From admissible probabilities to equivalent martingale measures

The article Holtan (2007) integrates option valuation theory with practical pricing in non-life insurance, analyzing both no-arbitrage and martingale approaches, as well as complete and incomplete market settings from a supply-and-demand perspective.

According to the precise definition given by Harrison and Pliska (1983), a financial market is complete if and only if there exists a unique equivalent martingale measure associated with the underlying stochastic process. In this context, any contingent claim can be fully hedged if and only if the martingale measure is unique Shreve (2004), reinforcing the link between market completeness and full risk coverage.

A discrete martingale is a stochastic process $ (X_t)_{t=0,1,2,\dots}$ adapted to a filtration $ (\mathcal{F}_t)$, with finite expectation, satisfying the property of temporal fairness:

\mathbb{E}[X_{t+1}\mid \mathcal{F}_t] = X_t.

Briefly, a discrete Itô process¹,

X_{t_{k+1}}
= X_{t_k}
+ \mu(t_k,X_{t_k})\,\Delta t_k
+ \sigma(t_k,X_{t_k})\,\Delta W_{k+1}
\tag{Euler–Maruyama}

induces a martingale if $ \mu(t,X_t) = 0$.

The rigorous foundations remain those established in the literature:

• the characterization of Radon–Nikodym derivatives as the core tool for change of measure in pricing Shreve (2004),
• and the formalization of stochastic drift modification through Girsanov’s theorem Øksendal (2003).

Although passing through the Radon–Nikodym theorem and subsequently through Girsanov’s theorem is mathematically essential, it often shifts the attention of economists and actuaries away from the pricing mechanism itself toward technical details of measure transformation.

We attempt here to illustrate this concept through a simple example, in which the selection criterion acts like a magnet, attracting from the set of admissible probabilities those distributions that align with risk-neutral probabilities.

This perspective highlights the connection between the absence of arbitrage, actuarial valuation, and the central role of equivalent probabilities in price determination.

The simplification adopted should be understood as an expository choice, one that favors economic intuition without denying the necessity of the full mathematical framework.

As an example, consider a discrete process $ (X_n)$ defined recursively starting from $ X_0 = 0$. At each step:

• $ X_{n+1} = X_n + 1$ with probability $ p$,
• $ X_{n+1} = X_n - 1$ with probability $ q = 1 - p$.

Conditioned on $ X_n$, the expected value at the next step is

\mathbb{E}[X_{n+1} \mid X_n, … , X_1]
= p (X_n + 1) + q (X_n - 1)
= X_n + (p - q).

Hence, $ (X_n)$ is a martingale if and only if $ p = \frac{1}{2}$.

Similarly to the de Moivre process,

Y_n = \left( \frac{q}{p} \right)^{X_n}

is a martingale because:

\begin{aligned}
\mathbb{E}[Y_{n+1}\mid X_{1},\dots ,X_{n}]
&= p \left( \frac{q}{p} \right)^{X_{n}+1}
+ q \left( \frac{q}{p} \right)^{X_{n}-1} \\[6pt]
&= p \left( \frac{q}{p} \right) \left( \frac{q}{p} \right)^{X_{n}}
+ q \left( \frac{p}{q} \right) \left( \frac{q}{p} \right)^{X_{n}} \\[6pt]
&= q \left( \frac{q}{p} \right)^{X_{n}}
+ p \left( \frac{q}{p} \right)^{X_{n}} \\[6pt]
&= \left( \frac{q}{p} \right)^{X_{n}}
= Y_{n}.
\end{aligned}

For a process with only two states, there exists a unique equivalent martingale measure $ q = (q_d, q_u)$, discussed here. We now take one step further, to three admissible states:

Problem data Link to heading

Single–step model (0 → T), with support for $ S_T$ in three states:

• $ S_d = 50$,
• $ K = 110$ (call striKe),
• $ S_u = 130$.

Parameters:

• $ S_0 = 100$,
• $ r = 0.05$,
• $ T = 1$,
• $ p = (0.6,0.2,0.2)$
• ‼️ The given probabilities are descriptive, not pricing probabilities, so we do not use them to compute $ S_0$.

In an incomplete market, the absence of arbitrage opportunities implies the existence of a non-empty set of equivalent martingale measures $ \mathcal{Q}$, but it does not require the physical measure $ P$ to belong to this set.

