→ Series: From EMH to Adaptive Dynamics
Part 1 of 3 · published 2026-09-06
Martingale pricing, the Markov property, and geometric Brownian motion set up the constant-parameter assumption behind classical market efficiency. Fama's three-form taxonomy, the joint hypothesis problem, and Grossman-Stiglitz's limits on informational efficiency show why that assumption fails — and why Lo's Adaptive Markets Hypothesis replaces an equilibrium with an ecology.
Under classical frameworks of market efficiency, asset prices are modelled as a martingale process – that the conditional expectation of tomorrow's price with respect to all currently available information is simply equal to today's price.
Mathematically, as:
$$E[P_{t+1} \mid \mathcal{F}_t] = P_t$$
where:
$P_t$ = the price today
$P_{t+1}$ = the price tomorrow
$\mathcal{F}_t$ = all currently known information set
An 18th century French betting strategy, a martingale involved doubling the stake after each loss at a game of coin toss so that any win would recover all previous losses. The term was borrowed in mathematics as it captures the essence of a fair game. The Fundamental Theorem of Asset Pricing specifies an arbitrage-free market to possess a risk-neutral measure under which discounted asset prices are martingales. Put simply, if you remove the time value of money and adjust probabilities for risk, an asset's expected future price equals its current price, holding that it is a fair game with no expected profit.
$$P_t = E^{Q}\!\left[\frac{P_{t+1}}{1+r} \,\middle|\, \mathcal{F}_t\right]$$
where:
$P_t$ = the price today
$P_{t+1}$ = the price tomorrow
$E^Q[\,\cdot \mid \mathcal{F}_t]$ = conditional expectation given all information $\mathcal{F}_t$ available at time $t$, taken under the risk-neutral measure $Q$ (not the real-world probability measure $P$)
r = the risk-free rate
Today's price equals the risk-free-discounted expected future price, under a probability measure adjusted for risk. Equivalently, the discounted price process $\tilde P_t = P_t/(1+r)^t$ is a martingale under $Q$:
$$E^{Q}[\tilde P_{t+1} \mid \mathcal{F}_t] = \tilde P_t$$
The gambler's martingale was invented to beat a fair game through clever sizing. But it fails as doubling down can't change the underlying expected value of a fair game. Doubling down simply redistributes variance (small frequent wins, one catastrophic tail loss) – while the expectation stays flat. This exact failure is why the mathematical martingale property holds for asset prices – no transformation of a strategy can extract expected profit from a process whose defining property is that its conditional expectation never moves.
The martingale property tells us specifically that the expected value of tomorrow's price based upon everything known today is simply today's price. It says nothing about the distribution, the variance, or what information was needed to compute it. The expected price is just a single number summarizing the entire distribution. Consider an asset to have endured five consecutive, sharp drops. The price tomorrow may still equal today's price (satisfying the martingale property), but how wildly it may swing in either direction still depends upon the recent turbulence. If such a dependency can exist, the process is not fully described by its present state, and its past is still doing work. Classical market efficiency needs a stronger claim to rule this out; that the entire distribution of tomorrow's price can be reduced to today's price, with no residual historic information – this is the Markov property.
Andrey Markov, in the early 20th century, introduced a mathematical concept where future events depend solely upon the current state, ignoring completely how that state was reached. The Markov Property, Markov Chain, Markov Process all follow this 'memorylessness', where the future or the next state is independent of the past.
$$P(P_{t+1} \le x \mid \mathcal{F}_t^{P}) = P(P_{t+1} \le x \mid P_t)$$
where:
$P_t$ = today's price (or state)
$P_{t+1}$ = the price (or state) at the next time step
$\mathcal{F}_t^P$ = the information generated by the price history alone, $P_0, P_1, \ldots, P_t$
The entire conditional distribution of the next state given the whole history, is identical to the conditional distribution given only the present state.
