q}(A)=0\). They may assign very different probabilities to events, but they agree on what is impossible. Equivalence is essential - an EMM that killed a genuinely possible payoff state would let you construct an arbitrage.
First Fundamental Theorem
One direction is easy intuition: if an EMM \(\mathbb{Q}\) exists, then any zero-cost self-financing portfolio has discounted value a \(\mathbb{Q}\)-martingale starting at 0, so its terminal expectation is 0; a nonnegative payoff with zero expectation (under a measure equivalent to \(\Prob\)) must be zero - no arbitrage. The converse (no arbitrage \(\Rightarrow\) an EMM exists) is a separating-hyperplane argument in finite models and the deep Dalang–Morton–Willinger / Delbaen–Schachermayer results in general.
Second Fundamental Theorem
Counting intuition (finite one-period model with \(K\) states and \(A\) traded assets including the bond): replicating an arbitrary payoff means solving \(K\) equations in \(A\) unknowns. You can hit every target payoff iff the asset payoff vectors span \(\R^K\), i.e. \(A\ge K\) with full rank. The same rank condition makes the martingale equations pin down \(\mathbb{Q}\) uniquely. Fewer independent assets than states (\(A\lt K\)) leaves both the hedging system and the martingale system underdetermined - incompleteness and non-unique \(\mathbb{Q}\) arrive together.
| Property | Complete market | Incomplete market |
|---|---|---|
| EMM \(\mathbb{Q}\) | Unique | Infinitely many |
| Every claim replicable? | Yes | No (some claims cannot be hedged) |
| Price of a claim | Single number \(R^{-T}\E_{\mathbb{Q}}[V_T]\) | Interval \([\inf_{\mathbb{Q}},\ \sup_{\mathbb{Q}}]\) |
| Example | Binomial (2 states, 2 assets) | Trinomial (3 states, 2 assets); most real markets |
Why incompleteness gives a price interval
If many EMMs \(\mathbb{Q}\) are consistent with the traded prices, an unhedgeable claim has a different discounted expectation under each. No-arbitrage requires only that its price lie between the smallest and largest such expectation:
where \(\mathcal M\) is the set of EMMs. Within the interval, selecting one price requires an extra criterion (a utility, a calibration, a hedging objective) - the market alone does not decide.
Interactive: no-arbitrage price bounds in an incomplete market
- Saying ‘risk-neutral measure’ as if it were always unique; uniqueness is FTAP2 and holds only in complete markets.
- Using a \(\mathbb{Q}\) that assigns probability 0 to a possible state; then \(\mathbb{Q}\not\sim\Prob\) and an arbitrage can hide in that state.
- Expecting a single price in an incomplete market; no-arbitrage delivers only bounds, and a model/utility choice is needed to narrow them.
- Confusing ‘no arbitrage’ (EMM exists) with ‘complete’ (EMM unique) - they are two different theorems.
- Rank rule of thumb (finite one-period): complete \(\iff\) independent traded payoffs span the state space \(\iff\) \(\#\text{assets}\ge\#\text{states}\) with full rank.
- Real markets are incomplete (stochastic volatility, jumps, transaction costs); quoting a single option price implicitly selects one \(\mathbb{Q}\) via calibration.
- Stochastic-volatility and jump models (Phase 15, Gatheral) are the canonical incomplete markets where EMM choice is a modeling decision.
- Keep the logic crisp: FTAP1 is about existence (pricing at all), FTAP2 about uniqueness (one price vs a range).
Knowledge Check
Practical Exercise
Consider a one-period market with three equally-possible states and only a bond and a stock. (a) Argue from FTAP2 whether it can be complete. (b) If the stock can go to \(110,\,100,\,90\) with \(R=1\), find the family of EMMs. (c) Give the no-arbitrage price bounds for a claim paying \((1,0,0)\).
(a) With 3 states but only 2 traded assets, the payoff vectors cannot span \(\R^3\), so some claim is unhedgeable - by FTAP2 the market is incomplete and the EMM is not unique.
(b) EMM needs \(110q_u+100q_m+90q_d=100\) and \(q_u+q_m+q_d=1\), \(q\ge0\). Subtracting \(100\times\) the second: \(10q_u-10q_d=0\Rightarrow q_u=q_d=t\), \(q_m=1-2t\), \(t\in[0,\tfrac12]\).
(c) The claim pays 1 in the up state, so its price is \(q_u=t\in[0,\tfrac12]\) (with \(R=1\)). No-arbitrage bounds: \([0,\tfrac12]\). Any price strictly inside is consistent with some EMM; the market does not single one out.
Lesson Summary
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: FTAP1: no arbitrage \(\Leftrightarrow\) an equivalent martingale measure exists. FTAP2: an arbitrage-free market is complete \(\Leftrightarrow\) the EMM is unique.
A: Multiple EMMs are consistent with traded prices; an unhedgeable claim has a different discounted expectation under each, and no-arbitrage only forces the price between the inf and sup over EMMs.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 2,5 - Vol. I–II: risk-neutral pricing and the fundamental theorems in binomial and continuous models.
- Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 1-2 - Ch. 1–2: equivalent martingale measures, completeness, and the two fundamental theorems.