Sample Spaces and Sigma-Algebras

Everything in modern probability starts with one object: the probability triple $(\Omega ,\mathcal{F},P)$.

Symbol	Name	Role
$\Omega$	Sample space	All possible outcomes
$\mathcal{F}$	$\sigma$-algebra	Events we can measure
$P$	Probability measure	Assigns probabilities to events

This article covers $\Omega$ and $\mathcal{F}$. The measure $P$ comes in probability-measures-and-axioms.

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The Sample Space $\Omega$

The sample space is the set of all possible outcomes of an experiment or random process. Every element $\omega \in \Omega$ is a single, fully specified outcome --- one complete realization of the world.

Three Financial Examples

1. Single stock, one period (binary model)

$\Omega =\{{\omega }_{\text{up}},{\omega }_{\text{down}}\}$

The stock either goes up or goes down. Two outcomes. This is the simplest possible model, but already enough to derive the key ideas of risk-neutral pricing.

2. Stock over $n$ days (binomial tree)

$\Omega =\{u,d{\}}^n|\Omega |=2^n$

Each $\omega$ is a sequence of ups and downs: $\omega =(u,d,u,u,d,\dots )$. This is the Cox-Ross-Rubinstein (1979) model. With $n=252$ trading days, $|\Omega |=2^{252}\approx 7.2\times 10^{75}$ --- more outcomes than atoms in the observable universe, from a model with only two moves per day.

3. Continuous paths

$\Omega =\{\omega :[0,T]\to {\mathbb{R}}^{+}\mid \omega \text{ is continuous}\}$

Each outcome is an entire continuous price path. This is the space where Brownian motion lives, and where Black-Scholes operates. $\Omega$ is now uncountably infinite, which is why we need $\sigma$-algebras.

Engineering Analogy

Think of $\Omega$ as the space of all execution traces of a nondeterministic distributed system. Each $\omega$ is one complete trace --- every message delivered, every timeout fired, every node’s state at every instant. The system might take any of these paths; probability tells us which ones are more likely.

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Events

An event $A$ is a subset of $\Omega$: a collection of outcomes we might want to ask about.

• “The stock goes up”: $A=\{{\omega }_{\text{up}}\}$
• “A call option expires in the money”: $A=\{\omega \in \Omega :S_T(\omega )>K\}$
• “The portfolio loses more than 5%”: $A=\{\omega \in \Omega :R(\omega )<-0.05\}$

Not every subset of $\Omega$ can be an event. When $\Omega$ is uncountable, we need to be careful about which subsets we can assign probabilities to. This is where $\sigma$-algebras enter.

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The $\sigma$-Algebra $\mathcal{F}$

A $\sigma$-algebra $\mathcal{F}$ on $\Omega$ is a collection of subsets of $\Omega$ satisfying three axioms:

1. Contains the whole space: $\Omega \in \mathcal{F}$
2. Closed under complements: $A\in \mathcal{F}⟹A^c\in \mathcal{F}$
3. Closed under countable unions: $A_1,A_2,\dots \in \mathcal{F}⟹{\bigcup }_{i=1}^{\infty }{A}_i\in \mathcal{F}$

These axioms guarantee that $\mathcal{F}$ is closed under every set operation you could want: unions, intersections, complements, symmetric differences, countable combinations of these.

Why Not Just Use All Subsets?

There are two reasons we need $\sigma$-algebras rather than simply using $2^{\Omega }$ (all subsets):

The technical reason. When $\Omega$ is uncountable (like our continuous price path space), it is mathematically impossible to assign probabilities consistently to all subsets. Vitali (1905) showed that assuming the Axiom of Choice, there exist subsets of $[0,1]$ that cannot be assigned any measure compatible with translation invariance. The $\sigma$-algebra restricts us to the “nice” subsets where probability is well-defined.

THE reason for finance. $\sigma$-algebras model information. This is not just a technical convenience --- it is the conceptual core of the entire framework. A $\sigma$-algebra ${\mathcal{F}}_t$ represents what you know at time $t$. Events in ${\mathcal{F}}_t$ are questions you can answer; events outside it are questions you cannot yet resolve.

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Filtrations --- Information Evolving Over Time

A filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ is a family of $\sigma$-algebras that grows over time:

${\mathcal{F}}_0\subseteq {\mathcal{F}}_1\subseteq \cdots \subseteq {\mathcal{F}}_T$

Information only grows. You cannot un-learn something.

