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It is modeled that a market maker will post a limit order for a period of time and will maintain that order until the set period of time expires, or the order is filled, and then he will provide a new quote based on his inventory.

In the modern day HFT world, how long is this? Is it in the range of milliseconds or minutes? Is a static or dynamic time optimal?

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"The Speed of the Equity Markets" by U.S. Securities and Exchange Comission gives you the precise answer:

Distributions of quote lifetimes show that quotes can be posted for as long as many minutes, or canceled in as short as a microsecond -- the distributions we compute each quarter require 24,000 separate bins to reveal structural features over such a broad time. However, by selecting a few key points of those distributions we can convey an overall sense of the absolute and relative speeds of the market.

The chart below presents such a summary of quote lifetimes over the period of Q2 2013 for corporate stocks traded across all exchanges for which there is sufficient data to produce these metrics

enter image description here

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A lot of it is event based. If some of your risk limits are hit, you end up cancelling orders. Also, it might be strategy dependent. A market maker would not cancel his orders if they arent filled and the market (or order book) doesnt change. A alpha seeker on the other hand might cancel his order if his orders arent filled within a specified time. If you are talking abut replacing or modifying orders, then the dynamics are slightly different.

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Depends on the market. For Canadian equities the quotes typically are entered /cancelled and resubmitted all the time. If it’s a blue chip stock like RY.to from my experience watching the market depth it’s typically second by second, however market makers are competing with eachother for milliseconds or less. However when you deal with illiquid stocks trading on the venture, orders can stay there for a month without getting a fill. When you move into derivative markets, options are quoted on a very high frequency again because they’re quoted against an underlying.

Not sure about American markets, but I do know they’re more competitive so I’d assume there’s much more speed for their HFT market-making algos.

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Milliseconds is much closer to the reality, than minutes. It's rarely a static time threshold and usually depends on the current LOB state.

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