Why Incomplete Markets Give Price Bounds
In a complete market, a claim can be replicated by trading existing assets, so its price is pinned down by the cost of that replicating strategy. In an incomplete market, some claims cannot be replicated. Their prices need not be unique: different probability measures can agree on the prices of all traded assets while assigning different expectations to the claim. The range of those expectations gives the no-arbitrage pricing bounds.
We work with a finite one-period model on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), where \(\Omega=\{\omega_1,\ldots,\omega_n\}\), every state has positive physical probability, \(\mathcal{F}\) contains all subsets of \(\Omega\), and \(\mathcal{F}_0\) is trivial. The terminal information is \(\mathcal{F}_T=\mathcal{F}\). The risk-free account is the numeraire, so all asset prices and claim payoffs below are discounted to time zero. Write \(x_i^j\) for the discounted terminal price of traded asset \(j\) in state \(\omega_i\), and \(x_0^j\) for its time-zero price. A claim has discounted terminal payoff \(h_i\) in state \(\omega_i\). We use no Brownian motion or stochastic integrals in this finite-state analysis.
Assume \(\mathcal{Q}\ne\varnothing\); this is the finite-state no-arbitrage setting. The probabilities \(q_i\) belong to the pricing measure \(\mathbb{Q}\), not necessarily the physical measure \(\mathbb{P}\). A time-zero price \(p\) for the claim is compatible with no arbitrage precisely when it equals \(\sum_i q_i h_i\) for some \(q\in\mathcal{Q}\). This follows by applying the standard finite-state first fundamental theorem to the market with the claim added as a traded asset: under its hypotheses, no arbitrage is equivalent to the existence of an equivalent martingale measure for all traded assets. The theorem is part of the earlier material on the first fundamental theorem.
Bounds and the Endpoint Issue
The set \(\mathcal{Q}\) is defined using strict positivity, so its infimum and supremum need not be attained. To compute them, allow zero probabilities temporarily and define the closed feasible set $$ \overline{\mathcal{Q}}=\left\{q:\ q_i\ge 0,\ \sum_{i=1}^n q_i=1,\ \sum_{i=1}^n q_i x_i^j=x_0^j\text{ for every }j\right\}. $$ This set is compact: it is a closed subset of the bounded probability simplex. It contains \(\mathcal{Q}\), and its boundary points are limits of equivalent martingale measures. Indeed, fix \(q^0\in\mathcal{Q}\). For any \(q\in\overline{\mathcal{Q}}\) and \(0<\varepsilon<1\), the vector \(q^\varepsilon=(1-\varepsilon)q+\varepsilon q^0\) satisfies the same pricing constraints and has every coordinate strictly positive. Also \(q^\varepsilon\to q\) as \(\varepsilon\to0\).
Proof. The function \(q\mapsto\sum_iq_i h_i\) is continuous, and \(\overline{\mathcal{Q}}\) is nonempty and compact, so it has a minimum and a maximum there. Since \(\mathcal{Q}\subseteq\overline{\mathcal{Q}}\), its infimum is at least \(L\). If \(q^-\) minimizes over \(\overline{\mathcal{Q}}\), the vectors \((1-\varepsilon)q^-+\varepsilon q^0\) belong to \(\mathcal{Q}\), and their payoff expectations converge to \(L\). Thus the infimum over \(\mathcal{Q}\) is \(L\). The same argument with a maximizer proves that the supremum is \(U\). By the finite-state first fundamental theorem applied to the enlarged market, the prices that admit no arbitrage are exactly the expectations under measures in \(\mathcal{Q}\). Finally, suppose \(L<p<U\). By the infimum and supremum equalities, there are \(q^-,q^+\in\mathcal{Q}\) whose expectations lie respectively below and above \(p\). A convex combination \(\lambda q^-+(1-\lambda)q^+\), with \(0<\lambda<1\) chosen so that its expectation is \(p\), remains strictly positive and satisfies all asset-pricing constraints. Hence it belongs to \(\mathcal{Q}\) and prices the claim at \(p\). At either endpoint, the price is attainable exactly when some member of \(\mathcal{Q}\) has that expectation. \(\square\)
The theorem identifies a closed interval of limiting bounds, \([L,U]\), but does not say that both endpoints are prices. When the feasible equivalent measures can approach a boundary only by assigning zero probability to some state, an endpoint expectation is a limit rather than an expectation under an equivalent measure. Convexity guarantees all prices strictly between distinct bounds; it does not turn a limiting boundary measure into an equivalent one.
