Empirical Volatility: What the Data Actually Shows

Most people who model stochastic volatility start by writing down a nice-looking equation and then try to fit it to data. Wilmott thinks this is backwards. In Chapter 53, he starts with the data and builds the model from the ground up. It is a refreshingly practical approach. Instead of picking a model because it is mathematically convenient, he asks: what does volatility actually do?

The Dataset

Wilmott works with daily closing prices of the Dow Jones Industrial Average from January 1975 to August 1995. That is twenty years and over 5,000 observations. From these, he calculates 30-day rolling volatility of daily returns.

When you plot this volatility over time, you see something very far from constant. The volatility bounces around, sometimes sitting at 10%, sometimes jumping to 40% or more. There was a massive spike during the October 1987 crash. But even ignoring the crash, volatility is clearly a moving target.

The goal of this chapter is to find a stochastic differential equation that describes how volatility evolves. Something of the form:

dv = alpha(v) dt + beta(v) dX

where alpha is the drift (which direction volatility tends to move) and beta is the volatility of volatility (how much randomness there is in volatility itself). Both functions depend on the current level of volatility.

Finding the Volatility of Volatility

Here is the clever part. If you have a time series of volatility values, you can compute the day-to-day changes. The square of each day-to-day change gives you an estimate of the local variance. So you bucket the volatility values, and for each bucket, you compute the average squared change.

Plot the log of this average squared change against the log of the volatility level, and you get something close to a straight line. That is a very useful result. A straight line on a log-log plot means a power law relationship.

From the slope and intercept of this line, Wilmott deduces that the volatility of volatility scales roughly as a power of the current volatility level. The data gives specific numbers for the parameters. The exponent turns out to be about 3/2, meaning volatility of volatility grows faster than linearly with the level of volatility.

This is important information. It tells you that when volatility is already high, its fluctuations are also larger. Volatility does not just wander randomly at a constant pace. It becomes more wild when it is elevated. This matches what most traders observe in practice.

Finding the Drift

The drift is harder to estimate directly from short-term changes. Day-to-day movements are dominated by the random component, so the drift signal is buried in noise. Wilmott uses a different and smarter approach.

Instead of looking at short-term changes, he examines the long-term distribution of volatility. If you assume the model parameters are constant over time, then the long-term (steady-state) distribution of volatility can be estimated from the time series using the ergodic property. In simple terms: if you watch one process for a very long time, the histogram of its values converges to the true probability distribution.

So Wilmott plots the frequency distribution of the 30-day volatility values and fits a curve to it. The distribution looks approximately lognormal. He fits a lognormal density and extracts its parameters.

Now here is the connection. The steady-state distribution and the drift are related through the Fokker-Planck equation (also called the forward Kolmogorov equation). If you know the steady-state distribution and the volatility of volatility, you can solve for the drift. This is a straightforward calculation once you have the pieces.

The result is a drift function that pulls volatility back toward its long-run average. This mean reversion is something nearly everyone assumes in stochastic volatility models, and it is nice to see it emerge directly from the data rather than being imposed from above.

Why This Order Matters

Wilmott makes an important methodological point. He estimates the volatility of volatility from short-term changes and the drift from long-term behavior. This is not arbitrary.

The randomness in volatility is visible over the shortest timescales. Look at daily changes, and you see the noise. But the drift, the tendency to revert to a mean, only shows up over longer periods. If you tried to estimate the drift from daily changes, you would get garbage. The noise would swamp the signal.

On the flip side, if you tried to estimate the volatility of volatility from the long-term distribution alone, you would lose the information about how fast things jiggle. Each quantity is best estimated at the timescale where it dominates.

This is a lesson that goes well beyond volatility modeling. Whenever you are fitting a stochastic model, think about which parameters show up at which timescales, and design your estimation procedure accordingly.

Does It Work Out of Sample?

Wilmott tests the model on about 150 stocks and indices. For each underlying, he fits the model to historical data and then uses it to forecast volatility ranges over the next two months. He comes back after two months to see what actually happened.

The result: the distribution of forecast volatilities and actual volatilities match closely. This is exactly what you want. The model trained on the past gives reasonable predictions about the future. It does not mean it is perfect, but it means the model is not just fitting noise. The underlying dynamics seem to be captured reasonably well.

Three Ways to Use the Model

1. Direct Option Pricing

Use it as a two-factor model with stock price and volatility. The catch: you need the risk-neutral drift, not the real-world drift, which requires estimating the market price of volatility risk from option data.

2. Watching Volatility Evolve

Simulate the future distribution of volatility from today’s known value using the Fokker-Planck equation. The distribution starts as a spike and gradually spreads until it reaches the steady-state shape. Useful for stress testing and understanding exposure.

3. Confidence Levels for Uncertain Volatility Pricing

This is the use case Wilmott seems most excited about, and it connects directly to the uncertain volatility model from Chapter 52.

Recall that in the uncertain volatility framework, you specify a range for volatility and compute worst-case option prices. But how confident are you that volatility will stay inside your chosen range?

The stochastic volatility model answers this question. You can compute the probability that volatility stays within any given band over any time horizon. Mathematically, this is a first-exit-time problem. You solve a PDE with boundary conditions at the upper and lower bounds of your volatility range, and the solution gives you the probability of the volatility never leaving that range.

Wilmott calls this a “nice compromise.” You get the benefits of the uncertain volatility model (no need for a market price of risk, clean worst-case pricing) while also having a probabilistic statement about how likely your range assumptions are to hold. If you set a range that is very narrow, you get tight price bounds but low confidence. Widen the range, and your confidence goes up but so does the spread between best and worst prices.

This creates a principled way to choose your volatility range. You pick the range that gives you acceptable confidence, say 95%, and then price accordingly. If the market price falls outside your worst-case bound for that range, you have a trade with a 95% confidence of profit. That is actionable.

The Bigger Picture

Most academic work on stochastic volatility picks a model first and then calibrates. Wilmott picks the data first and lets the model emerge. The resulting model might not be as tractable as Heston, but it is more honest. Combined with the uncertain volatility framework from Chapter 52, it gives you data-driven volatility ranges, confidence levels, and worst-case prices that do not depend on unobservable quantities like the market price of risk.

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