Equivalent Martingale Measures in Finite-State Models

State Probabilities and Discounted Prices

In a finite-state market, a probability measure assigns a weight to each possible state. An equivalent martingale measure is a choice of strictly positive state weights under which discounted traded asset prices have the martingale property. Those weights give a pricing formula for claims that can be replicated by trading. The formula is conditional on the information currently available; an unconditional state-weighted sum is justified at date zero only when the initial information is trivial.

Consider dates $t=0,1,\ldots,N$ on a finite probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t=0}^N,\mathbb{P})$, with $\mathcal{F}_N=2^\Omega$ and $\mathbb{P}(\{\omega\})>0$ for every $\omega\in\Omega$. The filtration $(\mathcal{F}_t)$ describes the information available at each date. There is a bank account with deterministic value $A_t>0$, and $d$ risky assets with prices $S_t^1,\ldots,S_t^d$. Define the discounted asset prices by $\widetilde S_t^i=S_t^i/A_t$. All random variables in this finite model are integrable.

Definition: A probability measure $\mathbb{Q}$ on $(\Omega,\mathcal{F})$ is equivalent to $\mathbb{P}$, written $\mathbb{Q}\sim\mathbb{P}$, if the two measures have the same null events. In this finite model, equivalence means $q_\omega=\mathbb{Q}(\{\omega\})>0$ for every state $\omega$, with $\sum_{\omega\in\Omega}q_\omega=1$. An equivalent martingale measure is an equivalent probability measure $\mathbb{Q}$ under which each discounted asset price $\widetilde S^i$ is a martingale: $\mathbb{E}^{\mathbb{Q}}[\widetilde S_{t+1}^i\mid\mathcal{F}_t]=\widetilde S_t^i$ for every $t<N$ and $i=1,\ldots,d$.

Here $\mathbb{P}$ is the probability measure used to describe the original model, while $\mathbb{Q}$ is the risk-neutral measure used for pricing. Equivalence requires that no state assigned positive probability under $\mathbb{P}$ is discarded under $\mathbb{Q}$. It does not require $\mathbb{Q}=\mathbb{P}$. The martingale condition is imposed on discounted prices, not generally on undiscounted prices: interest earned by the bank account is removed by dividing by $A_t$.

Because every state has positive $\mathbb{Q}$-probability, the martingale condition can also be checked at each information set. For any atom $C$ of $\mathcal{F}_t$, its probability $\mathbb{Q}(C)$ is positive, and the condition is

\sum_{\omega\in C}q_\omega \big(\widetilde S_{t+1}^i(\omega)-\widetilde S_t^i(\omega)\big)=0.

Indeed, conditional expectation on the atom $C$ is the weighted sum over its states divided by $\mathbb{Q}(C)$. Thus the displayed equation is precisely the martingale condition multiplied by that positive probability. Checking these equations at every date, asset, and information set is equivalent to checking the conditional martingale equations.

From Replication to a Conditional Price

A trading strategy chooses, at each date $t$, holdings $\theta_t^i$ in asset $i$ using only information in $\mathcal{F}_t$. The strategy is self-financing when changes in portfolio value come only from gains and losses on the assets held; it does not add or withdraw cash after inception. In discounted units its wealth satisfies

\widetilde V_t=\widetilde V_0+ \sum_{u=0}^{t-1}\sum_{i=1}^d\theta_u^i \big(\widetilde S_{u+1}^i-\widetilde S_u^i\big), \qquad \widetilde V_t=\frac{V_t}{A_t}.

A claim with terminal payoff $X$ is attainable if some self-financing strategy has $V_N=X$. In a finite-state model, its strategy and every displayed wealth variable have finite values on finitely many states, so they are integrable.

Proposition: Suppose an equivalent martingale measure $\mathbb{Q}$ exists and a self-financing strategy replicates $X$. Then, for every $t=0,\ldots,N$, its value is $$ V_t=A_t\,\mathbb{E}^{\mathbb{Q}}\!\left[\frac{X}{A_N}\,\middle|\,\mathcal{F}_t\right]. $$ In particular, if $\mathcal{F}_0$ is trivial, then $V_0=A_0\mathbb{E}^{\mathbb{Q}}[X/A_N]$. If $\mathcal{F}_0$ is not trivial, the date-zero formula remains conditional: $V_0=A_0\mathbb{E}^{\mathbb{Q}}[X/A_N\mid\mathcal{F}_0]$.