Worked Example: One Asset and Three States
Let the discounted asset price today be \(2\), with terminal values \(1,2,4\). A claim pays \(0,1,4\) in those states. Find its pricing bounds and test whether the claim is replicable.
Write \(q_1,q_2,q_3\) for the probabilities under an equivalent martingale measure. The normalization and asset-pricing equations are \(q_1+q_2+q_3=1\) and \(q_1+2q_2+4q_3=2\). Subtracting the first equation from the second gives \(q_2+3q_3=1\). Set \(q_3=u\); then \(q_2=1-3u\) and, from normalization, \(q_1=2u\). Strict positivity requires \(0<u<\tfrac13\).
The claim expectation is $$ 0q_1+q_2+4q_3=1+u. $$ Thus its lower and upper bounds are \(1\) and \(\tfrac43\), respectively, and neither endpoint is attained by an equivalent martingale measure. For example, \(u=\tfrac16\) gives \(q=(\tfrac13,\tfrac12,\tfrac16)\); its asset expectation is \(\tfrac13+2(\tfrac12)+4(\tfrac16)=2\), and its claim expectation is \(\tfrac12+4(\tfrac16)=\tfrac76\).
To test replication, a portfolio in the discounted cash account and this asset would pay \(a+bS_T\). Matching the first two claim payoffs requires \(a+b=0\) and \(a+2b=1\), hence \(b=1\) and \(a=-1\). That portfolio pays \(3\), not \(4\), in the third state. The claim is therefore unreplicable. Check: the candidate measure above is strictly positive, has probabilities summing to \(\tfrac13+\tfrac12+\tfrac16=1\), and prices both the asset and claim at the stated values.
Finding Bounds by Optimizing Over Feasible Probabilities
In a finite-state model, each candidate measure is a vector of probabilities subject to linear equations and positivity restrictions. The discounted payoff expectation is also linear in that vector. One can therefore compute the limiting bounds by optimizing a linear function over \(\overline{\mathcal{Q}}\). The extremizers may lie on its boundary, which is exactly why the endpoint check matters. The next example shows how a payoff concentrated in one state can have limiting bounds even though every equivalent measure gives that state positive probability.
Worked Example: A Claim Paying Only in the Highest State
Suppose one discounted asset has current price \(\tfrac32\) and terminal values \(0,1,2,3\). A claim pays \(1\) in the state with asset value \(3\) and \(0\) in the other three states. Find its bounds.
For any equivalent martingale measure, \(q_3>0\), so the claim expectation is \(q_3>0\). Also, the asset expectation is \(\tfrac32\). The contribution from the highest state is \(3q_3\); since the other state values are nonnegative, \(3q_3\le\tfrac32\). The inequality is strict for an equivalent measure, because the other states have positive probabilities and include a state with positive asset value. Thus \(q_3<\tfrac12\).
Both limiting values are feasible in \(\overline{\mathcal{Q}}\). The vector \((\tfrac14,0,\tfrac34,0)\) sums to \(1\) and has asset expectation \(2(\tfrac34)=\tfrac32\), while its claim expectation is \(0\). The vector \((\tfrac12,0,0,\tfrac12)\) also sums to \(1\), has asset expectation \(3(\tfrac12)=\tfrac32\), and gives claim expectation \(\tfrac12\). Therefore the bounds are \(L=0\) and \(U=\tfrac12\); equivalent measures do not attain either endpoint.
