Uncertainty Is the Product
Finance is the business of uncertainty. Every price, every return, every default — fundamentally uncertain. Probability is the mathematical language for reasoning precisely about things you cannot predict exactly.
If you have studied maths at A-level or equivalent, you have already met the basics: coin flips, dice, maybe some normal distributions. The leap to quant finance is not about learning fancier probability — it is about applying these same concepts to markets, where the stakes are real and the distributions are messier.
Sample Spaces, Events, and Axioms
Every probability setup has a sample space ( \Omega ) — the set of all possible outcomes. For a stock, the sample space is every possible future price path. An event is a subset: "the stock closes above 100" is an event.
Probability assigns a number between 0 and 1 to each event. The three axioms (credited to Kolmogorov) are:
- ( P(A) \geq 0 ) for any event ( A )
- ( P(\Omega) = 1 ) — something must happen
- For mutually exclusive events: ( P(A \cup B) = P(A) + P(B) )
Everything else follows from these. The entire edifice of quantitative finance rests on these three simple rules.
Random Variables and Distributions
A random variable maps outcomes to numbers. The stock return tomorrow is a random variable — you do not know its value yet, but you can describe its distribution.
Discrete Distributions
Used when outcomes are countable: default/no default, number of trades, up/down in a binomial tree.
The Bernoulli distribution is the simplest: probability ( p ) of success, ( 1-p ) of failure. A step in a binomial option pricing model is Bernoulli.
Continuous Distributions
Used when outcomes form a continuum (like returns).
The normal distribution ( N(\mu, \sigma^2) ) is the workhorse:
[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) ]
Log returns are often modelled as normal, which means stock prices are lognormal — always positive, skewed right. This is the assumption behind Black-Scholes.
Other Distributions You Will Meet
| Distribution | Use in Finance |
|---|---|
| Lognormal | Stock prices |
| Student's t | Heavy-tailed returns |
| Exponential | Time between events (e.g., defaults) |
| Poisson | Number of events in a period (e.g., trades, jumps) |
| Chi-squared | Hypothesis testing |
Expectation and Variance
Expected value is the probability-weighted average:
[ E[X] = \sum_i x_i P(x_i) \quad \text{(discrete)} ] [ E[X] = \int x , f(x) , dx \quad \text{(continuous)} ]
In finance, expected value is everywhere: expected return, expected payoff of an option, expected loss.
Variance measures spread:
[ \text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2 ]
The square root of variance is standard deviation — which in finance we call volatility. It is the single most important risk measure.
Conditional Probability and Bayes' Theorem
Conditional probability asks: given that event ( B ) has occurred, what is the probability of ( A )?
[ P(A|B) = \frac{P(A \cap B)}{P(B)} ]
In finance: "Given that the market dropped 3% today, what is the probability of a further 2% drop tomorrow?" Credit rating transitions, regime detection, and risk management all use conditional probabilities.
Bayes' theorem flips conditionals:
[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} ]
This is how you update beliefs with new data. Bayesian methods are increasingly popular in quant finance for parameter estimation, signal detection, and risk modelling.
The Law of Large Numbers
"Flip a fair coin many times and the proportion of heads converges to 0.5."
More formally: the sample average converges to the expected value as the sample grows. This is why diversification works — portfolio return converges to the expected return as you add more uncorrelated assets.
It is also why backtesting works: a strategy's average return over many trades approximates its true expected return. (With important caveats about overfitting — see the algorithmic trading module.)
The Central Limit Theorem
The CLT is arguably the most important theorem in statistics:
The sum (or average) of many independent random variables is approximately normally distributed, regardless of their individual distributions.
Why this matters in finance:
- Portfolio returns (sums of individual asset returns) are approximately normal — even if individual returns are not
- This justifies using the normal distribution in VaR calculations
- It underpins the mathematical basis for diversification
The CLT has limits though. When distributions have fat tails (as financial returns often do), convergence is slower. The 2008 financial crisis was partly a failure to respect those limits.
Risk-Neutral Probability
Here is where probability gets truly clever. In derivatives pricing, we do not use real-world probabilities. We use a modified set of probabilities — risk-neutral probabilities — under which the expected return of every asset equals the risk-free rate.
This is not because we believe markets are risk-neutral. It is a mathematical trick: under these modified probabilities, the fair price of a derivative is simply its discounted expected payoff. No need to estimate real-world expected returns (which are notoriously hard).
[ C = e^{-rT} E^Q[\max(S_T - K, 0)] ]
where ( E^Q ) denotes expectation under the risk-neutral measure. This is the foundation of modern derivatives pricing, and it connects probability directly to no-arbitrage theory.
Building Intuition
Probability is one of those subjects where intuition matters as much as formulas. Tossing coins, simulating random walks in Python, and playing with distributions in NumPy builds that intuition far faster than reading proofs.
Quantt covers probability from foundations through to risk-neutral pricing, with interactive simulations and coding challenges that make abstract concepts concrete. It is designed for people who want to use probability, not just prove theorems about it.
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