On the distribution of stock price changes

How to read this walk-through. The paper is short but dense. The main idea can be taken from the section “The method in one line” and the example in section 1 without reading the formulas. Sections 5–8 are Mandelbrot's discussion, in which he himself limits Taylor's construction; these are the parts most often lost in retelling. The PDF of the paper is a scan without a text layer, so there are few verbatim quotes here, and all of them are taken only where the text on the scan reads reliably.

1. What the dispute is about

Mandelbrot begins by noting that there are at least four schools of thought on the distribution of price changes. The most numerous, the technical analysts, talk about trends, support levels and rebounds. Two others reject the technical view and agree that prices are a random walk, but differ on what distribution the steps have and in what time to measure them: physical time (days, hours) or operational time, which runs together with the flow of trades. Some find price changes normal, others see a stable Paretian distribution with infinite variance. The fourth group accepts the random walk as a first approximation and notes second-order effects.

The paper answers the dispute between the two random-walk schools. The answer in the abstract: "Price changes over a fixed number of transactions may have a Gaussian distribution." Changes over a fixed period may at the same time follow a stable Paretian distribution, and since the number of trades in any period is random, the statements do not necessarily contradict each other. The text puts it this way: "Basically, our point is this: the Gaussian random walk as applied to transactions is compatible with a symmetric stable Paretian random walk as applied to fixed time intervals."

The idea in plain terms. Suppose the price changes only at the moment of a trade, and each trade shifts it by a small random amount, independently of the past. In trades this is an ordinary Gaussian walk. In clock time the picture is different: one hour sees 10 trades, another 10,000. The price change over an hour is a sum of a random number of random steps. If the number of steps itself has a very heavy tail, hourly changes inherit it.

A worked example (our example). Each trade moves the price by $\pm 1$ cent with equal probability. An hour with 100 trades gives a typical move of $\sqrt{100} = 10$ cents, an hour with 10,000 trades gives 100 cents. With a constant number of trades, hourly changes would be normal. A random number of trades with a heavy tail makes their distribution heavy-tailed.

The paper consists of two parts by different authors. The mathematical observation belongs to Taylor (the footnote to the section “A relation between the Gaussian and other stable laws”); the discussion and the link to specialists belong to Mandelbrot.

2. The method in one line

The price in clock time is the Gaussian price in trades, into which the random number of trades up to that hour has been substituted.

What this is. Three processes nested one inside another.

What each symbol means.

• $X(v)$: price as a function not of calendar time but of the accumulated number (or volume) of trades $v$. A Gaussian process with independent increments, mean zero and $E[X(u)X(v)] = \sigma^2\min\{u, v\}$, that is, an ordinary Brownian walk.
• $T(t)$: how many trades, or what volume, has accumulated by calendar time $t$. Also a random process, but one that only increases.
• $Z(t)$: what we see, the price as a function of calendar time.

How to read it. Take a calendar time $t$, look at how many trades have accumulated by then, and substitute this number into the Gaussian process.

Terms. $Z$ is called subordinated to the process $X$, and $T$ the directing process. Mandelbrot attributes the technique itself to Bochner, who introduced the notion of subordination.

Why this matters. It separates “how the price reacts to a trade” from “how fast trades come”. In this model heavy tails live in the second.

3. Stable laws and the exponent α

The section “Review of stable laws” recalls what a stable distribution is. A process with independent stationary increments follows a stable law if and only if its characteristic function has a certain form with four parameters: $\alpha$ (the characteristic exponent, from 0 to 2), $\beta$ (skewness, from −1 to 1), $\gamma > 0$ (scale) and $\delta$ (location).

What the characteristic exponent α is. A number from 0 to 2 that sets how thick the tails are. Citing Gnedenko and Kolmogorov, the authors list:

• $\alpha = 2$: the normal distribution, $Z(t)$ is Brownian motion;
• $\alpha = 1$ and $\beta = 0$: the Cauchy distribution;
• $1 < \alpha < 2$: the mean is finite, the variance infinite; this is the “stable Paretian distribution” of the dispute.

Two more facts from the same section: a stable distribution is positive if and only if $\alpha < 1$, $\beta = 1$ and $\delta \ge 0$, and it is symmetric when $\beta = 0$.

What “no variance” means. Not “the variance is large”, but that the sample variance converges to nothing as the sample grows and keeps rising. For a practitioner this means that a volatility estimate from such data is unstable, and so is everything built on it (our explanation; the paper does not draw this conclusion).

The characteristic function of the Gaussian process $X$:

What this is. The characteristic function, that is, the average of $e^{i\xi X}$. It determines the distribution uniquely, and it is convenient to work with because for a sum of independent variables the functions multiply.

What each symbol means. $\xi$ (“xi”): the argument of the function. $\sigma^2$: the variance of a step per unit of operational time. $t$: how much operational time has passed.

Step by step. Multiply the square of the argument by the variance and by the time, take half of it with a minus sign, raise $e$ to that power.

