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