Imagine you are a trader who has just discovered that an asset is underpriced. In a market without noise, there's no way for you to take advantage of this discovery (Aumann 1976, Milgrom and Stokey 1982). The moment you try to buy a share, other traders will immediately realise that you must have discovered some good news, and so will not sell you a share at the old price (Grossman 1976).

## Noise

Noise undercuts this no-trade theorem. In a market with noise, there are always some shares being bought or sold for erratic, non-fundamental reasons. When you try to buy a share, other traders will not immediately decide that you have discovered good news, because your buy order might just be random noise. This allows you to trade on, and profit from, your discovery. Noise "makes financial markets possible" (Black 1986).

Where exactly does noise come from? Research has pointed to several mechanisms. Early papers suggested that noise comes from random supply shocks, or from liquidity traders with random cash demands. There's also a lot of research into noise traders whose random demand stems from irrational beliefs. Other papers have modelled noise as the result of agents' need to hedge random endowment shocks.

## Computational complexity

We would like to propose an alternative noise-generating mechanism: computational complexity (Chinco and Vos 2019). In modern financial markets, the same asset is often held for different reasons by funds that follow a wide variety of threshold-based trading rules. It's computationally infeasible to predict how the trading rules will interact with one another, and so the net demand coming from the interacting mass of funds will appear random – even if market participants are fully rational, and each trading rule is simple.

To show how computational complexity generates noise, we studied a theoretical model representing three common features of modern financial markets:

**There are many funds following a variety of different trading rules.** This statement is self-evident. It applies equally to hedge funds, mutual funds, pension funds, algorithmic traders, and index funds.
**The same asset is held by different funds for different reasons**. This is because the market is populated by such a large heterogeneous collection of funds. For example, one fund's value stock might be another fund's low-volatility stock.
**Many funds use threshold-based trading rules.** For instance, the PowerShares S&P 500 Low-Volatility ETF (SPLV) tracks a benchmark consisting of the 100 lowest-volatility S&P 500 stocks. This is a threshold-based trading rule because an arbitrarily small change in a stock's volatility can move it from 101st to 100th place on the low-volatility leaderboard. When this happens, SPLV will sell that stock and build a new position in another. It therefore affects two assets in equal-but-opposite ways.

When there are so many different funds, with overlapping holdings, using so many different threshold-based trading rules, a small change in asset A's price can cause one fund to buy asset A and sell asset B. This can then cause a second fund, using a different threshold, to sell asset B and buy asset C, and so on.

Under these widely-observed conditions, it's possible to determine *if* an unrelated asset Z will be affected by one of these rebalancing cascades, but the problem of determining *how* that asset will be affected – buy or sell? – will be computationally infeasible. Formally, it is an NP-hard problem. Put differently, in a large market, the authors show that it's computationally infeasible to predict the demand coming from a rebalancing cascade in response to any initial shock unless, at minimum, a polynomial-time algorithm can be found that solves every NP problem.

No such algorithm is known, and it is widely believed that no such algorithm exists.

But, proving this remains “the central unsolved problem of theoretical computer science” (Aaronson 2013). So and the sign of the resulting demand shock may as well be thought of as a coin flip – it may as well be noise. This is true even if agents are fully rational, and each fund involved in the cascade follows a simple deterministic trading rule.

## If, but not how

The key insight is not that financial markets can be computationally complex in a few special cases, or that the resulting shocks might appear random to unsophisticated agents. Rather, our model shows how common market features can combine to produce shocks which are computationally intractable for all agents.

It's the combination of these two elements – ubiquitous features, universal intractability – that is new and noteworthy. This suggests that computational complexity is an important noise-generating mechanism in modern financial markets.

We empirically verified these predictions using data from ETF Global on the daily holdings of exchange traded funds (ETFs) from January 2011 to December 2017. Stocks on the cusp of more ETF rebalancing thresholds experienced more noise. When a stock Z was sitting on an above-median number of thresholds, its ETF rebalancing volume was 2.06% higher in the five days immediately after a shock to an unrelated stock A. This increase is no more likely to be made up of buy orders than of sell orders.

These results suggest that it is possible to predict *if* stock Z will be affected by an ETF rebalancing cascade, but not *how* stock Z will be affected. In short, susceptibility to ETF rebalancing cascades predicts noise volatility.

## References

Aaronson, S (2013), *Quantum computing since Democritus,* Cambridge University Press.

Aumann, R (1976), "Agreeing to disagree", *Annals of Statistics* 4(6): 1236-1239.

Black, F (1986), "Noise",* Journal of Finance *41(3): 528-543.

Chinco, A and V Fos (2019), "The Sound of Many Funds Rebalancing", CEPR discussion paper 13561.

Grossman, S (1976), "On the efficiency of competitive stock markets where trades have diverse information", *Journal of Finance *31(2): 573-585.

Milgrom, P and N Stokey (1982), "Information, trade, and common knowledge", *Journal of Economic Theory* 26(1): 17-27.