The Markov Property is given discretely as:
$$P(X_{t+1} = j \mid X_t = i, X_{t-1}, \ldots, X_0) = P(X_{t+1} = j \mid X_t = i)$$
where:
$X_t$ = the state of the process at time $t$
$i, j$ = specific states the process can occupy
$X_{t-1}, \ldots, X_0$ = the full history of states prior to time $t$
The martingale property fixes the conditional mean of the next price, whereas the Markov property fixes which information the entire conditional distribution depends upon. A process can be a martingale without being Markov and vice-versa.
The Random Walk Theory, first applied to security prices by Louis Bachelier (1900) and later formalized in the finance literature, describes the discrete-time object from which Brownian motion is built: the random walk. A simple random walk is described as a path of random succession – a sequence of independent steps, none of which carries any information about the next. It states that share price fluctuations have the same distribution and are completely independent of one another. And thus, we cannot use the past to predict the future.
$$P_t = P_{t-1} + \varepsilon_t$$
$$\varepsilon_t \sim \text{i.i.d. } F, \quad E[\varepsilon_t] = 0, \quad \operatorname{Var}(\varepsilon_t) = \sigma^2$$
where:
$\varepsilon_t \sim \text{i.i.d. } F$ = the innovations are independent and identically distributed draws from some distribution $F$
$E[\varepsilon_t] = 0$ – increments have zero mean
$\operatorname{Var}(\varepsilon_t) = \sigma^2$ – increments have finite, constant variance
Two consequences follow. First, the independence of $\varepsilon_t$ implies that increments are unforecastable from any function of past increments – the empirical content of weak-form efficiency, and the basis of the autocorrelation and variance-ratio tests employed later in this article. Second, by the Central Limit Theorem, the cumulative sum of a large number of such increments converges to a Gaussian, regardless of the distribution $F$ of any individual shock. As the time step between increments shrinks toward zero while their number grows without bound, the random walk converges to a continuous-time limit – a result formalized as Donsker's invariance principle. That limit is Brownian motion.
Robert Brown, a botanist first stumbled upon Brownian motion in 1827 when he noticed that pollen particles suspended in water jiggled in random motions with no apparent cause, when viewed under a microscope. This was later proved by Einstein in 1905 where this erratic jiggling was proven to be the result of random collisions with invisible molecules of the surrounding fluid. Einstein connected the kinetic molecular theory of heat to the visible phenomenon and reasoned that these individual unseen liquid molecules constantly bombard the suspended microscopic particles; creating an uneven, shifting net force on opposite sides of each particle. Mathematician Norbert Wiener later provided a mathematical formalization of standard Brownian motion, translating erratic movements into a continuous-time stochastic process.
For a stochastic process $W_t$ to be considered a standard Wiener process, it must satisfy four strict conditions:
Put simply, Brownian motion constitutes the ultimate mathematical realization of a continuous-time framework embodying 'memorylessness'. Incremental movements being independent of one another mean the next move carries no memory of the last one – analogous to the Markov Property. Zero-mean increments mean zero built-in drift – analogous to the martingale property.
Plain Brownian motion, however, can simply not be applied in stock price modelling. Standard Brownian motion allows values to become negative, which is not possible for a stock's price. And its additive nature implies that a \$1 move would be the same for a stock trading at \$10 or \$1000.
Geometric Brownian Motion (GBM) fixes these implications by modelling the proportional change in prices and working with prices in the log-space.
$$dP_t = \mu P_t\, dt + \sigma P_t\, dW_t$$
where:
$P_t$ = the asset price at time $t$
$\mu$ = the drift rate (expected proportional return per unit time)
$\sigma$ = the volatility (standard deviation of proportional returns per unit time)
$dW_t$ = an infinitesimal increment of the underlying Wiener process
Solving the above stochastic differential equation, we get the price explicitly as:
$$P_t = P_0 \exp\!\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right]$$
This form guarantees the price to be greater than zero at all time intervals, implies log-normal prices, and that log-returns inherit the independent Gaussian increment structure of the underlying Wiener process. This precisely shows how the GBM produces a random walk; expressing each subsequent log-return as an independent, identically distributed draw, unrelated to the one before it; the same memorylessness expressed within Brownian motion.
