Simulating a Limit Order Book with a Vanishing Tick Size
Summary
The document discusses how to simulate the SFGK continuous double auction limit order book model when the theoretical tick size approaches zero. The original question concerns Poisson order arrivals over an unbounded price line and how discrete price levels might approximate that limit. The proposed practical approach restricts prices to a finite interval and samples a price uniformly within that range.
A sampled price is assigned to a discrete price level by dividing the interval into tick-sized buckets. As the tick size decreases, the number of available levels increases while the bounded range remains fixed. The response also notes that prices inside the spread need not follow the same bucketing rule: they can be kept as sampled or represented with finer levels. This is a simulation suggestion, not a validation of the model's limiting behavior. The document does not discuss how to select the interval bounds, establish convergence, or test whether uniform sampling matches the intended arrival distribution.
Key ideas
- The proposed simulation samples prices from a bounded uniform interval.
- Discrete price levels are formed by bucketing sampled prices according to tick size.
- Reducing tick size increases the number of levels in a fixed price range.
- The response suggests leaving prices inside the spread unbucketed or using finer pricing levels.
- The approach is presented as a practical suggestion without convergence analysis or parameter guidance.
Tags
Full text
# Simulation of SFGK Model for limit order books # Simulation of SFGK Model for limit order books am trying to simulate the SFGK model from the paper "Statistical theory of the Continuous Double Auction", Eric Smith, J. Doyne Farmer, Laszlo Gillemot and Supriya Krishnamurthy [1]. The model simulates the Limit order book for various parameter values. The arrival processes are assumed to be poisson. The simulations are carried out for a limiting value of the tick size which is zero. However, zero tick size would mean the entire real line. I am uncertain as to how one simulates poisson processes for the entire line. That is essentially the issue. If we do chose a finite set of values with the tick size being non-zero. Then how are the different tick sizes different? They really can be scaled version of unit tick size. I am trying to get information on how to go from finite tick sizes to the zero tick size limit. [1] https://arxiv.org/abs/cond-mat/0210475 ## Answer by Ramesh Kadambi (score 1) https://quant.stackexchange.com/a/74938 The solution is really much simpler than I realized. This is fairly simple to simulate really. I just wanted to post an answer for folks if interested. The idea here is to use a uniform distribution [-L,L]. Once you get the number you bucket it to a price level based on where it falls. Divide the range [-L,L] in tick sizes and you can let that tick size go to zero. The only thing that will change is the number of levels in the range [-L,L]. So the number of levels will be 2L/tick size. The prices inside the spread are not bucketed, keep them as is or allow for finer levels of pricing inside the spread. I will let the moderators decide if this is even useful to keep around or the post should be entirely deleted.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.