The assumption $ P \in \mathcal{Q}$ is equivalent to stating that:

• investors price directly under the physical distribution,
• risk premia are zero,
• risk preferences do not affect prices.

The forward price, obtained by compounding the current (spot) value at the risk-free rate, is:

F_{0\to T} = S_0 e^{rT} = 100 e^{0.05} \approx 105.1271096.

We thus know the expected value under any martingale measure induced by $ q$. Even so, the replicated random variable is neither unique nor easy to determine.

1. Incomplete market: the equivalent martingale measure is not unique Link to heading

We seek a probability measure $ q = (q_d, q_K, q_u)$ such that:

\left\lbrace
\begin{aligned}
q_d + q_K + q_u &= 1, \\
q_d S_d + q_K S_K + q_u S_u &= F_{0\to T}, \\
q_i &\ge 0, \quad i \in \{d, K, u\}.
\end{aligned}
\right.

With three states and only these two equations (plus non-negativity), one degree of freedom remains: there exist infinitely many admissible martingale measures $ q$.

For the given values, the system yields:

4q_d + q_K = 1.24365548
\quad\Rightarrow\quad
q_K = 1.24365548 - 4q_d \ge 0,

and from $ q_d+q_K+q_u=1$:

q_u = 1 - q_d - q_K = -0.24365548 + 3q_d \ge 0.

The constraints $ q_K\ge 0$ and $ q_u\ge 0$ impose the interval:

q_d \in \left[\frac{0.24365548}{3},\frac{1.24365548}{4}\right]=
[0.08121849, 0.31091387].

2. Call price and the no-arbitrage interval Link to heading

European call payoff: $ (S_T-K)^+$.

Here $ K=110$, therefore:

• at $ S_d=50$: the option expires worthless,
• at $ K=110$: the option expires worthless,
• at $ S_u=130$: the payoff is $ 20$.

Hence, the price depends only on the mass $ q_u$:

C_0(q) = e^{-rT}\cdot 20\cdot q_u.

Since $ q_u = -0.24365548 + 3q_d$ and $ q_d\in[0.08121849,0.31091387]$, we obtain:

q_{u,\min} = 0,\qquad
q_{u,\max} = -0.24365548 + 3\cdot 0.31091387 = 0.68908613.

Therefore, the maximal price becomes:

C_{\max} = e^{-0.05}\cdot 20\cdot 0.68908613 \approx 13.11.

3. Selection of a unique q via additional constraints Link to heading

Additional criteria select a single measure $ q$ from the (infinite) set of admissible martingale measures.

Minimum divergence from the prior (MEMM)

We choose $ q$ minimizing the KL divergence² relative to the admissible prior $ p=(0.6,0.2,0.2)$:

\min_q \sum_i q_i \log\frac{q_i}{p_i}
\quad \text{s.t. } \mathbb{E}_q[S]=F_{0\to T},\; \sum_i q_i=1,\; q_i\ge 0.

The solution has an exponential tilt form:

q_i(\lambda)=\frac{p_i e^{\lambda S_i}}{\sum_j p_j e^{\lambda S_j}},

where $ \lambda$ is determined from the condition $ \sum_i q_i(\lambda)S_i=F_{0\to T}$.

For our data (numerically):

q3(Y, S0=100, r=0.05, method='MEMM')⏎ 
═════════════════════════════════════
A random variable with |3⟩ events, (µ = 105.13 | σ = 31.83), entropy (+relative) 1.06(0.8847)
──
ᴿᵢ      50     110     130
ᵖᵢ 0.23630 0.29844 0.46526

The Call option price becomes:

C_{\text{MEMM}} \approx e^{-0.05}\cdot 20\cdot 0.46526 \approx 8.85.

Variance-Optimal Martingale Measure (VOMM)

q3(Y, S0=100, r=0.05, method='VOMM')⏎ 
═════════════════════════════════════
A random variable with |3⟩ events, (µ = 105.13 | σ = 30.98), entropy (+relative) 1.07(0.905)
──
ᴿᵢ      50     110     130
ᵖᵢ 0.22522 0.34277 0.43201

C_{\text{VOMM}} \approx e^{-0.05}\cdot 20\cdot 0.43201 \approx 8.22.