Proof. For each $u<t$, the holdings $\theta_u^i$ are $\mathcal{F}_u$-measurable and the discounted price increments have conditional expectation zero under $\mathbb{Q}$. Therefore the conditional expectation of each gain given $\mathcal{F}_u$ is zero. More explicitly, the process $\widetilde V_t$ in the self-financing equation is a martingale: conditional expectation of the next increment given $\mathcal{F}_t$ is $\sum_i\theta_t^i\mathbb{E}^{\mathbb{Q}}[\widetilde S_{t+1}^i-\widetilde S_t^i\mid\mathcal{F}_t]=0$. All terms are integrable, so iterating conditional expectations gives $\widetilde V_t=\mathbb{E}^{\mathbb{Q}}[\widetilde V_N\mid\mathcal{F}_t]$. Since the strategy replicates $X$, $\widetilde V_N=X/A_N$; substituting $\widetilde V_t=V_t/A_t$ gives the asserted formula. If $\mathcal{F}_0$ is trivial, conditional expectation given $\mathcal{F}_0$ is the constant unconditional expectation. If it is not trivial, that simplification is not valid. $\square$

When $\mathcal{F}_0$ is trivial, the formula can be written directly in state probabilities:

V_0=A_0\sum_{\omega\in\Omega}q_\omega\frac{X(\omega)}{A_N}.

When $\mathcal{F}_0$ is nontrivial, the state weights must instead be normalized within each date-zero information set. For an atom $C$ of $\mathcal{F}_0$, the conditional expectation is $\sum_{\omega\in C}q_\omega X(\omega)/A_N$ divided by $\sum_{\omega\in C}q_\omega$. In particular, the price can depend on which atom is observed.

Worked Examples

Worked Example: A One-Period Replication Price

There are three states, all with positive physical probability, and $\mathcal{F}_0$ is trivial. Let $A_0=1$, $A_1=11/10$, and let one risky asset have $S_0=10$ and terminal prices $S_1=(5,10,15)$ across states $\omega_1,\omega_2,\omega_3$. Verify that $q=(1/5,2/5,2/5)$ gives an equivalent martingale measure, then price the attainable payoff $X=2S_1-10$.

All three state weights are positive and sum to $1/5+2/5+2/5=1$, so this probability measure is equivalent to any physical measure that gives positive probability to all three states. The weighted terminal stock price is

\mathbb{E}^{\mathbb{Q}}[S_1] =\frac15(5)+\frac25(10)+\frac25(15) =1+4+6=11.

Thus $\mathbb{E}^{\mathbb{Q}}[S_1/A_1]=11/(11/10)=10=S_0/A_0$, which verifies the discounted martingale condition. The payoff values in the three states are $2(5)-10=0$, $2(10)-10=10$, and $2(15)-10=20$. Holding two shares and a bank-account position of $-100/11$ gives terminal value $2S_1-(100/11)(11/10)=2S_1-10=X$. Its initial cost is $2(10)-100/11=120/11$.

The state-weighted pricing formula gives $\mathbb{E}^{\mathbb{Q}}[X]=\frac15(0)+\frac25(10)+\frac25(20)=0+4+8=12$. Discounting yields $12/(11/10)=120/11$, the same value as replication. Check: in each state the portfolio payoff equals the specified $X$, and its cost equals the risk-neutral expectation after discounting.

Worked Example: Conditional Prices at an Intermediate Date

Let $A_0=A_1=A_2=1$, and let $\Omega=\{a,b,c,d\}$. The date-zero information is trivial; at date one the information sets are $C_1=\{a,b\}$ and $C_2=\{c,d\}$; date two reveals the state. Suppose a traded asset has $S_0=17/2$, $S_1=5$ on $C_1$ and $S_1=12$ on $C_2$, and $S_2=(2,6,10,14)$ in states $a,b,c,d$. Use the state weights $q=(1/8,3/8,1/4,1/4)$ to price the attainable payoff $X=S_2$.

The weights are all positive and sum to $1/8+3/8+1/4+1/4=1$. On $C_1$, the conditional mean of $S_2$ is $\big((1/8)(2)+(3/8)(6)\big)/(1/8+3/8)=(1/4+9/4)/(1/2)=5$. On $C_2$, it is $\big((1/4)(10)+(1/4)(14)\big)/(1/4+1/4)=(10/4+14/4)/(1/2)=12$. These are exactly the stated date-one prices. At date zero, the expectation is $(1/8)(2)+(3/8)(6)+(1/4)(10)+(1/4)(14)=1/4+9/4+10/4+14/4=34/4=17/2$, the stated initial price. Hence the asset is a martingale under these state weights, and holding one share attains $X$.