And yet, both $\mu$ and $\sigma$ are constants in the equation above. The model assumes the market's expected return and its uncertainty are fixed quantities, irrespective of whether the asset has been bullish for a year or crashing for a week. That assumption is quiet, almost invisible, sitting inside two Greek letters – and it is the single thread the rest of this series pulls on.
In his 1970 paper, Eugene Fama defines an efficient market to be one where security prices fully reflect all available information $\mathcal{F}_t$ at all times. Operationally, it is impossible to earn risk-adjusted abnormal profits by trading on $\mathcal{F}_t$. The EMH is formalized through the fair-game / expected-return model:
$$E[P_{t+1} \mid \mathcal{F}_t] = \left(1 + E[r_{t+1} \mid \mathcal{F}_t]\right) P_t$$
where:
$P_t$ = the price today
$P_{t+1}$ = the price tomorrow
$\mathcal{F}_t$ = all currently known information
$r_{t+1}$ = the return over the next period
This is the general form of the claim made at the opening of this article. The pure martingale, $E[P_{t+1} \mid \mathcal{F}_t] = P_t$, is the special case in which the expected return is zero – the risk-neutral idealization under which the Fundamental Theorem of Asset Pricing operates. Under the real-world measure, prices carry a risk premium and are not martingales (LeRoy, 1973; Lucas, 1978). What EMH requires is not that prices stand still on average, but that the deviation between the realized and the expected price is unforecastable. This is the fair-game property: the difference between the realized and expected price has zero conditional expectation.
$$z_{t+1} = P_{t+1} - E[P_{t+1} \mid \mathcal{F}_t], \qquad E[z_{t+1} \mid \mathcal{F}_t] = 0$$
where:
$z_{t+1}$ = the realized deviation between actual and expected price
$P_{t+1}$ = the realized price tomorrow
$E[P_{t+1} \mid \mathcal{F}_t]$ = the expected price tomorrow, given today's information
In the excess-return form:
$$x_{t+1} = r_{t+1} - E[r_{t+1} \mid \mathcal{F}_t], \qquad E[x_{t+1} \mid \mathcal{F}_t] = 0$$
where:
$x_{t+1}$ = the realized excess return, above what was expected
$r_{t+1}$ = the realized return over the next period
$E[r_{t+1} \mid \mathcal{F}_t]$ = the expected return, given today's information
Fama distinguishes the fair-game model from two stronger specifications. The submartingale model permits a positive expected drift while still ruling out exploitable predictability:
$$E[P_{t+1} \mid \mathcal{F}_t] \ge P_t$$
The random walk model, stronger still, requires that not only the conditional mean but the entire conditional distribution of returns be independent of $\mathcal{F}_t$:
$$f(r_{t+1} \mid \mathcal{F}_t) = f(r_{t+1})$$
The ordering matters for everything that follows: a random walk implies a fair game, but a fair game does not require a random walk. A process whose variance changes over time – periods of calm giving way to periods of turbulence – violates the random walk while leaving the fair-game property intact, since the conditional mean need not have moved. Efficiency in Fama's sense is therefore compatible with a market whose behaviour changes through time. It is the constant-parameter random walk of GBM, not efficiency itself, that such change contradicts.
The fair-game and random walk formulations are defined relative to an information set $\mathcal{F}_t$ – but Fama's (1970) taxonomy does not propose three competing efficiency theories, it rather applies a single theory to three successively larger information sets: $\mathcal{F}_t^{\text{weak}} \subset \mathcal{F}_t^{\text{semi-strong}} \subset \mathcal{F}_t^{\text{strong}}$.
Each form is the same claim – no exploitable predictability given $\mathcal{F}_t$, restated for a strictly larger $\mathcal{F}_t$.
