Stochastic Calculus: The Math Behind Random Markets
Chapter 4 is the toolbox chapter. Before we can price options, we need the mathematical machinery to handle random variables properly. The centerpiece is Ito’s lemma, the rule that replaces ordinary calculus when things are random. Wilmott goes out of his way to make this accessible, and honestly, it is not as scary as it sounds.
Most academic papers in finance bury the reader in mathematical rigor. Wilmott takes the opposite approach: understand the tools well enough to use them, and do not worry about the proofs unless you really want to.
Starting With Coin Tosses
Wilmott builds up to Brownian motion from a simple coin-tossing game. Throw a coin: heads, you get $1; tails, you lose $1. After six tosses, you might be up, down, or even.
Let $R_i$ be the outcome of the ith toss (+1 or -1). The total after i tosses is $S_i = R_1 + R_2 + \ldots + R_i$. Simple.
Three properties of this game turn out to be fundamental:
The Markov Property
The future depends only on where you are now, not how you got there. After 5 tosses, knowing $S_5$ is all you need to predict $S_6$. It does not matter if you got there by HHHHH or HTHTH or any other path. The random walk has no memory beyond its current state.
The Martingale Property
Your expected future winnings equal your current winnings. After 5 tosses, the expected value of $S_6$ is just $S_5$, because the next toss is equally likely to go either way. In financial terms, a fair game.
Quadratic Variation
Add up the squares of all the increments: $(S_1 - S_0)^2 + (S_2 - S_1)^2 + \ldots$. Since each increment is either +1 or -1, each squared increment is 1. After i tosses, the quadratic variation is exactly i. This boring-sounding property turns out to be the key to understanding why stochastic calculus works differently from regular calculus.
From Coin Tosses to Brownian Motion
Now Wilmott does something clever. He speeds up the game. Instead of 6 tosses, play n tosses in the same total time t. Each toss takes time $t/n$, and the bet size is $\sqrt{t/n}$ instead of $1.
Why that specific bet size? Because it keeps the quadratic variation equal to t regardless of how many tosses you play. The time step shrinks like $1/n$ but the bet size only shrinks like $1/\sqrt{n}$.
Now let $n$ go to infinity. The resulting continuous random walk is Brownian motion, denoted $X(t)$.
Its key properties:
- Continuous paths (no jumps)
- Markov: future depends only on present
- Martingale: expected future value equals current value
- Normality: over any time interval, increments are normally distributed with mean zero and variance equal to the time interval
- Quadratic variation equals elapsed time
That last point is crucial. In ordinary calculus, $dx^2$ is negligibly small compared to $dx$. In stochastic calculus, $dX^2 = dt$. This is not negligible. This is the entire reason stochastic calculus exists.
Stochastic Differential Equations
Instead of writing a regular differential equation like $dS = a(S) , dt$, we write:
$dS = a(S) , dt + b(S) , dX$
Think of it as: “the change in S has a predictable part ($a \cdot dt$) and a random part ($b \cdot dX$).” The bit in front of $dt$ is deterministic. The bit in front of $dX$ tells you how much randomness there is.
The stochastic integral is non-anticipatory: the function being integrated is evaluated at the left-hand point of each time step. In financial terms, you choose your portfolio and only then does the stock price move. You cannot use future information.
Ito’s Lemma: The Main Event
Here is the question that drives everything. If S follows a random process and you have some function $F(S)$, like an option value, what random process does F follow?
In regular calculus, if $F = X^2$, then $dF = 2X , dX$. Simple chain rule.
In stochastic calculus, this is wrong. Because $dX^2 = dt$ (not zero), you get an extra term:
$dF = 2X , dX + dt$
This is Ito’s lemma for the simplest case. The general version for a function $F(X)$ of Brownian motion is:
$dF = \frac{dF}{dX} dX + \frac{1}{2} \frac{d^2F}{dX^2} dt$
That extra $\frac{1}{2} \frac{d^2F}{dX^2} dt$ term is what makes stochastic calculus different from regular calculus. It exists because the quadratic variation of Brownian motion is not zero.
The Practical Rule of Thumb
Wilmott gives a shortcut that works in practice. Just use a regular Taylor series expansion but apply the rule:
$dX^2 = dt$
This is technically not rigorous, but it gives the correct result and you are unlikely to go wrong using it. He uses this trick throughout the rest of the book.
Why Ito Matters for Finance
Imagine you have a stock price S following:
$dS = \mu S , dt + \sigma S , dX$
And an option $V(S, t)$ on that stock. Ito’s lemma tells you exactly what stochastic differential equation V satisfies:
$dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} dt$
The randomness in V comes entirely from the $dS$ term. The amount of randomness in the option relative to the stock is $\frac{\partial V}{\partial S}$. This ratio will give us a recipe for eliminating randomness by buying an option and selling the right amount of stock. That is delta hedging, and that is Black-Scholes. But we are getting ahead of ourselves.
Some Example Random Walks
Wilmott finishes the chapter with four random walks that show up throughout finance:
Brownian Motion with Drift
$dS = \mu , dt + \sigma , dX$
A random walk that trends upward or downward. Problem: S can go negative. Not great for stock prices.
The Lognormal Random Walk
$dS = \mu S , dt + \sigma S , dX$
This is the important one. Both drift and randomness scale with S, so the stock can never reach zero or go negative. Apply Ito to $\log(S)$ and you find that $\log(S)$ follows a simple Brownian motion with drift. This is why it is called “lognormal.”
This is the standard model for stocks, currencies, and commodities.
Mean-Reverting Walk (Vasicek Model)
$dS = (\nu - \mu S) , dt + \sigma , dX$
If S gets too high, the drift pushes it back down. If S gets too low, the drift pushes it up. S tends to wander around the level $\nu/\mu$. Good for interest rates, which tend to fluctuate within a range rather than trending forever. Downside: S can still go negative.
Square Root Mean-Reverting Walk (CIR Model)
$dr = (\nu - \mu r) , dt + \sigma \sqrt{r} , dX$
Similar to Vasicek but the randomness scales with $\sqrt{r}$. As r approaches zero, the randomness shrinks. If $\nu$ is large enough, the process cannot reach zero at all, keeping interest rates positive. This is the Cox-Ingersoll-Ross model.
The Takeaway
Chapter 4 gives us one essential tool: Ito’s lemma. It tells us how functions of random variables behave. The practical recipe:
- Write down the Taylor series expansion of your function
- Replace $dX^2$ with $dt$
- Drop higher-order terms
That is it. With this tool and the lognormal random walk from Chapter 3, we have everything we need to derive the Black-Scholes equation in the next chapter.
The key insight: stochastic calculus is just regular calculus with one extra rule ($dX^2 = dt$). Once you internalize that rule, the rest follows naturally.
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