The Shape of Financial Time
Ross M. Miller
Miller Risk Advisors
January 9, 2006
[This commentary appears in the January/February
of Financial Engineering News.
This is the raw, unedited version of that
It is must be reassuring to those entering finance from more tangible
disciplines to encounter a familiar concept amidst all the strange and
seemingly arbitrary newness. That concept appears in the form of a
variable represented by the letter t (or T) and is commonly
known as "time."
It can take a while to discover that this t, financial time, is
not the same thing as the more familiar tthe
physical time that most of us inhabit and that our clocks measure. It is
not that clock time does not matter in finance; it does, just not as much
as one might imagine. Absolute times, such as the expiration date for an
option, are (unless an exchange or the SEC happens to decides otherwise)
usually remain absolute. The speed at which financial time passes is
The field of economics has long recognized that it is unnecessary for
financial time to pass at a uniform clip. In the Arrow-Debreu
formalization of general equilibrium theory that underlies much of modern
finance, the notion of financial time is only loosely linked to physical
time. While physical time apparently exists so that everything does not
happen all at once, financial time is necessary only as a window through
which risk and uncertainty can climb into economic theory. In the
Arrow-Debreu world, time flows however it wants as long as there is
general agreement as to when certain points in time occur. In other words,
financial time flies like a socially constructed Arrow.
While it was generally taken for granted that time in financial and
economic models was identical to physical time, after a while doubts
surfaced. In light of the mounting empirical evidence that asset returns
did not look like they were drawn from independent and identical normal
distributions as many popular financial models had assumed, something was
wrong and the treatment of time was one place to look for answers. If the
financial world truly obeyed physical law, it would make sense (via the
central limit theorem) for asset returns to either themselves be normally
distributed or else be derivable from a normal process. By changing the
shape of financial time from a deterministic straight line to something
more exoticnot necessarily continuous and
certainly not deterministicperhaps one could
reunite the financial and physical realms.
When a market suffers from one its occasional fits of volatility, for
example, it could simply be that the financial clock has begun to tick
faster. What clocks out to be a single hour's worth of time in the
physical world might very well translate into a day, a week, a month, or
even a year of financial time. Indeed, if one tinkers sufficiently with
the financial clock, one can generate almost any volatility pattern that
one desires without having to distort the underlying normality of the
process generating prices and returns.
While the idea that financial time might be only loosely related to
physical time has yet to make waves in the financial mainstream, it has a
perverse elegance and furthermore is laughably obvious to any trader. The
time signature of the symphony played by the interaction of the various
financial markets changes constantly. Some of these changes are
predictablemarkets often open to a scherzo before
settling down to a midday waltzothers are not.
Markets also watch the calendar, while physics knows no season.
If financial time and physical are meaningfully different, coming up
with a way of translating between them might explain a lot. Perhaps by
doing so, in one fell swoop many outstanding problems of finance could be
solved or at least be made more tractable. Unfortunately, such a Rosetta
stone for moving between financial time and physical time remains
In an obvious parallel to the Einsteinian physical world, financial
time appears to be a local phenomenonsomething
this is permitted within the Arrow-Debreu framework, but is precluded by
most financial models. In the absence of a single, universal financial
clock, each market must keep its own time.
A common device for ticking off local financial time is to have trading
volume serve as a metronome for the market. Hence, when a market is
active, more financial time ticks away than when traders are napping or
out to lunch. It would be convenient if the trades or trading volume were
the key to financial time; however, a recent paper by Laszlo Gillemot, J.
Doyne Farmer, and Fabrizio Lillo from the Santa Fe Institute entitled
"There's More to
Volatility than Volume" shows, things are not so simple.
Volatility, however, is not the only place that time matters in
finance. Consider any of the several yield curves, which are expressed in
functional form as r(t)-so that the interest rate, r,
depends on time (presumably physical time), t.
The yield curve makes operational the clich้ that "time is
money." Under the normal situation of a rising yield curve, we find
that from the vantage point of the present a unit of time is worth more
money the further out into the future that we go. In the case of the
Treasury yield curve, however, time is not all that is money. Liquidity is
Time for Treasury securities is warped by the funding whims of the U.S.
Treasury. On-the-run Treasury securities (those most recently issued at a
given maturity) tend to trade at a somewhat lower yield than off-the-run
(seasoned) Treasuries because they are more liquid. Such small
discrepancies are consistent with Mom, apple pie, and efficient markets.
During the early 1990s, however, a more significant warpage developed.
Back then, I happened to be doing some yield curve modeling for one of
the parties responsible for denting the yield curve, a major insurance
company. Taking the lead from the mortgage markets, the big insurers had
gotten into some serious slicing and dicing of cash flows and the further
out in time they were able to slice off cash flows, the better. Whenever a
new 30-year Treasury bond was issued, it would be gobbled up and its price
bid up, with the old 30-year Treasury cast off like last season's designer
For anyone trying to fit a model to Treasury yields, the 30-year
Treasury bond was the sort of nuisance that were it not so important would
have qualified as an outlier. It marred the otherwise beautifully Treasury
yield curve that sloped smoothly upward. There was nothing special about
30 years, if the Treasury, to the delight of the insurance industry, had
decided to issue a 40-year bond; it too would have commanded an
unnaturally low yield. Physical time may lack a well-defined terminus, but
for many big-money players, financial time ends with the last Treasury
This anomaly, and the arbitrage opportunity it presented, did not last
forever. Hedge funds capitalized on this opportunity until they had
competed it away. Nonetheless, temporal anomalies of one kind or another
continue to serve up feasts to those who hunt alpha for a living. In
finance, it is nice to know what time it is, but it is even better to know
what time is.
Copyright 2006 by Miller Risk Advisors and Financial