Weak-form efficiency sets $\mathcal{F}_t$ as the asset's own price and return history. If this form holds, the past cannot forecast the future beyond what a random-walk would already permit and technical analysis would hold no value. Semi-strong-form efficiency sets $\mathcal{F}_t$ to include all publicly available information – earnings reports, stock-splits, corporate announcements, etc. If this form holds, fundamental analysis of available information holds no value, because the information is already reflected in the price at the moment of its release.
Strong-form efficiency extends $\mathcal{F}_t$ to all information – public and private. If this form holds, no market participant – not even a corporate insider can extract any abnormal returns from $\mathcal{F}_t$. The three forms are not a taxonomy of market types but rather, a hierarchy of what has already been adapted to. Each form asks the same question to a larger and larger $\mathcal{F}_t$ – has price fully absorbed it?
The empirical record, however, complicates this picture. Weak-form tests such as autocorrelation, runs tests, and Lo & MacKinlay's (1988) variance-ratio tests find statistically significant deviations from the random walk in U.S. equity data, concentrated within small-cap and less liquid stocks and decaying over time as those inefficiencies are arbitraged away. Semi-strong-form tests shift the focus towards the velocity at which price readjusts. Fama, Fisher, Jensen & Roll (1969) found stock prices adopting instantaneous adaptation, absorbing stock-split information in a clean jump. Conversely, Ball & Brown (1968) and Bernard & Thomas (1989) documented a non-instantaneous response to earnings announcements: prices kept drifting in the direction of the surprise for weeks post-announcement – now known as post-earnings-announcement drift. Strong-form efficiency was never defended as an empirical reality, even by Fama himself. Jaffe (1974) and later studies consistently find insiders earning abnormal returns, a gap which was closed only by regulatory enforcement (SEC Rule 10b-5) rather than by market mechanisms.
The pattern across all three forms is the same: efficiency is not binary. It is graded, and its completion – where it completes at all – takes measurable time.
Testing any form of efficiency requires maintaining a model $M$ to define a 'normal' return:
$$E[r_{t+1} \mid \mathcal{F}_t] = E_M[r_{t+1} \mid \mathcal{F}_t]$$
where:
$r_{t+1}$ = the realized return over the next period
$\mathcal{F}_t$ = all currently known information at time $t$
$E[\cdot \mid \mathcal{F}_t]$ = the true conditional expected return, given $\mathcal{F}_t$
$M$ = the maintained equilibrium asset-pricing model used to define 'normal'/expected returns
$E_M[\cdot \mid \mathcal{F}_t]$ = the expected return implied by model $M$, given $\mathcal{F}_t$
Any abnormal return is then defined only relative to $M$:
$$u_t = r_t - E_M[r_t \mid \mathcal{F}_t]$$
where:
$u_t$ = the abnormal return at time $t$
$r_t$ = the realized return at time $t$
$E_M[r_t \mid \mathcal{F}_t]$ = the return expected under model $M$, given $\mathcal{F}_t$
A statistically significant $u_t$ is therefore always ambiguous: it may reflect a genuine inefficiency, or simply an incorrect $M$. This is the Joint Hypothesis Problem – any empirical test of efficiency is always a joint test of both market efficiency and the chosen pricing model. Fama himself formalized it in his 1991 restructured taxonomy, and the historical record bears it out directly: the size and value anomalies that initially appeared to violate the CAPM in the 1980s were reclassified as compensation for priced risk once the Fama-French three-factor model replaced the CAPM as the standard benchmark $M$. No finite sequence of tests can cleanly confirm or falsify efficiency, since every rejection can always be absorbed by revising $M$ instead of revising beliefs about the market.
Two conclusions follow. First, Fama's taxonomy only specifies what information price has absorbed, and says nothing about how, or at what velocity, the absorption occurs. Lo (2017) makes this distinction explicit, and the empirical record above bears it out: the variation was never in whether adaptation occurred, but in its speed – instantaneous for stock splits, drawn out over weeks for earnings surprises, never closed at all for insider information without legal intervention. Second, the binary question the taxonomy invites – is the market efficient or not? – cannot be answered cleanly even in principle. Every test of efficiency is bundled with an assumed pricing model $M$, and a rejection can always be attributed to a flawed $M$ rather than a genuine inefficiency; no sequence of tests can separate the two. The question remains structurally unresolvable within EMH's own framework.
