The Limits of Arbitrage

Using the example of two otherwise-identical Bund futures contracts trading on different exchanges at different prices, the authors show that even a textbook-perfect convergence trade requires the arbitrageur to post good-faith margin money and remain exposed to interim mark-to-market losses if the mispricing widens before it narrows: “even if eventually the prices of the two contracts converge and the arbitrageur makes money, in the short run he loses money and needs more capital. The model of capital-free arbitrage simply does not apply.” If the arbitrageur has limited capital, he “may run out of money and have to liquidate his position at a loss” even though the trade is a sure winner at maturity – turning what should be riskless arbitrage into what the authors call “risk arbitrage,” the form taken by most real bond and equity market arbitrage.

The authors argue that “the millions of little traders are typically not the ones who have the knowledge and information to engage in arbitrage”; in practice, arbitrage is “conducted by relatively few professional, highly specialized investors who combine their knowledge with resources of outside investors to take large positions,” so that “brains and resources are separated by an agency relationship.” This agency structure – money from investors with limited market-specific knowledge, invested by specialists with that knowledge – is the feature the rest of the paper’s model is built to analyze, and it is precisely this separation (not the mere fact that arbitrage requires capital) that the authors say generates the paper’s central results.

PBA refers to the fact that, because investors cannot observe or verify an arbitrageur’s strategy or ability directly, they rationally allocate and withdraw capital based on the arbitrageur’s past returns rather than on the (unobservable) current expected return of his trades; the paper models investors’ fund supply as an increasing function G of the arbitrageur’s gross past return, G(1)=1, G’>0, G’’≤0. Because poor performance could reflect either bad luck (a deepening noise-trader shock) or genuinely inferior ability, and investors cannot tell the two apart, “the seemingly perverse behavior of taking money away from an arbitrageur after noise trader sentiment deepens, i.e., precisely when his expected return is greatest, is a rational response to the problem of trying to infer the arbitrageur’s (unobserved) ability…from past returns.” The authors note this signal-extraction logic implies responsiveness to past performance is greater when ability varies a lot across arbitrageurs, and smaller when noise-trader-sentiment variance is large relative to variation in ability.

The model has three periods and three agent types – noise traders, risk-neutral specialized arbitrageurs, and outside investors who supply capital to arbitrageurs – trading a single asset whose fundamental value V is known only to arbitrageurs and becomes public at t=3 (so there is no long-run fundamental risk, isolating “pure” arbitrage). Noise traders generate a demand shock S_t each period, with S_1 known at t=1 but S_2 uncertain (it may deepen to S > S_1, or correct to zero); arbitrageurs have limited resources F_t, assumed insufficient to fully correct mispricing (F_t < S_t). The key addition relative to De Long et al. (1990), which stressed noise-trader risk itself as a deterrent to arbitrage, is that here the arbitrageur’s own capital base F_t is endogenously tied to past fund performance via PBA, so capital available to exploit a worsening mispricing can shrink exactly when the mispricing (and hence the opportunity) grows.

Proposition 1 shows there is a threshold probability q* (where q is the probability the mispricing corrects at t=2 rather than deepens) such that arbitrageurs fully invest their capital at t=1 (D_1 = F_1) when q is below q*, but hold back reserves (D_1 < F_1) when q exceeds it; the authors describe the full-investment case as “extreme circumstances.” In their numerical example (V=1, F_1=0.2, a=1.2, S_1=0.3, S_2=0.4), q* = 0.35: for q below that, arbitrageurs invest all 0.2 and the first-period price is 0.9, with the price falling to 0.7636 if the shock deepens; for q = 0.5 (above q*), arbitrageurs hold back some funds, investing only 0.1743 at t=1 (so p_1 = 0.8743, a less aggressive initial trade) in exchange for having more capital (0.1766) available if the shock does deepen (p_2 = 0.7766) – illustrating that both corner and interior equilibria are plausible depending on parameters.

Proposition 3 shows that if arbitrageurs are fully invested at t=1 and the noise-trader shock deepens at t=2, then (for performance sensitivity a>1) their period-2 resources fall short of their period-1 investment, both in dollar terms and as a share of the asset’s price – “arbitrageurs bail out of the market when opportunities are the best.” Proposition 4 sharpens this: at the fully invested equilibrium, prices fall more than one-for-one with the noise-trader shock (dp₂/dS < −1), and this overreaction intensifies as PBA sensitivity (a) rises. The authors contrast this explicitly with “more standard models, in which arbitrageurs are most aggressive when prices are furthest away from fundamentals,” and connect it to Friedman’s (1953) argument that destabilizing arbitrage would require arbitrageurs to lose money on average: “our model is consistent with Friedman in that, on average, arbitrageurs make money and move prices toward fundamentals. However,…the times when they lose money are precisely the times when prices are far away from fundamentals, and…the trading by arbitrageurs has the weakest stabilizing effect” at exactly those times.

The authors tie their liquidation result to work on market liquidity (Shleifer and Vishny 1992; Stein 1995): in their model, an asset is “liquidated involuntarily at a time when the best potential buyers – other arbitrageurs of this asset – have limited funds and external capital is not easily forthcoming,” so the resulting fire sale pushes the price even further below fundamental value, holding the underlying noise-trader shock fixed. The broader implication they draw is that limited market “resiliency” – the sensitivity of the price decline to the underlying shock – means arbitrage fails to bring prices close to fundamental value precisely in the extreme circumstances where correcting mispricing matters most.

The authors argue arbitrage capital concentrates where value can be ascertained with confidence and realized quickly and where fundamental risk is low – the bond market, where future cash flows are nearly certain, and the foreign exchange market, especially when central banks try to defend non-market exchange rates, letting arbitrageurs “tell that prices are not equal to fundamental values and to profit quickly.” By contrast, “both the absolute and the relative values of different securities are much harder to calculate” in stock markets, so arbitrage opportunities there are harder to identify and verify. They add a risk-based reason specific to their model: because specialized arbitrageurs are undiversified, idiosyncratic volatility deters them even though it would not deter a well-diversified investor, and markets where fundamental uncertainty resolves slowly can have a high long-run but low short-run ratio of expected “alpha” to volatility – exactly the unfavorable risk-reward profile, over the horizons that matter for reputations and performance-based capital flows, that stocks often present.

The efficient-markets approach “explains away” an apparent anomaly by arguing excess returns must be compensation for exposure to some unmeasured systematic risk factor; the authors argue this response rests on “a highly implausible assumption of many diversified arbitrageurs,” when in reality concentrated, specialized, undiversified arbitrageurs care about total risk, including idiosyncratic risk, not just systematic risk. Applied to the value/glamour anomaly – where Fama and French (1992) posit an unidentified “distress factor” – the authors instead point to investor extrapolation of past earnings growth as the source of mispricing, and to the volatility of the arbitrage trade itself as the reason it persists: a diversified long position in value stocks has historically beaten the S&P 500 only “roughly a 60-40” proposition over one year (though much more reliably over five years), so the trade is volatile and slow to pay off over the short horizons that matter for a fund with a short track record. They illustrate the mechanism with commercial bank stocks in 1990-1991: as prices fell, “many traditional value arbitrageurs invested heavily,” but “the prices kept falling, and many value arbitrageurs lost most of their funds under management,” forcing liquidations that added further downward pressure – after which “the returns on banking stocks have been very high, but many value funds did not last long enough to profit from this recovery.”