Check: the conditional prices average back to the initial price: $q(C_1)5+q(C_2)12=(1/2)5+(1/2)12=17/2$. This calculation uses the probability of each information set under $\mathbb{Q}$, not an unweighted average of the two date-one values.

Worked Example: Why Nontrivial Initial Information Requires Conditioning

Take two states $a,b$, with $\mathbb{P}(a)=\mathbb{P}(b)=1/2$, and let $\mathcal{F}_0=\mathcal{F}_1=2^\Omega$. Let $A_0=A_1=1$, with no risky assets, and let $X(a)=1$, $X(b)=0$. Under $\mathbb{Q}=\mathbb{P}$, the bank account is discounted to $1$ and is a martingale; the measure is equivalent to $\mathbb{P}$. Because the state is already known at date zero, holding $X$ units of the bank account replicates the claim, and its date-zero value is $V_0=X$.

The correct formula gives $\mathbb{E}^{\mathbb{Q}}[X\mid\mathcal{F}_0]=X$, so it produces $V_0(a)=1$ and $V_0(b)=0$, as required. The unconditional expectation is $\mathbb{E}^{\mathbb{Q}}[X]=(1/2)(1)+(1/2)(0)=1/2$, which is not the date-zero value in either state. Check: the payoff is already known at date zero, so the replicating portfolio can choose its bank-account holding using that information; a single unconditional average discards that information.

Why Attainable Claims Have a Consistent Price

Corollary: If a claim $X$ is attainable and an equivalent martingale measure exists, every equivalent martingale measure gives the same conditional price for $X$ as the replicating strategy. With trivial $\mathcal{F}_0$, this gives the same date-zero price under every such measure. Moreover, a self-financing strategy with zero initial value cannot have a nonnegative terminal gain that is strictly positive in at least one state.

Proof. Apply the Proposition under any equivalent martingale measure. In each case the conditional expectation of the discounted terminal payoff equals the discounted value of the same replicating strategy at time $t$. Therefore all such measures give that strategy’s value. For the last assertion, suppose a self-financing strategy starts with zero and has discounted terminal gain $G\geq0$ in every state, with $G>0$ in at least one state. Under any equivalent martingale measure, discounted wealth is an integrable martingale, so $\mathbb{E}^{\mathbb{Q}}[G]=0$. But every state has positive $\mathbb{Q}$-probability, and at least one state has $G>0$, implying $\mathbb{E}^{\mathbb{Q}}[G]>0$, a contradiction. Thus no such gain exists. $\square$

The corollary is a pricing implication of the existence of an equivalent martingale measure; it does not assert that every claim is attainable or that the measure is unique. When multiple equivalent martingale measures exist, their expectations can differ for claims outside the set of attainable payoffs. For attainable claims, replication fixes the price, and the conditional expectation formula must agree with it under each measure.

Common Pitfall: Dropping the Conditioning

A tempting but incorrect step is to write $V_0=A_0\mathbb{E}^{\mathbb{Q}}[X/A_N]$ without checking whether $\mathcal{F}_0$ is trivial. The two-state example shows the error concretely: its unconditional expectation is $1/2$, while the replicating value is $1$ in state $a$ and $0$ in state $b$. The corrected statement is $V_0=A_0\mathbb{E}^{\mathbb{Q}}[X/A_N\mid\mathcal{F}_0]$; only when $\mathcal{F}_0$ is trivial does this reduce to the unconditional formula.

Also distinguish the risk-neutral probability weights from prices themselves. The $q_\omega$ are positive probabilities summing to one, and the claim’s discounted payoff is averaged with those weights. The bank-account factor $A_N$ then converts the discounted price to the appropriate units. An equivalent martingale measure must satisfy both requirements: positive weight on every state and the martingale equations for every traded discounted asset.

Check Your Understanding

Use the definitions and conditional pricing formula to answer the following questions.

In a one-period model with $A_0=1$, $A_1=6/5$, and $S_0=10$, what equation must positive state weights satisfy for the discounted stock price to be a martingale?
Why does assigning probability zero to a state of positive $\mathbb{P}$-probability violate equivalence?
If a claim is replicated by a self-financing strategy, why must every equivalent martingale measure give the replicating strategy’s value as the conditional expectation of its discounted payoff?
State the date-zero pricing formula when $\mathcal{F}_0$ is trivial, and the formula when it is not trivial.
In the two-state example with the state already known at date zero, what are the conditional and unconditional expectations of $X$, and why are they different?