The joint hypothesis problem shows that EMH's equilibrium cannot be cleanly tested. Grossman & Stiglitz (1980) show something stronger: that it cannot exist. Their argument arrives at an "equilibrium degree of disequilibrium", in which perfect informational efficiency and costly information are jointly inconsistent. Competitive equilibrium refers to prices at which all arbitrage profits are eliminated. But if information-gathering – and the trading that exploits it – is costly, an equilibrium with zero profit gives informed traders no compensation for that cost. If prices fully reveal all information, there is no incentive to pay for it; and if nobody pays for information, prices cannot reveal it. Efficiency, taken literally, undermines the very activity that produces it.
A risky asset pays:
$$u = \theta + \eta$$
where:
$\theta$ = the fundamental signal, observable only at cost $c$
$\eta$ = unobservable noise, independent of $\theta$
Informed traders observe $\theta$ directly and uninformed traders observe only price $P$
Under constant absolute risk aversion (CARA) preferences, informed demand is:
$$X^{I}(P, \theta) = \frac{\theta - RP}{a\sigma_\eta^2}$$
where:
$R = 1 + r$, the gross safe return
$a$ = risk aversion
Market clearing (with λ fraction of informed traders) yields an equilibrium price that is a linear function of the statistic $w(\lambda)$:
$$w_\lambda(\theta, x) = \theta - \frac{a\sigma_\eta^2}{\lambda}\left(x - E[x^{*}]\right)$$
where:
$x$ = random per-capita asset supply (a noise/liquidity shock).
$E[x^*]$ = the mean per-capita supply
Informationally, the price is equivalent to observing $\theta$ through noise of magnitude $\propto \sigma_\eta^2/\lambda$ – the noisier the supply, the less price reveals, for a given $\lambda$.
However, as $\sigma_\eta^2 \to 0$ (a perfect signal) or if $\operatorname{Var}(x) \to 0$ (no noise), price would reveal $\theta$ correctly – removing any incentive to pay the information cost $c$. And if nobody pays, price reveals nothing – paying becomes worthwhile again. In these limits, no self-consistent $\lambda$ exists. Away from them, the equilibrium fraction of informed traders is pinned by the condition that informed and uninformed traders are equally well off:
$$e^{2ac} = 1 + n\left(1 - \operatorname{Corr}^2(P, \theta)\right)$$
where:
$n = \sigma_\theta^2/\sigma_\eta^2$, signal precision relative to noise
$\operatorname{Corr}^2(P,\theta)$ = the squared price-signal correlation (informativeness)
Equilibrium informativeness is pinned down by $c$, $n$, and $a$ alone – noise variance only enters indirectly, through its effect on $\lambda$. The result is a market that is always partially – and never fully – efficient. Some inefficiency is not a failure of the equilibrium; it is its precondition.
Grossman–Stiglitz establishes that mispricing must exist. It does not explain why mispricing, once present, is not immediately arbitraged away by rational traders. The limits-to-arbitrage literature supplies that answer, beginning with the risk that sentiment itself poses to anyone betting against it. De Long, Shleifer, Summers & Waldmann (1990) isolate a risk that irrational traders' beliefs fail to correct or worsen before reverting – even with zero fundamental uncertainty. In steady state, price breaks down as:
$$P_t = 1 + \frac{\nu(\rho_t - \rho^{*})}{1+r} + \frac{\nu \rho^{*}}{r} - \frac{2a\nu^2 \sigma_\rho^2\, r}{(1+r)^2}$$
where:
$P_t$ = price of the risky asset, whose fundamental value is normalised to 1
$\rho_t \sim N(\rho^*, \sigma_\rho^2)$ = noise traders' misperception of next-period price
$\rho^*$ = average bullishness (a permanent bias)
$\nu$ = fraction of noise traders in the population
$a$ = risk aversion
r = the risk-free rate
The final term, $\dfrac{2a\nu^2\sigma_\rho^2\,r}{(1+r)^2}$, is the core result: even with $\rho^{*}=0$, the unpredictability of noise-trader sentiment pushes the price permanently below fundamental value. Because the risk that deters arbitrageurs scales with $a\nu\sigma_\rho^2$, noise traders can earn higher expected returns (for intermediate $\rho^*$) than those betting against them. Rational arbitrageurs demand compensation simply for bearing the risk that sentiment moves further against them before it reverts.