To produce the results above, VOMM selects the martingale measure $ q$ that minimizes the quadratic deviation from the physical measure $ p$:

\min_{q \in \mathcal{Q}}
\mathbb{E}_{p}
\Biggl[
\Bigl(
\frac{q}{p} - 1
\Bigr)^2
\Biggr].

Equivalently, it minimizes the variance of the Radon–Nikodym density, with mean $ \sum_i p_i \frac{q_i}{p_i} = \sum_i q_i = 1.$

The variance-optimal martingale measure is the solution of:

q^\star
=
\arg\min_{q \in \mathcal{Q}}
\sum_{i=1}^3 \frac{(q_i - p_i)^2}{p_i}.

The objective function represents a weighted quadratic distance between $ q$ and $ p$, interpretable as a measure of the energy of deviation from the initial admissible probabilities.

Introducing Lagrange multipliers $ \alpha$ and $ \beta$ for the normalization and martingality constraints, the first-order conditions yield the linear relation:

q_i
=
p_i\bigl(1 + \alpha + \beta S_i\bigr),
\qquad i=1,2,3.

The coefficients $ \alpha$ and $ \beta$ are uniquely determined by imposing the two constraints:

\sum_{i=1}^3 q_i = 1
\;\Longrightarrow\;
1+\alpha+\beta\sum_{i=1}^3 p_i S_i = 1
\;\Longrightarrow\;
\alpha = -\beta\,\mathbb{E}_p[S],

and the martingale condition then determines $ \beta$:

\sum_{i=1}^3 q_i S_i = F_{0\to T}
\;\Longrightarrow\;
\mathbb{E}_p[S] + \alpha\,\mathbb{E}_p[S] + \beta\,\mathbb{E}_p[S^2] = F_{0\to T},

which yields equivalently:

\beta=\frac{F_{0\to T}-\mathbb{E}_p[S]}{\mathrm{Var}_p(S)}.

Unlike MEMM, where probabilities have an exponential form, the VOMM solution is affine in the states $ S_i$.

Maximum entropy

We choose $ q$ maximizing Shannon entropy. This is equivalent to applying the first method with prior $ p=(1/3,1/3,1/3)$.

For our data (numerically):

q3(Y, S0=100, r=0.05, use_prior=F)⏎ 
═══════════════════════════════════
A random variable with |3⟩ events, (µ = 105.13 | σ = 30.68), entropy (+relative) 1.07(0.9064)
──
ᴿᵢ      50     110     130
ᵖᵢ 0.22137 0.35816 0.42047

Thus:

C_{\text{MaxEnt}} \approx e^{-0.05}\cdot 20\cdot 0.420 \approx 7.99.

In the absence of a prior measure, we may choose the MaxEnt (uniform) distribution $ p_i =1/3$ as a starting point. We observe, however, that when this distribution already satisfies the martingale conditions, applying MEMM/VOMM produces no further calibration: the optimal solution coincides with the initial measure; no new information forces a different replication of probabilities.

Conclusion Link to heading

Using the methods above, we started from an initial random variable and, by exploiting admissible correlations, replicated a version with high informational content. That is all. The rest is philosophical noise.

• The set of martingale measures $ q$ is not uniquely determined (incomplete market).
• In this example, the call price depends only on the risk-neutral probability of the upper state $ q_u$.
• No-arbitrage interval:

C_0 \in [0,\; 13.11].

Three principled selections:

• MEMM (min KL relative to $ p$): $ C_{\text{MEMM}}\approx 8.85$.
• VOMM (minimizes volatility): $ C_{\text{VOMM}}\approx 8.22$.
• MaxEnt (maximum entropy): $ C_{\text{MaxEnt}}\approx 7.99$.

This highlights the variation of support and the fact that, in the absence of completeness, price is not unique without an additional criterion. Put differently: you wanted options, you got options.

References Link to heading

@online{Cornaciu2025Equiv,
  author   = {Cornaciu, Valentin},
  orcid    = {0000-0001-9239-7145},
  title    = {From Admissible Probabilities to Equivalent Martingale Measures},
  year     = {2026},
  date     = {2026-01-20},
  url      = {https://rcor.ro/posts/2025-12-26-from-admissible-probabilities-to-equivalent-martingale-measures/},
  abstract = {We analyze incomplete markets with three states, derive the martingale 
  measure set, and compare MEMM, VOMM, and MaxEnt as principled pricing selectors.}
}