Financial Mathematics · Lecture 1 of 12

Probability: The Language of Uncertainty

Before we can price anything, we need a precise way to talk about "what might happen". Sample spaces, random variables, expectation and the law that makes randomness predictable in bulk.

Black–Scholes is, at heart, a statement about averages. The price of an option will turn out to be an expected value of its payoff, computed under a special probability measure and discounted. So the series starts where every derivation eventually leads back to: probability.

Sample spaces, events and probability

A probability model has three ingredients, written $(\Omega, \mathcal{F}, \mathbb{P})$:

• $\Omega$, the sample space: every possible outcome. For one coin flip, $\Omega = \{H, T\}$. For tomorrow's stock price, $\Omega$ is a continuum.
• $\mathcal{F}$, the events: the questions we're allowed to ask ("is the price above 100?"). Formally a σ-algebra, closed under complements and countable unions.
• $\mathbb{P}$, the probability measure: assigns each event a number in $[0,1]$, with $\mathbb{P}(\Omega)=1$ and additivity over disjoint events.

Random variables and distributions

A random variable $X:\Omega\to\mathbb{R}$ attaches a number to each outcome, such as the payoff of a bet or tomorrow's return. Its distribution is summarised by the cumulative distribution function

Discrete variables have a probability mass function $p(k)=\mathbb{P}(X=k)$. Continuous ones have a density $f(x)=F'(x)$ with $\mathbb{P}(a \lt X\le b)=\int_a^b f(x)\,dx$.

Expectation and variance

The expectation is the probability-weighted average:

The last formula, the "law of the unconscious statistician", is the one we'll use constantly: an option's payoff is $g(S_T)$, and its value will be $e^{-rT}\mathbb{E}^{\mathbb{Q}}[g(S_T)]$.

Expectation is linear: $\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]$ always, even when $X$ and $Y$ are dependent. The variance measures spread:

For independent variables variances add: $\operatorname{Var}(X+Y)=\operatorname{Var}X+\operatorname{Var}Y$. This one fact is why volatility scales with $\sqrt{t}$, as we'll see shortly.

The coin-flip model of a stock

Let each day the price move up or down by one unit with probability $p$ and $1-p$. Let $\xi_i=\pm1$ be the $i$-th move and $X_n=\sum_{i=1}^n \xi_i$ the total. The number of up-moves $U$ is binomial:

Each $\xi_i$ has variance $1$ when $p=\tfrac12$, so by independence $\operatorname{Var}(X_n)=n$ and the typical size of $X_n$ is $\sqrt{n}$, not $n$. Ten times more days means only about $3.16\times$ more spread. Keep this in mind: it becomes $\sigma\sqrt{T}$ in the Black–Scholes formula.

Two limit theorems

The law of large numbers: the sample average of i.i.d. variables converges to their mean, $\frac1n\sum X_i\to\mathbb{E}X$. This is why Monte Carlo pricing (Lecture 12) works.

The central limit theorem: if the $X_i$ are i.i.d. with mean $\mu$ and variance $\sigma^2$, then

In the figure, drag $n$ upward and the jagged binomial bars snap onto the bell curve. That's the CLT at work. Many tiny, independent shocks add up to something Gaussian whatever the shape of each shock. This is the deepest justification for modelling (log) returns as normal, and it's how Brownian motion arises from coin flips in Lecture 6.