Concrete Example: Two-Day Binomial Model

Let $\Omega =\{uu,ud,du,dd\}$ where the first letter is day 1 and the second is day 2.

At time 0 (before any trading):

${\mathcal{F}}_0=\{\varnothing ,\Omega \}$

You know nothing. The only questions you can answer are trivial: “did something happen?” ($\Omega$) and “did nothing happen?” ($\varnothing$). This $\sigma$-algebra has 2 elements.

At time 1 (after day 1 closes):

${\mathcal{F}}_1=\{\varnothing ,\{uu,ud\},\{du,dd\},\Omega \}$

You observed whether day 1 went up or down, but you cannot yet distinguish $uu$ from $ud$. The atoms of ${\mathcal{F}}_1$ are $\{uu,ud\}$ (“day 1 was up”) and $\{du,dd\}$ (“day 1 was down”). This $\sigma$-algebra has 4 elements.

At time 2 (after day 2 closes):

${\mathcal{F}}_2=2^{\Omega }=\{\varnothing ,\{uu\},\{ud\},\{du\},\{dd\},\{uu,ud\},\{uu,du\},\dots ,\Omega \}$

You observed everything. Every outcome is distinguishable. This $\sigma$-algebra has 16 elements ($2^4$, since $|\Omega |=4$).

The nesting ${\mathcal{F}}_0\subseteq {\mathcal{F}}_1\subseteq {\mathcal{F}}_2$ captures the flow of information through time.

Why Filtrations Matter in Finance

Filtrations are not a formalism you tolerate --- they are the concept that makes derivative pricing coherent:

• Derivative pricing at time $t$ depends on ${\mathcal{F}}_t$. The price of an option reflects what the market knows now, not what it will know tomorrow.
• An adapted process is one that only uses available information: $X_t$ is ${\mathcal{F}}_t$-measurable. A trading strategy must be adapted --- you cannot use tomorrow’s prices to make today’s trades (no lookahead bias).
• A martingale satisfies $E[X_{t+1}\mid {\mathcal{F}}_t]=X_t$: the best forecast of tomorrow’s value, given everything known today, is today’s value. Under the risk-neutral measure, discounted asset prices are martingales. This is the theoretical engine of derivatives pricing.

Engineering Analogy

A filtration is a causal ordering. In distributed systems, Lamport timestamps define what events are causally before others. A filtration does the same thing for information: ${\mathcal{F}}_t$ is the set of all events that are causally accessible at time $t$. An adapted process is one that respects causal ordering --- no reading from the future.

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Two Measures, One Structure: A Preview

Here is the key structural insight that the rest of quantitative finance builds on.

Given the same $(\Omega ,\mathcal{F})$, we can define two different probability measures:

Measure	Name	Meaning
$P$	Physical (real-world) measure	How outcomes actually occur
$Q$	Risk-neutral measure	Hypothetical measure where all assets earn the risk-free rate in expectation

The physical measure $P$ is what you’d estimate from historical data. The risk-neutral measure $Q$ is what the market implies through prices. They agree on what’s possible (same null sets) but disagree on what’s likely.

Girsanov’s theorem provides the bridge between $P$ and $Q$. The fundamental insight of modern derivatives pricing:

Pricing = computing expectations under $Q$, not $P$.

This is why the $(\Omega ,\mathcal{F},P)$ framework matters: it cleanly separates the structure (what can happen) from the measure (how likely each thing is), and lets us swap measures while keeping the structure fixed.

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One-Period Binomial Model: $P$ vs $Q$ in Action

Let’s make this concrete with the simplest possible example.

Setup

Parameter	Value
$S_0$ (initial stock price)	100
$S_{\text{up}}$	110
$S_{\text{down}}$	90
$K$ (strike price)	105
$r$ (risk-free rate)	0.02
$p$ (real-world probability of up)	0.60

The Key Calculation

The risk-neutral probability $q_{\text{up}}$ is the unique value that makes the discounted stock price a martingale:

$E^Q\left[\frac{S_1}{1+r}\right]=S_0$

Solving:

$q_{\text{up}}=\frac{S_0(1+r)-S_{\text{down}}}{S_{\text{up}}-S_{\text{down}}}=\frac{100\times 1.02-90}{110-90}=\frac{12}{20}=0.60$

