The Dis­tri­b­u­tion of Price Fluc­tu­a­tions

Are price fluc­tu­a­tions in the fi­nan­cial mar­kets nor­mal­ly dis­trib­ut­ed? If I un­der­stand his­to­ry cor­rect­ly, it was French math­e­mati­cian Louis Bache­li­er who was the first to ex­plore this top­ic over 100 years ago. While Bache­lier’s work as­sumed that price move­ments were nor­mal­ly dis­trib­ut­ed, a math­e­mati­cian named Benoit Man­del­brot made some in­ter­est­ing ob­ser­va­tions that sug­gest oth­er­wise.

In the 1960s, Man­del­brot stud­ied his­tor­i­cal cot­ton prices and no­ticed the dis­tri­b­u­tion of price changes did not ex­hib­it prop­er­ties char­ac­ter­is­tic of a nor­mal dis­tri­b­u­tion. Thank­ful­ly, com­put­er tech­nol­o­gy and the avail­abil­i­ty of his­tor­i­cal price da­ta have im­proved dras­ti­cal­ly since the 1960s. Do­ing this sort of analy­sis is much eas­i­er to­day than it was in the past. In this post, I set out to per­form my own analy­sis of the dis­tri­b­u­tion of price fluc­tu­a­tions across a va­ri­ety of dif­fer­ent mar­ket­s.

An­a­lyz­ing the Da­ta

The first set of da­ta I want to look at is the dai­ly clos­ing price his­to­ry of an S&P 500 in­dex-track­ing fund. Specif­i­cal­ly, I’m us­ing the SP­DR S&P 500 ET­F. The tick­er sym­bol is SPY. This is cur­rent­ly one of the most heav­i­ly trad­ed in­stru­ments on the New York Stock Ex­change. Be­low is a chart of the dai­ly clos­ing prices over the past 21 years:

No­tice in the chart above that the ver­ti­cal ax­is rep­re­sents the nat­ural log­a­rithm of the clos­ing price, not the ac­tu­al clos­ing price it­self. We are con­cerned with the vari­ance of price change per­cent­ages here and not the change in re­al price, so we’re go­ing to an­a­lyze the log­a­rith­mic val­ues in this study in­stead of the ac­tu­al price val­ues. The chart be­low shows the dif­fer­ences in the log­a­rith­mic price val­ues from one day to the nex­t:

The price moves up and down seem­ing­ly at ran­dom. The mag­ni­tude of each price move varies from one day to the nex­t. Are these vari­a­tions in price move­ment nor­mal­ly dis­trib­ut­ed? It’s hard to tell just by look­ing at the chart. What stands out, how­ev­er, is that the vari­ance seems to be het­eroskedas­tic; there are pe­ri­ods of rel­a­tive calm fol­lowed by clus­ters of large moves up and down. While a study of het­eroskedas­tic­i­ty is be­yond the scope of this post, it is def­i­nite­ly a pe­cu­liar­i­ty worth point­ing out.

Ig­nor­ing these pock­ets of high and low vari­abil­i­ty, the ques­tion re­main­s: is the over­all vari­a­tion in price move­ment nor­mal­ly dis­trib­ut­ed? Would the da­ta bet­ter con­form to a dif­fer­ent prob­a­bil­i­ty dis­tri­b­u­tion in­stead? Let’s put the da­ta in a his­togram and see what it looks like:

The his­togram is over­laid with two prob­a­bil­i­ty den­si­ty func­tion­s, one rep­re­sent­ing a nor­mal dis­tri­b­u­tion and the oth­er rep­re­sent­ing a Laplace dis­tri­b­u­tion. The pa­ra­me­ters for each dis­tri­b­u­tion were es­ti­mat­ed us­ing the max­i­mum like­li­hood method. You can see my post ti­tled Nor­mal and Laplace Dis­tri­b­u­tions for de­tails on how to com­pute the max­i­mum like­li­hood es­ti­mates.

So are the price fluc­tu­a­tions nor­mal­ly dis­trib­ut­ed? Eye­balling the chart above, it ap­pears not. At least for this da­ta set. With­out mea­sur­ing it ob­jec­tive­ly, it cer­tain­ly looks like the Laplace dis­tri­b­u­tion is a bet­ter fit. But what hap­pens if we per­form this ex­per­i­ment on dif­fer­ent da­ta set­s? Will we get the same re­sults for dif­fer­ent as­set type­s? What if we study in­tra­day da­ta in­stead of dai­ly clos­ing prices? The fol­low­ing sec­tions present the re­sult of this analy­sis across a mix of dif­fer­ent mar­kets and time frames.

