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Order books

sekai.finance.lob is a limit order book for one instrument, in integer price ticks, plus a stochastic model of the background traders that keep it alive. Together they are the market behind LimitOrderBook-v0 and the multi-agent LimitOrderBookMarket, and they work on their own for simulation studies.

OrderBook

A continuous double auction with price-time priority: an incoming order matches the opposite side while the prices cross, best price first and first in, first out within a price level, and always executes at the resting (maker) order's price. Whatever a limit order cannot fill rests in the book; whatever a market order cannot fill is dropped.

import numpy as np
from sekai.finance import BACKGROUND, OrderBook, Side

book = OrderBook()
book.limit(Side.ASK, 101, 5)  # background sellers (owner BACKGROUND by default)
book.limit(Side.ASK, 102, 10)
book.limit(Side.BID, 99, 4)
mine, fills = book.limit(Side.BID, 99, 3, owner=7)  # joins the queue behind 4 lots
print(book)  # OrderBook(time=0, best_bid=99, best_ask=101, orders=4)
print(book.mid(), book.spread(), book.queue_ahead(mine), book.volume_at(Side.BID, 99))  # 100.0 2 4 7

A market buy of 8 lots takes the whole level at 101 and part of the next:

order_id, fills = book.market(Side.BID, 8, owner=7)
for fill in fills:
    print(fill.price, fill.quantity, fill.maker_owner, fill.taker_owner)
print(book.best_ask(), book.last_price)  # 102 102

101 5 -1 7
102 3 -1 7

A sell limit at 99 for 5 lots crosses the bid: it fills the 4 background lots that arrived first, then 1 of owner 7's 3, and fills completely (so it returns no resting id):

order_id, fills = book.limit(Side.ASK, 99, 5, owner=3)
print(order_id, [(fill.quantity, fill.maker_owner) for fill in fills])  # None [(4, -1), (1, 7)]
print(book.get(mine))  # Order(order_id=4, side=<Side.BID: 1>, price=99, quantity=2, owner=7, time=0.0)
Method Does
limit(side, price, quantity, owner=BACKGROUND) match, then rest the remainder; returns (order_id or None, fills)
market(side, quantity, owner=BACKGROUND) match and drop the remainder; returns (order_id, fills)
cancel(order_id) remove a resting order; False if it is no longer resting
reduce(order_id, quantity) lower a resting order's quantity, keeping its place in the queue (0 cancels)
best_bid(), best_ask(), mid(), spread() top of book in ticks (None if a side is empty)
depth(levels) (bid_prices, bid_sizes, ask_prices, ask_sizes), best level first, zero-padded
volume_at(side, price), n_levels(side) aggregate volume at a price; number of price levels
queue_ahead(order_id) volume that must trade at the order's price before it starts to fill
get(order_id), orders_of(owner), n_orders_of(owner) resting orders (live records: read, never mutate)
clear() empty the book and reset time, last_price and the id counter

Side.BID is +1 and Side.ASK -1 (the sign of the owner's position change on a fill). A Fill records the price, quantity, both order ids and owners, the taker's side and the time; an Order its id, side, price, remaining quantity, owner and arrival time. owner is any integer: BACKGROUND (-1) for background flow, agent ids otherwise. The book itself allows self-trades (an order may fill against a resting order of the same owner); the environments prevent them for their agents (below). book.time is a plain attribute the caller (or the background flow) advances; it stamps orders and fills. book.last_price is the last fill's price.

print(book.depth(3))  # (array([99,  0,  0]), array([2, 0, 0]), array([102,   0,   0]), array([7, 0, 0]))
print(book.reduce(mine, 1), book.queue_ahead(mine), book.cancel(mine), book.cancel(mine))  # True 0 True False

Complexity. Prices are kept in sorted lists, so creating or removing a price level is an \(O(\log L)\) search over \(L\) levels; matching costs \(O(1)\) per fill. Cancelling is \(O(1)\) by lazy deletion: a cancelled order is zeroed in place and dropped when it reaches the front of its queue (or when cancelled orders come to outnumber live ones at a level), while every level's aggregate volume stays exact.

Zero-intelligence order flow

ZeroIntelligenceFlow drives a book with randomly arriving orders in the spirit of Smith, Farmer, Gillemot & Krishnamurthy (2003) and Cont, Stoikov & Talreja (2010): traders with no strategy at all, whose aggregate behaviour nevertheless produces a spread, a depth profile and a diffusing price. Three kinds of event arrive as independent Poisson processes:

  • Limit orders, at rate \(\lambda_L\) (limit_rate) per side. A buy is placed \(k\) ticks below the best ask and a sell \(k\) ticks above the best bid, with \(k \in \{1, \dots, K\}\) (\(K\) = max_offset) and \(\Pr(k) \propto e^{-\alpha (k - 1)}\) (\(\alpha\) = offset_decay; 0 is uniform). So a limit order never crosses on arrival, and \(k = 1\) improves on, or joins, the best quote.
  • Market orders, at rate \(\lambda_M\) (market_rate) per side, taking the opposite side.
  • Cancellations, at rate \(\delta\) (cancel_rate) per resting background order, each removing a uniformly random one. Agents' orders are never cancelled by the flow.

Order sizes are \(1 + \text{Poisson}(\bar m - 1)\), so their mean is \(\bar m\) (mean_size). With \(n\) resting background orders the next event comes after an exponential time with total rate

\[ \Lambda = 2\lambda_L + 2\lambda_M + \delta\, n , \]

which is recomputed after every event. If an event empties one side of the book, a background limit order is placed there at once, so the book stays two-sided.

