Probability Space

Motivation: why this matters in quant finance

The pricing formula

V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[H]

looks like a single expectation, but it hides three modelling choices. There must be a set of possible states $\Omega$ , a collection of events $\mathcal{F}$ that can be assigned probabilities, and a probability law $\mathbb{Q}$ under which the payoff $H$ can be averaged. The same payoff can also be studied under the real-world measure $\mathbb{P}$ for forecasting and risk. Risk-neutral pricing only makes sense because both measures live on a shared event structure.

Bertsekas begins probability with a practical modelling warning: a sample space must contain mutually exclusive, collectively exhaustive outcomes, and it should include enough detail for the questions being asked without adding irrelevant detail. In finance this is not cosmetic. A one-period binomial model, a credit portfolio, and a Brownian path model answer different questions because they choose different $\Omega$ 's and different event collections.

Without the probability-space triplet:

$\mathbb{P}(A)$ is not defined because there is no agreed event $A$ .
$\mathbb{E}[X]$ is not defined because there is no probability law to integrate against.
Statements such as "discounted prices are martingales under $\mathbb{Q}$ " have no formal domain.

This lesson is the static foundation for random variables, expectation and variance, conditional expectation, and the later filtered framework used by Brownian motion.

The informal idea

A probability space answers three questions.

What can happen? The sample space $\Omega$ is the list, set, or path space of all possible outcomes.
Which questions are legitimate? The event collection $\mathcal{F}$ says which subsets of $\Omega$ can be assigned probabilities.
How much probability is assigned? The probability measure $\mathbb{P}$ gives each event a number in $[0,1]$ in a way that respects disjoint unions.

For a die roll, the model is tiny: $\Omega = \{1,2,3,4,5,6\}$ , events are subsets of $\Omega$ , and a fair die gives each singleton probability $1/6$ . For a stock path, $\Omega$ may be a set of continuous functions $[0,T] \to \mathbb{R}$ , and events are questions like "did the path ever cross the barrier?" or "is $S_T > K$ ?" The same structure survives, but the sample space is no longer enumerable.

The key modelling point is that

\Omega

is chosen for a purpose. If the question is the terminal payoff of a European call, it may be enough to model

S_T

. If the question is a barrier option, the whole path matters. If the question is whether a trading strategy is allowed, the timing of information matters and the probability space must later be enriched into a filtration.

Formal definitions

Sample space

The sample space

\Omega

is a non-empty set of mutually exclusive outcomes. Exactly one

\omega \in \Omega

is realised when the experiment is performed. A subset

A \subseteq \Omega

is an event when it is included in the event collection used by the model.

Event collection

For finite or countable models, it is often harmless to take every subset of

\Omega

as an event. For uncountable models, some pathological subsets cannot be assigned probabilities consistently. Modern probability therefore uses a sigma-algebra

\mathcal{F}

: a collection of subsets of

\Omega

closed under complements and countable unions. The separate sigma-algebras note develops this object in detail.

Probability law

A probability measure on

(\Omega,\mathcal{F})

is a function

\mathbb{P}:\mathcal{F}\to[0,1]

satisfying:

Non-negativity: $\mathbb{P}(A) \ge 0$ for every $A\in\mathcal{F}$ .
Normalisation: $\mathbb{P}(\Omega)=1$ .
Countable additivity: if $A_1,A_2,\ldots$ are disjoint events, then

\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty}\mathbb{P}(A_n).

The triple

(\Omega,\mathcal{F},\mathbb{P})

is a probability space.

Key properties

Finite models are determined by singleton probabilities

If $\Omega=\{\omega_1,\ldots,\omega_n\}$ and the probabilities $p_i=\mathbb{P}(\{\omega_i\})$ are specified with $p_i\ge0$ and $\sum_i p_i=1$ , then every event has probability

\mathbb{P}(A)=\sum_{\omega_i\in A} p_i.

This is the arithmetic behind binomial trees and finite-state credit models.

Disjoint additivity drives all probability algebra

For disjoint $A$ and $B$ ,

\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B).

From this follow $\mathbb{P}(\emptyset)=0$ , $\mathbb{P}(A^c)=1-\mathbb{P}(A)$ , monotonicity, and inclusion-exclusion:

\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B).

The pair

(\Omega,\mathcal{F})

says what can be discussed; the measure says how likely those events are. A real-world measure

\mathbb{P}

and a risk-neutral measure

\mathbb{Q}

may assign different probabilities to the same event

A\in\mathcal{F}

. This is why change of measure changes probabilities without changing the underlying path space.

Model choice is part of the mathematics

Bertsekas stresses that the sample space should distinguish the outcomes relevant to the modeller. A model for a one-day return distribution may not contain intraday paths. A model for a barrier option must. Probability theory does not choose $\Omega$ for you; it tells you how to reason once the model is chosen.

Worked examples

Example 1: a one-period pricing model

Let $S_0=100$ and suppose $S_T$ is either $110$ or $90$ . Take

\Omega=\{u,d\},\qquad \mathcal{F}=2^{\Omega}.

A physical estimate might be $\mathbb{P}(u)=0.6$ . With zero interest for simplicity, the risk-neutral probability $q=\mathbb{Q}(u)$ must satisfy

100=q\cdot110+(1-q)\cdot90,

so $q=1/2$ . Both $\mathbb{P}$ and $\mathbb{Q}$ are probability measures on the same $(\Omega,\mathcal{F})$ ; they disagree only on weights. A call with strike $100$ has time-zero price

\mathbb{E}^{\mathbb{Q}}[(S_T-100)^+]=q\cdot10+(1-q)\cdot0=5.

Example 2: choosing enough detail

If a desk only needs the terminal price bucket $S_T\in\{\text{down},\text{flat},\text{up}\}$ , a three-state space may be enough. If the desk prices a knock-out option, that same $\Omega$ is too coarse: the terminal bucket does not say whether the path crossed the barrier before maturity. The probability space must be rebuilt around paths or at least around states that record barrier hits.

Example 3: probability laws from area

For a continuous model, probabilities may be assigned by area or density. If Romeo and Juliet arrive independently and uniformly during an hour, $\Omega=[0,1]^2$ . The event "they arrive within 15 minutes of each other" is a diagonal band in the unit square. Its probability is the area of that band. This is the same measure idea used later when a density $f_X$ assigns probability to intervals through integration.

Common confusions and pitfalls

"The sample space is unique." No. The same physical situation admits many sample spaces. The right one depends on the questions the model must answer.

"An event is any sentence about the world." In the model, an event is a set of outcomes in

\mathcal{F}

. If the sample space does not record a detail, the model cannot ask about it.

"The probability law is the model." It is only one part. A probability law without a sample space and event collection has no domain.

"A zero-probability event is impossible." In continuous models, individual outcomes often have probability zero while still being possible. What matters is whether the event is empty, not whether its probability is zero.

"Risk-neutral and real-world probabilities require different sample spaces." Usually they do not. They are different measures on the same measurable space, which is what lets pricing compare

\mathbb{P}

-estimation and

\mathbb{Q}

-valuation.

Where this goes next

Sigma-Algebras: Explains why the event collection must be closed under countable logical operations.
Random Variables: Turns outcomes into numerical quantities such as prices, returns, and payoffs.
Expectation and Variance: Defines averaging over a probability space and measuring dispersion.
Independence and Conditioning: Builds probability laws after partial information is revealed.
Filtrations and Information: Adds time-indexed information to the static probability space.
Change of Measure: Studies how $\mathbb{P}$ and $\mathbb{Q}$ relate on the same event structure.

References

Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific. Ch. 1 §1.1 (Sets), §1.2 (Probabilistic Models).