Ex­change-Trad­ed Funds

To get some broad­er in­sights re­gard­ing the be­hav­ior of price move­ments, I want to take a look at some more ex­change-trad­ed funds with a few dif­fer­ent un­der­ly­ing as­set type­s. Let’s ex­am­ine the dai­ly clos­ing prices for the fol­low­ing ETF­s:

Each da­ta set con­tains at least 10 years’ worth of data. Us­ing the same tech­nique as be­fore, we can plot the his­togram of dai­ly price fluc­tu­a­tions and over­lay the fit­ted nor­mal and Laplace den­si­ty func­tion­s. Here are the chart­s:

As you can see, the his­togram seems to fit the Laplace dis­tri­b­u­tion bet­ter than the nor­mal dis­tri­b­u­tion most of the time. But not al­ways. There are some cas­es that ap­pear to fit some­where in be­tween the nor­mal dis­tri­b­u­tion and the Laplace dis­tri­b­u­tion. A cur­so­ry look at UNG, for ex­am­ple, might sug­gest the vari­a­tion is nor­mal­ly dis­trib­ut­ed.

In­di­vid­ual Stocks (Dai­ly)

In­stead of look­ing at broad stock mar­ket in­dex­es, let’s see what hap­pens if we ex­am­ine in­di­vid­ual stock­s. Let’s con­sid­er the dai­ly clos­ing prices for the fol­low­ing stock­s:

Here are the chart­s:

Sub­jec­tive­ly, it seems like the Laplace dis­tri­b­u­tion is a bet­ter over­all fit.

In­di­vid­ual Stocks (In­tra­day)

What hap­pens if we study in­tra­day stock prices in­stead of dai­ly clos­ing prices? Let’s per­form the analy­sis on the same set of stocks us­ing one-minute price da­ta in­stead of end-of-day prices. Here are the chart­s:

Each da­ta set con­tains five days’ worth of da­ta. Us­ing in­tra­day da­ta in­stead of dai­ly data does­n’t seem to change the out­come with re­spect to the gen­er­al shape of the dis­tri­b­u­tion.

Volatil­i­ty In­dex

I’m cu­ri­ous what hap­pens if we ap­ply this analy­sis to the val­ues of the CBOE Volatil­i­ty In­dex. This in­dex is a mea­sure of im­plied volatil­i­ty based on the mar­ket price of S&P 500 op­tion­s. While not a trad­able in­stru­ment in its own right, there are trad­able de­riv­a­tives based on this in­dex. Here is a chart of dai­ly clos­ing val­ues cov­er­ing a span of about 19 years:

Con­sis­tent with the oth­er da­ta sets used in this post, the “price” val­ues on this chart are the log­a­rith­mic val­ues of the volatil­i­ty in­dex, not the ac­tu­al in­dex val­ues. The VIX is quot­ed in per­cent­age points, so the use of log­a­rith­mic val­ues may or may not be the best ap­proach here, de­pend­ing on how one might go about trad­ing volatil­i­ty de­riv­a­tives. I sus­pect that study­ing the be­hav­ior of volatil­i­ty in­stru­ments might war­rant some spe­cial con­sid­er­a­tion, but that’s be­yond the scope of the post. For our pur­pos­es here, let’s treat the in­dex the way we would any oth­er trad­able in­stru­men­t. The dai­ly “price” dif­fer­ences look like this:

The vari­ance seems to be a bit more con­sis­tent than that of the price dif­fer­ences of the S&P 500 ETF shown ear­lier. What does the his­togram look like? Not much dif­fer­ent than the in­di­vid­ual stocks and ETFs ex­am­ined ear­lier:

It looks like the changes in im­plied volatil­i­ty vary from one day to the next ac­cord­ing to a dis­tri­b­u­tion that might be some­where be­tween a nor­mal dis­tri­b­u­tion and a Laplace dis­tri­b­u­tion. Here is a his­togram of five days’ worth of one-minute in­tra­day val­ues:

The in­tra­day da­ta ap­pear to fit the Laplace dis­tri­b­u­tion bet­ter than the nor­mal dis­tri­b­u­tion. I was ex­pect­ing the val­ues of the volatil­i­ty in­dex to have dif­fer­ent char­ac­ter­is­tics than val­ues from the oth­er da­ta set­s, but that does­n’t seem to be the case.

For­eign Ex­change

Do the ex­change rates be­tween dif­fer­ent fi­at cur­ren­cies ex­hib­it the same prop­er­ties as the da­ta sets stud­ied above? Let’s con­sid­er the ex­change rate be­tween the eu­ro and the US dol­lar. Here is a chart of the dai­ly ex­change rate val­ues over a pe­ri­od of about 18 years:

As be­fore, we’re us­ing the log­a­rith­mic val­ues here in­stead of the re­al val­ues. Here is a plot of the dai­ly dif­fer­ences in log­a­rith­mic ex­change rate val­ues:

There does seem to be some het­eroskedas­tic­i­ty, but it’s not as pro­nounced as what we saw with the S&P 500 ET­F. Here is the his­togram:

Here we see a fa­mil­iar pat­tern. It takes the shape rough­ly of a Laplace dis­tri­b­u­tion.