Parameter Default
limit_rate 10.0 \(\lambda_L\), limit orders per unit time per side
market_rate 2.0 \(\lambda_M\), market orders per unit time per side
cancel_rate 0.5 \(\delta\), cancellations per resting order per unit time
max_offset 10 \(K\), the farthest limit-order distance in ticks
offset_decay 0.5 \(\alpha\), decay of the offset distribution per tick
mean_size 3.0 \(\bar m\), mean order size

initialise(book, rng, *, mid, levels=10, orders_per_level=3) clears a book and seeds it symmetrically around mid; advance(book, rng, until) simulates every event up to time until and returns the fills, including those against agents' resting orders:

from sekai.finance import ZeroIntelligenceFlow

flow = ZeroIntelligenceFlow()
rng = np.random.default_rng(0)
flow.initialise(book, rng, mid=10_000)
print(book)  # OrderBook(time=0, best_bid=9999, best_ask=10001, orders=60)
fills = flow.advance(book, rng, until=20.0)
print(book, len(fills))  # OrderBook(time=20, best_bid=10000, best_ask=10001, orders=37) 129

What the parameters do

The steady state is set by the balance between orders arriving (limit orders) and leaving (fills and cancellations). The table measures it over ten sessions of 300 time units after a warm-up of 20, from the standard deviation of the mid's change over one time unit (ticks per \(\sqrt{\text{time}}\); it scales diffusively), the mean spread, the mean number of resting orders and the number of fills per unit time:

def measure(flow, sessions=10, length=300, warmup=20):
    moves, spreads, depth, n_fills = [], [], [], 0
    for seed in range(sessions):
        rng = np.random.default_rng(seed)
        book = OrderBook()
        flow.initialise(book, rng, mid=10_000)
        flow.advance(book, rng, warmup)
        mids = [book.mid()]
        for t in range(1, length + 1):
            n_fills += len(flow.advance(book, rng, warmup + t))
            mids.append(book.mid())
            spreads.append(book.spread())
            depth.append(len(book))
        moves.append(np.diff(mids))
    return np.concatenate(moves).std(), np.mean(spreads), np.mean(depth), n_fills / (sessions * length)


print(f"{'flow':20s} {'mid vol':>8s} {'spread':>7s} {'orders':>7s} {'fills':>6s}")
for label, params in {
    "defaults": {},
    "cancel_rate=0.1": {"cancel_rate": 0.1},
    "cancel_rate=1.0": {"cancel_rate": 1.0},
    "cancel_rate=2.0": {"cancel_rate": 2.0},
    "limit_rate=20": {"limit_rate": 20.0},
    "market_rate=4": {"market_rate": 4.0},
    "offset_decay=0.1": {"offset_decay": 0.1},
    "offset_decay=2.0": {"offset_decay": 2.0},
}.items():
    vol, spread, orders, fills = measure(ZeroIntelligenceFlow(**params))
    print(f"{label:20s} {vol:8.2f} {spread:7.2f} {orders:7.1f} {fills:6.1f}")

flow                  mid vol  spread  orders  fills
defaults                 0.50    1.08    32.1    6.5
cancel_rate=0.1          0.05    1.00   158.0    6.7
cancel_rate=1.0          1.31    1.21    16.3    6.4
cancel_rate=2.0          2.60    1.54     8.6    6.1
limit_rate=20            0.05    1.00    72.2    6.6
market_rate=4            1.31    1.29    24.8   13.0
offset_decay=0.1         1.45    1.61    32.2    6.6
offset_decay=2.0         0.04    1.00    31.9    6.7

Volatility comes from the best quotes being consumed faster than they are replenished:

  • More cancellation (or more market orders) thins the book, so the best levels empty more often and the mid moves more; the spread widens slowly.
  • More limit orders thicken the queues at the best prices until the mid barely moves.
  • Offset decay concentrates new orders near the touch (large \(\alpha\), deep best levels, a frozen mid) or spreads them out (small \(\alpha\), thin best levels, more volatility and a wider spread).
  • The fill rate is driven by market_rate (each market order may fill several resting orders); mean_size scales volumes without changing the price dynamics much.

The defaults give a liquid, one-tick market whose mid diffuses by about half a tick per unit time: with decisions every time unit, an agent sees the price move on most steps without being swamped by it.

In the environments

LimitOrderBook-v0 (sekai.envs.finance.LimitOrderBookEnv) puts one agent (owner 0) in a book driven by a ZeroIntelligenceFlow (pass your own with flow=). Each episode initialises the book at mid_price and lets the flow run for warmup time units; then, every decision_interval, the agent may send a market order, then rests or moves a bid and an ask, and the flow advances to the next decision. A market order first cancels the agent's own resting quote on the opposite side (a buy cancels its ask, a sell its bid), so the agent never trades with itself: self-trade prevention, as in the cancel-resting mode of most exchanges; the quote is then placed again if the action asks for it, at the back of its queue. Quote levels count from the best price among the other traders' orders (level 1 joins it, level \(k\) rests \(k - 1\) ticks behind), and a quote already at the requested level is left alone, including a level-1 quote that is itself the best price, alone there; so repeating an action moves a quote only to follow the other traders' best price, never off the best. A quote left at the same price keeps its queue position, so queue_ahead matters and appears in the observation; moving it sends it to the back. The live book is env.unwrapped.book.

LimitOrderBookMarket puts several agents (owners 0, 1, ...) in the same book, each deciding on its own clock: an event-driven multi-agent environment in which one agent's market order can fill another's resting quote (but not its own: self-trade prevention applies to every agent). All the actions of a step are checked before any is carried out. The observations, actions and rewards, the baselines and the multi-agent stepping rules are on the environments page.

See also