Why this matters. The normal distribution is recognized by the square of the argument in the exponent. A stable one with exponent $\alpha$ is recognized by $\lvert\xi\rvert^{\alpha}$.

4. Taylor's derivation

Taylor takes the directing process $T(t)$ to be a positive stable process with exponent $0 < \alpha < 1$, $\beta = 1$ and $\delta = 0$ (equation 3 of the paper). The characteristic function of $Z$ is then computed in two steps: first for a known $T(t)$, then averaged over $T(t)$.

What this is. For a known number of trades $T$ the price is Gaussian with variance $\sigma^2 T$. What remains is to average the Gaussian characteristic function over the random $T$.

Step by step. Substitute $\frac12\xi^2\sigma^2$ as the argument into the characteristic function of $T$. The power $\alpha$ of $\xi^2$ gives $\lvert\xi\rvert^{2\alpha}$.

An example (ours). If $\alpha = 0.85$, the exponent of the price in clock time equals $2 \times 0.85 = 1.7$: this is a stable Paretian distribution with a mean but without a variance. If $\alpha = 0.5$, the exponent is 1; this is the Cauchy distribution, which does not even have a mean.

Why this matters. This is the proof of the reconciliation itself. A Gaussian walk in trades plus a jumpy number of trades gives a symmetric stable law with exponent $2\alpha < 2$ in clock time.

Taylor's result: $Z(t)$ is a symmetric stable process with exponent $2\alpha < 2$, whose skewness and location parameters are zero. A footnote honestly notes that the step is formal: a complex argument is substituted into the formula for the characteristic function derived for a real argument. The authors write that this step is easy to justify and refer to Feller.

The interpretation from the same section: $X(v)$ is the price on the scale of trading volume, $T(t)$ the accumulated volume or number of trades by physical time $t$, and $Z(t)$ the price in physical time.

5. Mandelbrot's discussion: the infinite moments have not gone anywhere

The second half of the paper is Mandelbrot's discussion. He recalls his 1963 paper: price changes over fixed periods are approximately independent and approximately stable Paretian, that is, with infinite variance. He accepts Taylor's remark but immediately clarifies what it is not.

Taylor's directing process is itself non-Gaussian and stable, with an infinite mean, and hence also an infinite variance. Therefore, Mandelbrot writes, Taylor's remark "does not belong to the category of attempts to avoid stable Paretian distributions": it is not an attempt to rescue finite variance but another way of introducing the hypothesis of infinite moments into the study of prices. Heavy tails remain in the model; they simply sit in the trade-count process.

Why this matters. Retellings of the paper often reduce it to the formula “heavy tails are an artifact of the clock; switch to trade time and they will disappear”. Mandelbrot says something else in the same article: the tails are not an artifact, they are transferred to the flow of trades, and the infinite mean of that flow is no less inconvenient than the infinite variance of the price.

A footnote to the same section mentions two other authors. Granger noticed that $X(T)$ can be Gaussian while $Z(t)$ is very heavy-tailed, but did not note that for this $T(t)$ must be approximately a subordinator. Brada and coauthors showed in detail that price changes between trades are short-tailed and approximately Gaussian, and Mandelbrot adds that this property is ensured by the SEC instructions for specialists (section 7).

6. Real and “virtual” trades

The subordinator is almost surely discontinuous and increases by positive jumps. The jump size $U$ has a power-law (Paretian) conditional tail. Mandelbrot reads this as follows: if $U > 1$, trades come in bursts. He calls the last trade in a burst “final” and the rest “virtual”.

Hence an inconvenient consequence. If the volumes on the “virtual” trades are negligible, the price change between two final trades is a sum of $U$ independent Gaussian variables with unit variance. The variance of such a sum equals the mean of $U$, and that is infinite. The distribution of changes between final trades coincides with the distribution of jumps of the original process with infinite variance. Mandelbrot's conclusion: the Bochner representation "brings nothing new, unless nonnegligible amounts are traded on the so-called" “virtual” trades.

How to understand this. In observed trades Taylor's model does not promise a Gaussian price. If a burst of trades is seen as one trade, the heavy tail is also seen in the changes between observed trades. Gaussianity in trades appears only when something is actually traded inside the burst.

One more caveat from a footnote: strictly speaking, $T(t)$ is an integer and therefore cannot be exactly a subordinator. As a first approximation the jumps of $T$ are treated as quantized but not smoothed, and $X(T)$ as a discrete-time process with zero mean and unit step variance. Taylor's derivation, according to Mandelbrot, is practically unaffected by such quantization.

7. The role of the specialist

In his 1963 paper Mandelbrot wrote that discontinuities of the price process are hardly visible in trade-by-trade data. They either hide where the market is closed or trading is halted, or they are smoothed out by specialists, who, under SEC instructions, "ensure the continuity of the market" by taking part in trades themselves.