The claim is not replicable: an affine payoff \(a+bS_T\) that is zero at asset values \(0\) and \(1\) must have \(a=0\) and \(b=0\), so it cannot equal \(1\) at asset value \(3\). The theorem also shows that every price strictly between \(0\) and \(\tfrac12\) is attainable under some equivalent martingale measure. Check: the two displayed boundary vectors satisfy the price constraint exactly, but their zero coordinates confirm that neither is equivalent to a physical measure assigning positive probability to every state.
Worked Example: Two Assets and a Claim on Their Joint Outcomes
There are four states, labelled by terminal discounted asset values \((0,0),(2,0),(0,2),(2,2)\). Each asset has current price \(1\). A claim pays \(1\) in states \((0,0)\) and \((2,2)\), and \(0\) in the other states. Find its bounds.
Let the state probabilities under an equivalent martingale measure be \(q_{00},q_{20},q_{02},q_{22}\). The first asset's pricing equation gives \(2(q_{20}+q_{22})=1\), and the second gives \(2(q_{02}+q_{22})=1\). Put \(q_{22}=a\). Then \(q_{20}=q_{02}=\tfrac12-a\); normalization gives \(q_{00}=a\). Strict positivity is equivalent to \(0<a<\tfrac12\). The claim expectation is \(q_{00}+q_{22}=2a\), so its limiting bounds are \(0\) and \(1\), neither attained by an equivalent measure. For instance, \(a=\tfrac3{10}\) gives probabilities \((\tfrac3{10},\tfrac15,\tfrac15,\tfrac3{10})\); they sum to \(1\), each asset expectation is \(2(\tfrac15+\tfrac3{10})=1\), and the claim expectation is \(\tfrac35\).
The claim is unreplicable. A portfolio in cash and the two assets has payoff \(c+b_1x_1+b_2x_2\). Matching the payoff \(1\) at \((0,0)\) gives \(c=1\); matching \(0\) at \((2,0)\) and \((0,2)\) gives \(b_1=b_2=-\tfrac12\). That portfolio pays \(-1\) at \((2,2)\), not the required \(1\). Check: the feasible parameter \(a=\tfrac3{10}\) lies strictly between \(0\) and \(\tfrac12\), and the resulting claim price \(\tfrac35\) lies strictly between its bounds.
A Common Pricing Error
A frequent mistake is to use the physical probabilities \(\mathbb{P}\) to price a claim. No-arbitrage bounds are obtained from equivalent martingale measures, which must reproduce the prices of traded assets. Physical probabilities describe the actual model of outcomes; they need not satisfy those asset-pricing equations.
For the first example, suppose \(\mathbb{P}\) assigns probability \(\tfrac13\) to each state. The physical expected payoff is \(\tfrac13(0+1+4)=\tfrac53\). It is incorrect to call \(\tfrac53\) the no-arbitrage price on that basis: the martingale-measure expectations lie between \(1\) and \(\tfrac43\), so \(\tfrac53\) is above the upper bound. The correction is to impose the traded asset's pricing constraint before computing a claim expectation. The physical expectation is not itself a substitute for that constraint.
These bounds matter because they distinguish what the traded assets force from what remains uncertain when the claim cannot be replicated. The bounds do not select a unique price or describe every preference-based valuation. They identify the prices compatible with no arbitrage under an equivalent martingale measure. In the next step of the analysis, superhedging asks a different question: what is the least cost of a portfolio that covers the claim in every state?
Check Your Understanding
Use the definition of \(\mathcal{Q}\) and keep the distinction between its strictly positive probabilities and the nonnegative probabilities in \(\overline{\mathcal{Q}}\) explicit.
- In the first worked example, find the equivalent martingale measure corresponding to \(u=\tfrac14\), and calculate its claim expectation.
- Explain why a minimizing vector in \(\overline{\mathcal{Q}}\) need not give an attainable no-arbitrage price.
- In the second worked example, verify the asset expectation and claim expectation for each of the two boundary probability vectors.
- In the two-asset example, find the measure and claim expectation for \(a=\tfrac14\).
- State why a physical expected payoff need not be a no-arbitrage price.