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I have written a simulation that matches buy and sell orders, keeps track of an order book and simulates trades. My first pass at order submission was to generate random orders around the bid/ask spread. This does not produce a random walk in trade prices. What happens is the orders are distributed evenly but around the bid/ask spread they get traded out but beyond this they start clumping together thereby forming 'price resistance'for want of a better word. Is there an algorithm I can use that will result in a random walk for trade prices?

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Analyzing this question theoretically, I think the following strategy will result in a random walk.

  1. the average size of limit orders is much smaller than the average size of market orders;
  2. the size of market order is random, and subjects to a kind of distribution;
  3. market buy orders and market sell orders arrive at random but evenly;
  4. the size of limit order subjects to a kind of distribution.

I do not know your strategy by detail, so I am not sure if your have considered the above items already. But by guess, it may be the first requirement that bothers you. Too many limit orders on the best-bid / best-ask seems tend to prevent market prices from change. In fact, as long as limit orders at best-bid / best-ask have the opportunity to be exhausted, the simulation strategy will generate a random walk, you just need a longer time to watch the price to walk away. Any way, it can still be called a "random walk".

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You have intense academic research on orderbook dynamics simulations, just cite:

  1. Econophysics: Empirical facts and agent-based models, by Anirban Chakraborti, Ioane Muni Toke, Marco Patriarca, Frederic Abergel (Arxiv 2010)
  2. High Frequency Simulations of an Order Book: a Two-Scales Approach by: Charles-Albert Lehalle, Olivier Guéant, Julien Razafinimanana, In Econophysics of Order-Driven Markets (2010) edited by: F. Abergel, B. K. Chakrabarti, A. Chakraborti, M. Mitra
  3. The Price Impact of Order Book Events, by Rama Cont, Arseniy Kukanov, Sasha Stoikov (Arxiv 2011)

(1) is a generic survey, (2) provides you a way to build consistent simulations and different scales (i.e. controlling the volatility, the bid-ask spread and their relation along the whole day), (3) is more local but allow to simulate a consistent way the order flow dynamics and the price one.

One simple way to improve you simulations is to:

  • have a model for the bid-ask spread
  • choose Point processes (Poisson or better: Hawkes) to model the input and output in the queues
  • use Rama's model to generate a price: when a queue (Bid or Ask) equals zero, move the price (down or up)

If you want to have proofs that the diffusive limit of a point process will be a Brownian motion, you can read for example:

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