The wide availability of high-frequency data for many financial instruments stimulates a upsurge interest in statistical research on the estimation of volatility. Jump-diffusion processes observed with market microstructure noise are frequently used to model high-frequency financial data. Yet, existing methods are developed for either noisy data from a continuous diffusion price model or data from a jump-diffusion price model without noise. We propose methods to cope with both jumps in the price and market microstructure noise in the observed data. They allow us to estimate both integrated volatility and jump variation from the data sampled from jump-diffusion price processes, contaminated with the market microstructure noise. Our approach is to first remove jumps from the data and then apply a noise-resistent method to estimated the integrated volatility. The asymptotic analysis and the simulation study reveal that the proposed wavelet methods can successfully remove the jumps in the price processes and the integrated volatility can be estimated as well as the case with no presence of jumps in the price processes. In addition, they have outstanding statistical efficiency. The methods are illustrated by applications to two high-frequency exchange rate data sets.

This paper reviews recent theoretical and empirical research on high-frequency trading (HFT). Economic theory identifies several ways that HFT could affect liquidity. The main positive is that HFT can intermediate trades at lower cost. However, HFT speed could disadvantage other investors, and the resulting adverse selection could reduce market quality.

Over the past decade, HFT has increased sharply, and liquidity has steadily improved. But correlation is not necessarily causation. Empirically, the challenge is to measure the incremental effect of HFT beyond other changes in equity markets. The best papers for this purpose isolate market structure changes that facilitate HFT. Virtually every time a market structure change results in more HFT, liquidity and market quality have improved because liquidity suppliers are better able to adjust their quotes in response to new information.

Does HFT make markets more fragile? In the May 6, 2010 Flash Crash, for example, HFT initially stabilized prices but were eventually overwhelmed, and in liquidating their positions, HFT exacerbated the downturn. This appears to be a generic feature of equity markets: similar events have occurred in manual markets, even with affirmative market-maker obligations. Well-crafted individual stock price limits and trading halts have been introduced since. Similarly, kill switches are a sensible response to the Knight trading episode.

Many of the regulatory issues associated with HFT are the same issues that arose in more manual markets. Now regulators in the US are appropriately relying on competition to minimize abuses. Other regulation is appropriate if there are market failures. For instance, consolidated order-level audit trails are key to robust enforcement. If excessive messages impose negative externalities on others, fees are appropriate. But a message tax may act like a transaction tax, reducing share prices, increasing volatility, and worsening liquidity. Minimum order exposure times would also severely discourage liquidity provision.

High-frequency trading, as distinct from other forms of algorithmic trading, does not enhance social welfare and can lead to instability. Mandating short, regular trading delays would prevent high-frequency trading while retaining the benefits of algorithmic trading.

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