Here he clarifies how the specialist changes the data. First, the specialist smooths out small aimless drifts, and the price change between trades is zero more often than it would be otherwise. Second, the specialist replaces most large changes with series of small and medium ones whose amplitudes are so interdependent that almost all are of the same sign. In some markets this is additionally ensured by legal daily limits on price changes.

A footnote takes the mechanism apart with an example. After a trade at price $Z_i$ important news arrives, and supply below the level $Z^0$ disappears. The specialist sells from inventory and takes part in the following trades, and this pause for reflection, as Mandelbrot writes, removes almost all large values from the distribution of $Z_{i+1} - Z_i$. The only large changes left are those across the last trade of a session or across a trading halt, and their contribution drowns in the huge number of trades within the session.

Mandelbrot also proposes a hypothesis: it is tempting to identify Bochner's “virtual” trades with the specialists' trades, although specialists probably see where the price is going and cover the necessary distance in fewer than $U$ independent Gaussian steps. The section's conclusion: "the Bochner representation is concretely plausible and a program for empirical study of specialists."

What can be seen from this today (our reading). The mechanism of an intermediary that spreads a jump out over time produces series of small one-directional moves. Part of the observed autocorrelation over short intervals may be generated by the structure of the market rather than by the behavior of participants. The paper says this about NYSE specialists of the 1960s; carrying it over to modern markets requires checking.

8. Why trade data help little in practice

The last section of the paper explicitly limits its practical usefulness. Mandelbrot agrees that a short tail (and even Gaussianity) of changes between successive quotations is fully compatible with Paretian behavior of changes over a fixed period. But, he writes, it is hard to see how this distribution is useful to an individual investor. The investor has no trade clock. The investor does not know whether their trade will be the next one on the market or the fiftieth. Their order, placed at price $Z_i$, will be executed at price $Z^0$, and $Z^0 - Z_i$ is a complicated mixture of quantities of the form $Z_{i+j} - Z_i$ over all $j$. And "Mixtures of Gaussian distributions are known to have fatter tails than the Gaussian, and these tails can be very fat indeed."

For economists, trade data are also inconvenient: if price is one of the variables of a time-series model, there is no sense in measuring it at intervals between trades when the other variables are measured at fixed intervals.

Mandelbrot also makes the reverse caveat: describing the price in ordinary, uniformly flowing time does not solve the problem either, because the moment one's own trade is executed is not known in advance, and the mixture reappears. In Bachelier's model, where changes over a fixed period are Gaussian, the same difficulties with the mixture arise. The stable Paretian distribution has a useful property: the asymptotics of the distribution are the same for a mixture of periods and for each period separately. "Thus, mixing has no effect upon considerations related to that asymptotic behavior."

9. What is shaky in the paper

The authors' own caveats

No experimental test. Mandelbrot calls Taylor's alternative "illuminating though restricted to the symmetric case and as yet devoid of direct experimental backing".

Only the symmetric case. The construction gives a symmetric stable law; it does not explain asymmetry in price changes.

A formal step in the derivation. Substituting a complex argument into a formula for a real one is justified by a reference to Feller but is not carried out in the text.

The number of trades is an integer. $T(t)$ cannot be exactly a subordinator; the construction holds as a first approximation.

“Virtual” trades. If no appreciable volume is traded on them, the construction gives nothing new.

Practical usefulness is limited, section 8: the investor has no trade clock, and the mixture brings back fat tails.

What stands out on reading

The numbers and formulas were not recomputed, and we did not look for discrepancies with the text.

The form of the trade-count process is postulated. An exponent $\alpha$ between 0 and 1 for the directing process is an assumption that the paper neither supports nor measures in any way.

The market of 1967. Specialists, daily limits, no electronic trading. The smoothing mechanism works differently today, and the conclusions of section 7 cannot be carried over without checking.

A scan without a text layer. Verbatim quotes in this walk-through are taken only where the text on the scan reads reliably; the formulas were worked through from the rendered pages.

10. What follows in practice

Trade or volume bars are a testable hypothesis, not a matter of taste. If the heavy tails of hourly returns are generated by the unevenness of the flow of trades, slicing by number of trades should reduce them noticeably. This is a cheap check on your own data.

A shortened tail on trade bars does not mean the risk has gone. Mandelbrot says plainly that the heaviness of the tails is transferred to the flow of trades, and executing an order at an unknown moment again gives a mixture with fat tails. The risk of a position held for hours is measured in hours.

Bursts of trades matter more than the average rate. If trades come in bursts, a bar of fixed duration contains incomparable amounts of information, and it is the bursts that determine the tail.

An intermediary may create part of the autocorrelation over short intervals. Smoothing jumps with series of small moves of the same sign gives a trend-like structure that is unrelated to the behavior of participants. This is a hypothesis to test, not a conclusion of the paper about modern markets.

Risk computed via variance is unstable on data with an exponent below 2. If daily changes are closer to a stable Paretian distribution, the sample volatility does not converge, and hard thresholds should not be built on it (our explanation of the notion of “infinite variance”).

11. Testable ideas