Noise trader risk explains why arbitrage is risky. Shleifer & Vishny (1997) explain why it is also capital-constrained – and why the two interact. Real arbitrage requires capital and carries risk, and it is conducted by professionals trading external capital; textbook arbitrage requires neither capital nor risk-bearing. Outside investors, unable to evaluate a strategy directly, rationally use past returns as their only signal and exit on losses – despite those losses signalling a more attractive trade.
$$\frac{dP_2}{dS} < -1$$
where: $S$ = the noise-trader sentiment shock and
$P_2$ = price after capital withdrawal
Once arbitrageurs are fully invested, price falls more than one-for-one with the sentiment shock. Capital flight amplifies the very mispricing arbitrageurs were correcting, and the effect worsens as investors' redemption sensitivity ($\alpha$) rises:
$$\frac{\partial^2 P_2}{\partial \alpha \, \partial S} < 0$$
where: $\alpha$ = sensitivity of arbitrageurs' funds under management to past performance
Arbitrageurs remain profitable across a trade's life – consistent with Friedman's (1953) argument that destabilising speculators are driven out – yet in a subset of episodes they are forced into retreat at exactly the moment the mispricing is widest, deepening the dislocation they exist to correct. Shleifer and Vishny reconcile Friedman with observed reality: arbitrage works on average and fails precisely when it is most needed.
Long-Term Capital Management (LTCM) – a hedge fund founded in 1994 by John Meriwether, whose partners included Nobel laureates Myron Scholes and Robert Merton – ran convergence-arbitrage trades: bets that pricing gaps between closely related securities such as sovereign spreads and swap spreads would narrow towards their historical/theoretical relationships. Structurally, these trades are the Shleifer–Vishny setting made real: near-zero fundamental risk in the long run, financed with outside capital. At its height, the firm controlled over \$100B, had \$5B in assets, and derivative positions worth over \$1T – with over \$125B in borrowed assets. On August 17, 1998, Russia devalued the rouble and defaulted on its domestic debt. The global flight to quality that followed made previously uncorrelated spread trades move together in the same direction – destroying the diversification assumed by LTCM's model. Roughly \$1.9B, around 45% of remaining capital, was erased in August alone – including about \$550M in a single day. Fearing a global financial collapse, the Federal Reserve organised a \$3.625B recapitalisation from fourteen banks on September 23, 1998.
In retrospect – including by Lo himself in the 2004 AMH paper – LTCM's spread positions are widely judged to have been rational, with many of the underlying spreads eventually converging. The failure was one of survival, not analysis: correct trades, wrong timing on capital.
LTCM is the limits-to-arbitrage argument made concrete, and it exposes what EMH's equilibrium framing leaves out. Whether a mispricing is corrected depends not on whether it is "really" a mispricing, but on who is present to correct it, how much capital they command, how their investors behave under losses, and whether they survive long enough to be right. Grossman-Stiglitz shows that inefficiency is necessary; De Long et al. and Shleifer-Vishny show that it persists; LTCM shows that the correcting mechanism itself can be destroyed by the very dislocation it exists to correct. What these results describe is not an equilibrium at all, but an ecology: a market whose efficiency rises and falls with the composition, capital, and survival of its participants. That is the picture Lo (2004) formalises as the Adaptive Markets Hypothesis.
Andrew Lo, in his AMH papers (2004, 2005) generalizes all three results into a single claim: efficiency isn't a fixed equilibrium property, rather a population-dynamics outcome, set by: