Probability in Finance: What We Know, What We Price and What We Assume

Financial modelling begins with a question about what is known, what is uncertain and which probabilities are relevant. A return forecast, an option value and a stress loss may use the same market variables while answering different questions. Treating their probability assumptions as interchangeable produces convincing calculations with the wrong interpretation.

This reference develops a common language for those calculations. It connects events and distributions to conditional expectation, information timing and convergence. Its purpose is to make model assumptions inspectable; it makes no market call.

Scenarios, events and observable quantities

A probability space is written

$$(\Omega,\mathcal F,\mathbb P).$$

Here $\Omega$ is the set of possible scenarios, $\mathcal F$ is the collection of events to which probabilities can be assigned, and $\mathbb P$ is the probability measure. A scenario can be a complete future path of prices, rates and defaults. An event is a question with a yes-or-no answer, such as whether a loss exceeds a threshold.

The event collection is a sigma-algebra: it contains the whole scenario space and is closed under complements and countable unions. This structure ensures that combining meaningful questions produces another meaningful question. Probability is nonnegative, assigns one to the entire space and adds across mutually exclusive events.

A random variable $X$ maps each scenario to a numerical outcome. Measurability means that questions about this number, such as $X\leq x$, belong to the event collection. A random variable is therefore more precise than an unspecified “random number”: its relationship to observable events is part of its definition.

A loss distribution may be discrete, continuous or mixed. A portfolio with no default in most scenarios and a range of positive losses after default has a point mass at zero and a continuous component elsewhere. A density alone cannot represent the point mass. Conversely, a continuously distributed variable can have zero probability at every individual value while having positive probability over an interval.

Distribution functions and tail questions

The cumulative distribution function is

$$F_X(x)=\mathbb P(X\leq x).$$

It is nondecreasing and right-continuous. Its limiting values are zero and one at the two extremes. For $a<b$, the interval probability is $F_X(b)-F_X(a)$.

If a density $f_X$ exists, interval probabilities are areas under it:

$$\mathbb P(a<X\leq b)=\int_a^b f_X(x)\,dx.$$

The height of a density is not itself a probability. This distinction matters when inspecting a fitted loss curve or comparing distributions with different units. A density expressed per percentage point has different numerical heights from the same density expressed per unit return, even though the underlying probabilities agree.

For a loss variable $L$, a quantile-based risk threshold at confidence level $\alpha$ can be defined as

$$\operatorname{VaR}_\alpha(L)=\inf\{\ell:F_L(\ell)\geq\alpha\}.$$

In words: Find the smallest loss threshold whose cumulative probability meets or exceeds the chosen confidence level.

The infimum definition also handles distributions with jumps. It prevents the casual use of an inverse function where an ordinary inverse does not exist. The threshold says where the tail begins; it does not describe the severity of all losses beyond it.

Expectations retain the whole weighting scheme

For an integrable random variable, expectation is a probability-weighted average:

$$\mathbb E_{\mathbb P}[X]=\int_\Omega X(\omega)\,d\mathbb P(\omega).$$

In words: Average the outcomes using the probabilities assigned to their possible scenarios.

In a discrete case this becomes a weighted sum. In a continuous case with density it becomes an integral over possible values. The probability measure is part of the calculation, even when suppressed in notation.

Consider the retained teaching example in which loss takes values 0, 1 and 2 units with probabilities

Teaching loss distribution with the original outcome probabilities

Expected loss is

$$\mathbb E[L]=0(0.7)+1(0.2)+2(0.1)=0.4.$$

No scenario has a loss of 0.4 units. Expected loss is an average over repeated scenarios, not necessarily the most likely realised outcome. The same caution applies to expected terminal wealth and expected default counts.

Another teaching density is $f(x)=2x$ for $0\leq x\leq1$. It integrates to one, and

$$\mathbb E[X]=\int_0^1 2x^2\,dx=\frac23.$$

The higher mean than a uniform distribution’s one-half reflects the greater weight assigned to high outcomes. This example makes the weighting mechanism visible without adding market assumptions.

Finite expectation is a substantive condition. Heavy-tailed distributions may have a finite mean but infinite variance, or neither. A formula using sample variance does not become reliable merely because software returns a number.

Information arrives through a filtration

A filtration $(\mathcal F_t)$ records information available by each time $t$. For $s\leq t$, it satisfies $\mathcal F_s\subseteq\mathcal F_t$. An adapted process $X_t$ is measurable with respect to the information available at its own time.

This is the mathematical counterpart of avoiding look-ahead bias. A strategy can use the price just observed; it cannot legitimately use tomorrow’s closing price to decide today’s trade. Revised macroeconomic data, final constituent lists and corrected historical records can all introduce future information into an apparently historical analysis.

For trading gains, predictable processes provide a more precise timing convention: the position over the next price movement is selected using information available before that movement. A simple strategy chooses holdings at time $t_i$ and keeps them until $t_{i+1}$. Its gain is the chosen holding multiplied by the subsequent price change.

A stopping time $\tau$ is a random time for which the event $\{\tau\leq t\}$ can be determined from $\mathcal F_t$. First passage of a continuously observed price through a barrier is the standard example. The time of the highest price over a future year generally is not a stopping time: identifying it requires seeing the remainder of the year.

Conditional expectation is an information-dependent estimate

The conditional expectation $\mathbb E[X\mid\mathcal G]$, where $\mathcal G$ is an information set, is a $\mathcal G$-measurable random variable that preserves averages over events in $\mathcal G$. For square-integrable variables it is the best mean-square predictor using that information.

The tower property says that conditioning first on a richer information set and then on a poorer one gives the same result as conditioning directly on the poorer set. For $\mathcal H\subseteq\mathcal G$,

$$\mathbb E[\mathbb E[X\mid\mathcal G]\mid\mathcal H] =\mathbb E[X\mid\mathcal H].$$

In words: Averaging a more informed estimate using less information gives the estimate based directly on that lesser information.

In a valuation engine, this supports backward induction: future values can be calculated using future information and then averaged back to today’s information. In forecasting, it clarifies why the forecast changes when new observations arrive without implying that the original forecasting method was internally inconsistent.

Physical probabilities and pricing probabilities

The physical measure $\mathbb P$ describes a model of real-world outcomes. A pricing measure $\mathbb Q$ is associated with a chosen numeraire and no-arbitrage valuation assumptions. It is not a prediction that investors actually expect every asset to earn the risk-free rate.

With constant risk-free rate $r$, maturity $T$ and payoff $H_T$, a basic pricing expression is

$$V_t=\mathbb E_{\mathbb Q}[e^{-r(T-t)}H_T\mid\mathcal F_t].$$

Its validity depends on the model and contractual setting. Stochastic rates require stochastic discounting. Dividends, collateral terms, default and incompleteness can require further structure. A unique pricing measure is a stronger conclusion than mere absence of arbitrage.

When $\mathbb Q$ is absolutely continuous with respect to $\mathbb P$, a likelihood ratio $Z=d\mathbb Q/d\mathbb P$ reweights scenarios. For suitable $X$,

$$\mathbb E_{\mathbb Q}[X]=\mathbb E_{\mathbb P}[ZX].$$

Absolute continuity prevents an event impossible under $\mathbb P$ from receiving positive probability under $\mathbb Q$. Equivalence requires the converse as well. Neither condition says that the two measures assign the same probabilities to ordinary events.

Convergence must match the intended use

Different convergence claims support different uses

Almost-sure convergence and mean-square convergence each imply convergence in probability, but neither is universally stronger than the other. Convergence in probability implies convergence in distribution. Stronger implications require additional assumptions.

The distinction prevents two common mistakes. First, distributional convergence of a scaled random walk to Brownian motion does not automatically identify a pathwise limit on the original probability space. Second, convergence of simulated terminal distributions does not guarantee accurate path-dependent barrier or stopping calculations.

Passing a limit through an expectation also requires justification. Dominated convergence uses an integrable dominating variable; monotone convergence applies to increasing nonnegative variables. Uniform integrability is often the missing condition when very rare, increasingly large outcomes prevent expected values from following apparently well-behaved pathwise limits.

A conditional-loss example

Consider an explicitly illustrative two-state economy. The normal state has probability 0.8 and the stressed state probability 0.2. Suppose a particular loss event has conditional probability 1% in the normal state and 8% in the stressed state. The unconditional probability is

$$P(\text{loss event})=0.8(0.01)+0.2(0.08)=0.024.$$

It is 2.4%, not the simple average of 1% and 8%. The state weights matter. If new information establishes that the stressed state has occurred, the relevant conditional probability becomes 8%; the earlier unconditional number was not necessarily wrong when it was made.

The same example illustrates a distinction between an event and information about that event. A signal can increase the probability of stress without revealing the state with certainty. Bayesian updating then changes the weights using the signal’s reliability. Treating every warning signal as definitive stress would substitute a different information model.

For several exposures, a shared state creates dependence. Even if their losses are independent conditional on the state, they become dependent after averaging over the uncertain state. This is a simple explanation of why portfolio losses can cluster without one obligor directly causing another’s default.

Rare outcomes can defeat an apparently converged average

Let $U$ be uniformly distributed on $(0,1)$, and define the mathematical example $X_n=n\mathbf1_{\{U<1/n\}}$. For almost every fixed $U>0$, eventually $U$ exceeds $1/n$, so $X_n$ becomes zero. Thus $X_n$ converges almost surely to zero.

Its expectation nevertheless remains one for every $n$: the probability $1/n$ is multiplied by the value $n$. The expectation does not converge to the expectation of the pathwise limit. Increasingly rare but increasingly severe values prevent the required uniform integrability.

This is not a market scenario. It is a precise counterexample to the claim that a well-behaved-looking sequence of outcomes automatically has well-behaved expected values. In a risk engine, analogous behaviour can arise when importance-sampling weights are unstable or a heavy-tailed model produces rare observations that dominate the average.

The operational response is to inspect the tail contribution and moment assumptions. A repeated simulation may show little variation simply because none of its runs encountered the relevant tail. Stability of a displayed average is evidence about those runs, not proof that the expected value is reliably identified.

Independence is stronger than zero correlation

Correlation measures linear co-movement when the relevant second moments exist. Independence means the joint distribution factorises across all events. The latter implies zero covariance under finite moments, but the reverse implication generally fails.

For an illustrative symmetric variable $X$, the variable $X^2$ is entirely determined by $X$, yet its covariance with $X$ can be zero because the relevant odd moments cancel. Calling the variables independent would therefore be incorrect. This distinction matters when a model replaces nonlinear dependence with a correlation matrix and then interprets the absence of linear association as absence of common risk.

Putting the specification into use

A useful model record states the scenario variables, probability measure, information timestamps, payoff or loss definition and required convergence criterion. It also identifies which moments exist and which numerical approximations are being used.

These details turn probability from decorative notation into a control on interpretation. A forecast should answer a physical-probability question; a price should identify its pricing assumptions; a backtest should respect information timing. Keeping those distinctions explicit is the foundation for reliable quantitative work.