Why Stock Prices Move Randomly (And Why That Matters)
Chapter 3 is where the real modeling begins. Wilmott takes us from “stock prices look random” to “here is the specific mathematical model for that randomness.” By the end of this chapter, we have the fundamental equation that drives almost everything in quantitative finance.
Three Ways People Try to Understand Markets
Wilmott outlines three approaches to analyzing financial markets:
Fundamental analysis tries to figure out what a company is really worth by studying balance sheets, management, patents, competitors. Sounds smart, but it is extremely hard and, as Keynes supposedly said, “the market can stay irrational longer than you can stay solvent.” Even a perfect model of a company’s value does not help you if the market disagrees.
Technical analysis ignores everything about the company and just looks at price patterns. Draw trendlines, spot shapes, make predictions. Most academic evidence suggests this does not work.
Quantitative analysis is the approach this book takes. Treat prices as random, choose the best models for that randomness, and build a rigorous framework on top. This is what has powered portfolio theory, derivatives pricing, and risk management for the past 50+ years.
Why Randomness Matters: Jensen’s Inequality
Before building the model, Wilmott shows why we cannot just ignore randomness. The example is elegant.
A stock is at $100 today. In one year it could be $50 or $150, equally likely. What is a call option with strike 100 worth?
Approach 1: The expected stock price is $100 (average of 50 and 150). The call payoff at $100 is zero. So the option is worth zero?
Approach 2: If the stock goes to $50, the call pays zero. If it goes to $150, the call pays $50. The expected payoff is $25.
These two approaches give different answers because of Jensen’s inequality. For any convex function f of a random variable S:
$E[f(S)] \geq f(E[S])$
The expected value of the payoff is not the same as the payoff of the expected value. The order matters. This is why we cannot just use the “average” future stock price to value options. The randomness itself has value because options have that curved (convex) payoff. More uncertainty means more option value.
This single insight explains why volatility is so central to option pricing.
Building the Random Walk Model
When you invest, you care about returns (percentage growth), not absolute dollar changes. A $10 rise matters a lot more on a $100 stock than on a $1000 stock. So Wilmott focuses on modeling returns:
$\text{Return} = \frac{S_{i+1} - S_i}{S_i}$
Looking at real data (he uses an Argentine company called Perez Companc as an example), daily returns look like noise. They are roughly normally distributed with a small mean and a much larger standard deviation.
That last point is important: on a daily basis, the noise (randomness) completely dominates the trend. You have to wait months or years to see the trend emerge from the noise.
How Returns Scale With Time
This is where the model gets precise. The mean return scales linearly with the time step:
$\text{Mean return} = \mu \cdot \delta t$
Where $\mu$ is the drift rate (annualized expected return) and $\delta t$ is the time step.
But the standard deviation of returns scales with the square root of time:
$\text{Standard deviation} = \sigma \cdot \sqrt{\delta t}$
Where $\sigma$ is the volatility (annualized standard deviation of returns).
Why the square root? Because when you add up many independent random variables, the variance (not the standard deviation) adds up. Variance is proportional to time, so standard deviation goes as the square root of time.
This square root scaling is not arbitrary. Wilmott shows it is the only scaling that gives a sensible model in the continuous-time limit. Any other scaling would either make the asset trivially constant or blow up to infinity.
The Random Walk Equation
Putting drift and volatility together, we get:
$\delta S = \mu S \delta t + \sigma S \phi \sqrt{\delta t}$
Where $\phi$ is a random number drawn from a standard normal distribution (mean 0, standard deviation 1).
This says: the change in stock price has two parts. A predictable drift ($\mu S \delta t$) and a random shock ($\sigma S \phi \sqrt{\delta t}$). Both are proportional to the current stock price S, which means the model works in percentage terms. A stock at $200 moves twice as much in absolute terms as a stock at $100, but the percentage moves are the same.
Drift vs. Volatility
The drift $\mu$ tells you the average direction. It is hard to measure because it is small compared to the noise. In the Perez Companc example, the annualized drift was about 73%, but daily noise was so large you could barely see it.
Here is a key insight from Wilmott that will become very important later: in classical option pricing theory, the drift plays almost no role. It drops out of the Black-Scholes equation entirely. So even though drift is hard to measure, this does not matter for pricing options.
The volatility $\sigma$ is the star of the show. It measures the amount of randomness. It is “the most important and elusive quantity in the theory of derivatives.” Wilmott says he will come back to it again and again. He is not kidding.
Measuring Volatility in Practice
The simplest estimate: take historical closing prices, compute daily returns, calculate the standard deviation, and annualize by multiplying by $\sqrt{252}$ (since there are roughly 252 trading days per year).
The problem: volatility is almost certainly not constant. It changes with economic conditions, seasons, market events. Using a fixed window of past data creates a plateauing effect: if there is one big price drop, your volatility estimate will stay elevated for as long as that drop is in your window, then suddenly drop when it falls out.
Wilmott acknowledges this is a real problem and promises more sophisticated volatility estimates later in the book.
The Wiener Process: Going Continuous
The discrete random walk we built works with time steps. But for the elegant math of continuous-time finance, we need to go to the limit where the time step shrinks to zero.
Wilmott introduces the Wiener process notation. Instead of $\phi \sqrt{\delta t}$, we write $dX$, a random variable with mean zero and variance $dt$:
$E[dX] = 0, \quad E[dX^2] = dt$
The continuous-time model becomes:
$dS = \mu S , dt + \sigma S , dX$
This is our first stochastic differential equation (SDE). It is the most widely accepted model for equities, currencies, commodities, and indices. This one equation is the foundation of so much finance theory.
Do Fund Managers Beat the Market?
Wilmott throws in a fun data point. He shows data on what percentage of UK funds outperformed the all-share index over 1, 3, 5, and 10-year horizons. The answer: the vast majority do not. Over 10 years, almost no one beats the market consistently.
His personal take: this does not prove markets are random, but it is suggestive enough that most of his personal stock exposure is through an index-tracker fund. Hard to argue with that.
The Takeaway
Chapter 3 builds the mathematical model that the rest of the book depends on. The key results:
- Jensen’s inequality shows why randomness itself has value for options. You cannot ignore it.
- Returns (not absolute prices) are what we model, and they are approximately normally distributed.
- The mean return scales with time $(\mu \cdot dt)$, while standard deviation scales with the square root of time $(\sigma \cdot \sqrt{dt})$.
- The continuous-time model $dS = \mu S , dt + \sigma S , dX$ is the starting point for everything.
- Volatility is the most important parameter. Drift barely matters for option pricing.
Next chapter: we need the mathematical tools to actually work with this stochastic differential equation. That means Ito’s lemma and stochastic calculus.
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