Serial Autocorrelation: When Today's Return Predicts Tomorrow's

One of the core assumptions behind most of quantitative finance is that stock returns are independent. What happened yesterday tells you nothing about today. The stock went up ten days in a row? Irrelevant. Tomorrow is a fresh coin toss. But is this really true? Chapter 65 of Wilmott’s book looks at the evidence for serial autocorrelation in stock returns and asks what happens to our models when returns are not independent.

What Serial Autocorrelation Means

Serial autocorrelation measures whether today’s return is correlated with yesterday’s return. You take a time series of returns, subtract the mean, divide by the standard deviation, and then calculate the correlation between the series and itself shifted by one day. In Excel, you literally use the CORREL function with two arrays: the return series and the same series lagged by one period.

Positive autocorrelation means trending behavior. If the stock went up today, it is more likely to go up tomorrow than a pure coin toss would predict. This is what technical analysts and momentum traders believe in.

Negative autocorrelation means mean-reverting behavior. If the stock went up today, it is slightly more likely to come back down tomorrow. Think of it as the market taking profits after a move.

The Evidence in Real Data

Wilmott shows a chart of the moving 252-day, one-day lagged serial autocorrelation in the S&P 500. The average level is around 2%. Before the year 2000, the average had been above 4%. More recently, the autocorrelation has drifted into negative territory.

Is this significant? It is hard to say. The numbers are small, and sampling error could explain them entirely. Maybe there is no real autocorrelation and what we see is just noise. Or maybe there is something structural going on.

Wilmott raises an interesting question. Are the ubiquitous hedge funds changing the traditional behavior of markets? Positive autocorrelation can be interpreted as trend-following behavior. Negative autocorrelation could be the signature of systematic profit-taking. As more hedge funds pursue momentum strategies, they might be consuming the very trends they depend on, pushing autocorrelation toward zero or even negative values.

The Telegraph Equation

Here is where the math gets genuinely different from everything else in the book. Wilmott builds a random walk model with serial autocorrelation and derives the equation governing its probability density function. The result is called the telegraph equation, and it has a fundamentally different character from the diffusion equations we have seen everywhere else.

The setup is simple. A variable y starts at the origin. At each time step, it goes up or down by a fixed amount. But the probability of going up depends on what happened last time. If the previous move was up, the probability of going up again is 0.5 + phi. If the previous move was down, the probability of going up is 0.5 - phi. The parameter phi controls the strength of the autocorrelation.

When phi is positive, you get trending random walks. Simulations show the variable drifting away from the origin in clear directional moves. When phi is negative, you get oscillatory behavior with the variable bouncing back and forth rapidly.

To track this mathematically, Wilmott introduces two probability functions: p+(y, t) for the probability of being at y having just risen, and p-(y, t) for the probability of being at y having just fallen. Each depends on the other through the conditional probabilities.

After expanding in Taylor series for small time steps and spatial increments, he combines the equations and eliminates one of the unknowns. The result is a second-order equation in both space and time.

The telegraph equation has the form: a * (second derivative of p with respect to time) + (first derivative of p with respect to time) = c * (second derivative of p with respect to y).

This equation has a second derivative with respect to time. Every other equation in the book has been parabolic, with only a first derivative in time. That makes them diffusion equations. The telegraph equation, when phi is positive, is hyperbolic. It is a wave equation.

The word “momentum” is exactly right here. In physics, wave equations describe systems with inertia and momentum. The second time derivative is what gives the telegraph equation its wave-like character. The probability distribution propagates with a finite speed instead of spreading instantaneously like a diffusion.

Wilmott makes a nice connection to Newton’s laws. Force equals rate of change of momentum. The second time derivative is the acceleration term. When returns have positive autocorrelation, the probability distribution has “inertia” that carries it forward.

What This Means for Pricing Derivatives

Now for the surprise. Despite this fundamentally different probability structure, the pricing equation for derivatives is still Black-Scholes in the continuous-time limit. The telegraph equation does not change option pricing.

Why? Because delta hedging eliminates any dependence on the probability of the stock going up or down. The hedge ratio depends on the option’s sensitivity to the stock price, not on the stock’s directional tendencies. Since autocorrelation only affects the conditional probability of up and down moves, and hedging removes that dependency, we end up right back at risk-neutral pricing and the diffusion equation.

This is a profound result. The real-world dynamics of the stock can be genuinely different from a random walk (with trending or mean-reverting behavior), but the risk-neutral pricing framework does not care. As long as you can hedge continuously, the pricing equation is unchanged.

Practical Implications

If you only hedge discretely (which is the reality for everyone), then serial autocorrelation does matter. You can modify your delta to account for the expected autocorrelation, along the lines of the discrete hedging framework covered in earlier chapters.

Wilmott also points out an interesting trading opportunity. In a market with significant negative autocorrelation and the common pattern of implied volatility rising when stocks fall, you can time the sale of put options. When the market drops, implied volatility spikes, so you bring in more premium. Then negative autocorrelation gives you a probable bounce back, making those puts less likely to pay out.

This is not a free lunch. It depends on the autocorrelation being real and persistent, which is far from guaranteed. But it is a good example of how understanding the statistical properties of returns can inform trading strategy even when the pricing model itself does not change.

An Exotic Derivative on Autocorrelation

Wilmott ends the chapter with a challenge. How would you value and hedge an exotic derivative that pays off based on the realized serial autocorrelation in a stock price? Or the serial autocovariance?

You would still price these in a risk-neutral framework, probably using Monte Carlo simulation. The theoretical value of such a contract would always be zero (because risk-neutral returns are independent), but it would be valuable to anyone with a view on autocorrelation. If you believe returns are positively autocorrelated, you would buy a contract that pays the realized autocorrelation. If the autocorrelation materializes, you profit. If not, you lose.

These are exotic contracts that do not trade widely, but they illustrate a general principle. Any statistical property of returns can potentially be packaged into a derivative. The pricing is straightforward in theory even if the calibration and hedging present challenges in practice.

Key Takeaways

Three things from this chapter. First, there is some empirical evidence for serial autocorrelation in stock returns, but it is small and may be disappearing as markets become more efficient. Second, the mathematical structure of a serially autocorrelated random walk is fundamentally different from a diffusion. It follows a wave equation (the telegraph equation) with momentum, not a heat equation with pure diffusion. Third, despite this dramatic difference in the real-world dynamics, continuous-time derivative pricing is unchanged because hedging eliminates the dependency on directional probabilities. The autocorrelation matters for discrete hedging and trading strategy, but not for the pricing equation itself.

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