Cur­ren­cy Pairs (Dai­ly)

To see if the pat­tern hold­s, let’s take a look at the dai­ly ex­change rates of a few more cur­ren­cy pairs. Here is the list:

Here are the chart­s:

And here we see the fa­mil­iar pat­tern once again.

Cur­ren­cy Pairs (In­tra­day)

For the same cur­ren­cy pairs as above, let’s ex­am­ine the dis­tri­b­u­tion of price fluc­tu­a­tions for one-minute in­tra­day da­ta over a span of 24 hours:

No­tice that in some of these charts there is a large spike in the con­cen­tra­tion of price changes at or around ze­ro. I sus­pect this is be­cause there are cer­tain times of day when these in­stru­ments are thin­ly trad­ed and don’t move very much.

Cryp­tocur­ren­cies

A num­ber of dif­fer­ent cryp­tocur­ren­cies have emerged in the past few years. And some of them have made enor­mous price moves. Can we ex­pect the price fluc­tu­a­tions of these dig­i­tal as­sets to ex­hib­it the same char­ac­ter­is­tics as stocks and cur­ren­cies? Let’s ex­am­ine a few of the most pop­u­lar ones:

Here are the chart­s:

For all three of these da­ta set­s, the vari­a­tion in dai­ly price moves does not con­form to the nor­mal dis­tri­b­u­tion. In fac­t, the dis­tri­b­u­tion of price move­ments seems to be even more clus­tered around the cen­ter than would be ex­pect­ed for a Laplace dis­tri­b­u­tion.

Draw­ing Con­clu­sions

Can we draw any con­clu­sions from this ex­per­i­men­t? I think it’s safe to say price fluc­tu­a­tions are not al­ways nor­mal­ly dis­trib­ut­ed. In all da­ta sets ex­am­ined, the kur­to­sis is more lep­tokur­tic than a nor­mal dis­tri­b­u­tion to some de­gree. But would it be ap­pro­pri­ate to use a Laplace dis­tri­b­u­tion to mod­el price move­ments?

Vance Har­wood from Six Fig­ure In­vest­ing wrote an in­ter­est­ing ar­ti­cle as­sert­ing that the Laplace dis­tri­b­u­tion should be used in­stead of the nor­mal dis­tri­b­u­tion to mod­el stock price move­ments. You can read it here:

• Pre­dict­ing Stock Mar­ket Re­turn­s—Lose the Nor­mal and Switch to Laplace

His analy­sis in­cludes a deep­er study of the tails of the dis­tri­b­u­tion. While he ac­knowl­edges the ob­served price da­ta in­clude a high­er fre­quen­cy of large moves than what would be ex­pect­ed from a Laplace dis­tri­b­u­tion, I agree with his as­sess­ment that the Laplace dis­tri­b­u­tion is a bet­ter al­ter­na­tive com­pared to the nor­mal dis­tri­b­u­tion.

The Busi­ness Fore­cast­ing blog au­thored by Clive Jones in­cludes a whole se­ries of ar­ti­cles on the top­ic of price change dis­tri­b­u­tion­s:

• The Dis­tri­b­u­tion of Dai­ly Stock Mar­ket Re­turns
• The Laplace Dis­tri­b­u­tion and Fi­nan­cial Re­turns
• The Nas­daq 100 Dai­ly Re­turns and Laplace Dis­trib­uted Er­rors
• Mi­crosoft Stock Prices and the Laplace Dis­tri­b­u­tion
• Dis­tri­b­u­tions of Stock Re­turns and Oth­er As­set Prices

This se­ries cites plen­ty of ev­i­dence that fa­vors the use of the Laplace dis­tri­b­u­tion to mod­el price move­ments. Nonethe­less, as I have il­lus­trat­ed in this post, there are still some cas­es where the his­togram does not ex­hib­it the pointy-shaped peak char­ac­ter­is­tic of the Laplace dis­tri­b­u­tion.

Based on his study of his­tor­i­cal cot­ton prices, Man­del­brot claims that price fluc­tu­a­tions are best de­scribed by a fam­i­ly of sta­ble dis­tri­b­u­tions. These dis­tri­b­u­tions are more round­ed at the peak than the Laplace dis­tri­b­u­tion. I cur­rent­ly don’t know enough about this class of prob­a­bil­i­ty dis­tri­b­u­tions to draw my own con­clu­sion­s, but I think it’s an area wor­thy of fur­ther study.

An­oth­er pos­si­bil­i­ty is that the dis­tri­b­u­tion of price fluc­tu­a­tions could take the form of a gen­er­al­ized nor­mal dis­tri­b­u­tion. This is a fam­i­ly of dis­tri­b­u­tions that can take the shape of a nor­mal dis­tri­b­u­tion, a Laplace dis­tri­b­u­tion, or some­thing else de­pend­ing on the val­ue of a shape pa­ra­me­ter. I think this might be an­oth­er area worth fur­ther in­ves­ti­ga­tion.

Ac­com­pa­ny­ing source code is avail­able on GitHub.