The call option payoff is $C_1=\max (S_1-K,0)$:

• If up: $\max (110-105,0)=5$
• If down: $\max (90-105,0)=0$

Risk-neutral price:

$C_0=\frac{1}{1+r}{E}^Q[C_1]=\frac{q_{\text{up}}\cdot 5+(1-q_{\text{up}})\cdot 0}{1.02}=\frac{0.60\times 5}{1.02}\approx 2.94$

Real-world expected payoff (discounted):

$\frac{p\cdot 5+(1-p)\cdot 0}{1.02}=\frac{0.60\times 5}{1.02}\approx 2.94$

In this example, $p$ and $q$ happen to coincide numerically. Change the parameters and they diverge --- but the price is always determined by $q$, never by $p$.

Python Implementation

"""One-period binomial model: P vs Q measures."""
 
 
def binomial_one_period(
    *,
    s0: float,
    s_up: float,
    s_down: float,
    strike: float,
    r: float,
    p_real: float,
) -> dict[str, float]:
    # Risk-neutral probability (unique martingale measure)
    q_up = (s0 * (1 + r) - s_down) / (s_up - s_down)
    q_down = 1 - q_up
 
    # Verify no-arbitrage condition: 0 < q_up < 1
    assert 0 < q_up < 1, f"No-arbitrage violated: q_up={q_up}"
 
    # Option payoffs
    payoff_up = max(s_up - strike, 0)
    payoff_down = max(s_down - strike, 0)
 
    # Pricing under Q (the CORRECT price)
    price_q = (q_up * payoff_up + q_down * payoff_down) / (1 + r)
 
    # Expected payoff under P (NOT a valid price)
    expected_p = (p_real * payoff_up + (1 - p_real) * payoff_down) / (1 + r)
 
    # Verify martingale property: E^Q[S1/(1+r)] = S0
    martingale_check = (q_up * s_up + q_down * s_down) / (1 + r)
 
    return {
        "q_up": q_up,
        "q_down": q_down,
        "price_Q": price_q,
        "expected_P": expected_p,
        "martingale_check": martingale_check,
        "S0": s0,
    }
 
 
result = binomial_one_period(
    s0=100, s_up=110, s_down=90, strike=105, r=0.02, p_real=0.60
)
 
print("Risk-neutral probability (q_up):", f"{result['q_up']:.4f}")
print("Option price (Q-measure):       ", f"{result['price_Q']:.4f}")
print("Expected payoff (P-measure):    ", f"{result['expected_P']:.4f}")
print("Martingale check E^Q[S1/(1+r)]: ", f"{result['martingale_check']:.4f}")
print("S0:                             ", f"{result['S0']:.4f}")
print(f"\nMartingale property holds: {abs(result['martingale_check'] - result['S0']) < 1e-10}")

The critical insight: real-world belief ($p=0.60$) is IRRELEVANT for pricing. Two traders can completely disagree about whether the stock will go up (one thinks $p=0.9$, the other $p=0.3$) and yet agree on the option price, because the price is determined by $q$, which is derived purely from the current stock price, the possible outcomes, and the risk-free rate.

The risk-neutral probability $q_{\text{up}}$ is the unique probability that makes the discounted stock price a martingale under $Q$. It encodes the no-arbitrage condition, not anyone’s beliefs.

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Historical Note

The axiomatic framework $(\Omega ,\mathcal{F},P)$ was established by Andrey Kolmogorov in 1933, in his monograph Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability). Before Kolmogorov, probability was a collection of techniques without a unified foundation. His axiomatization --- built on Lebesgue’s measure theory --- put probability on the same rigorous footing as the rest of modern mathematics and made it possible to define conditional expectation, stochastic processes, and everything that followed.

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Checkpoint Questions

1. In the binomial model $\Omega =\{u,d{\}}^n$, describe the elements (atoms) of ${\mathcal{F}}_1$. How many sets does ${\mathcal{F}}_1$ contain?

2. Why can’t we always use $2^{\Omega }$ as our $\sigma$-algebra? Give both the technical reason and the financial reason.

3. At Gottex, you optimized no-arbitrage yield curves subject to constraints. In the language of this article, what role does the no-arbitrage condition play? How does it relate to the existence of a risk-neutral measure $Q$?

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Next: probability-measures-and-axioms --- Kolmogorov’s axioms and how to construct measures on the spaces we’ve just defined.