as-market-maker

Python implementation of the Avellaneda-Stoikov (2008) optimal market-making strategy, with Poisson and Hawkes self-exciting fill models, head-to-head Monte Carlo against two baselines, and a γ-sensitivity sweep that recovers the expected interior-optimum Sharpe ratio.

Avellaneda, M. & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance 8(3), 217-224.

$ pytest tests/                      ->  15 passed in 0.48s

$ python examples/run_sim.py         (200 Monte Carlo paths per strategy)

Poisson (memoryless) fills
  AvellanedaStoikov     PnL =  +57.31 +/-  5.85   Sharpe =  9.80   <|q|> = 1.03
  Symmetric (q-blind)   PnL =  +60.97 +/- 10.86   Sharpe =  5.61   <|q|> = 4.21
  Constant spread       PnL =  +60.97 +/- 11.83   Sharpe =  5.15   <|q|> = 4.37

Hawkes (self-exciting) fills
  AvellanedaStoikov     PnL = +215.29 +/- 12.35   Sharpe = 17.43   <|q|> = 0.78
  Symmetric (q-blind)   PnL = +220.89 +/- 29.30   Sharpe =  7.54   <|q|> = 10.12
  Constant spread       PnL = +232.16 +/- 26.08   Sharpe =  8.90   <|q|> = 9.56

AS trades a few cents of expected P&L for ~2x the Sharpe by skewing quotes away from accumulating inventory. Under self-exciting Hawkes fills (realistic order-flow clustering) the gap widens to >2x.

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The math

The control problem

A market maker continuously quotes a bid S_t^b and ask S_t^a on a risky asset with mid-price S_t evolving as a Brownian motion:

dS_t = σ · dW_t

The trader's cash x_t and inventory q_t evolve when their quotes are hit. Fills arrive as point processes with intensity that decays exponentially with quote distance from mid:

λ^a(δ^a) = A · exp(-k · δ^a), λ^b(δ^b) = A · exp(-k · δ^b)

Wealth at terminal T is W_T = x_T + q_T · S_T. The maker maximizes the CARA-utility of terminal wealth:

max E[ -exp(-γ · W_T) ]

with risk aversion γ > 0.

The HJB equation and its closed-form approximation

Standard dynamic programming yields the Hamilton-Jacobi-Bellman equation for the value function u(s, x, q, t). Avellaneda & Stoikov derive an approximation valid in the high-frequency limit (the value function is well-approximated by an exponential ansatz). Two closed-form quantities fall out:

(1) The reservation price — the maker's indifference price given they currently hold q shares of inventory:

r(s, q, t) = s − q · γ · σ² · (T − t) [eq. 9 in the paper]

The skew is linear in inventory and shrinks to zero as t → T (no future inventory risk to hedge).

(2) The total quoted spread (ask side + bid side together):

δ^a + δ^b = γ · σ² · (T − t) + (2/γ) · ln(1 + γ/k) [eq. 10]

Decomposed into:

• Diffusion / risk-aversion term γσ²(T−t): vanishes at terminal time
• Liquidity / competition term (2/γ)·ln(1 + γ/k): scales with how quickly fill intensity decays — denser quotes if k large

Quote placement

In the high-frequency approximation, quotes are placed symmetrically around the reservation price r (not around the mid s):

bid_price = r − (δ^a + δ^b)/2 ask_price = r + (δ^a + δ^b)/2

All inventory skew enters through r. If the maker is long, r < s, so their midpoint of quotes is below the public mid — they're trying to unload by inviting buyers (ask near mid, bid far below).

Limit behaviors (all unit-tested in tests/test_model.py)

Limit	Predicted behavior
q → 0	r → s, symmetric quotes about mid
q > 0 (long)	r < s, quotes skew DOWN
q < 0 (short)	r > s, quotes skew UP
t → T	diffusion term → 0, spread minimum
σ → ∞	spread widens linearly
k → ∞	liquidity term → 0, spread tightens

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Fill models

Poisson (PoissonFillModel)

λ(δ) = A · exp(-k · δ), memoryless. This is the AS paper's assumption and yields a clean closed-form analysis.

Hawkes (HawkesFillModel)

Self-exciting:

λ(t, δ) = A · exp(-k · δ) · (1 + Σᵢ α · exp(-β · (t − tᵢ)))

Each fill bumps the intensity by α, which then decays at rate β. This captures the empirical fact that order arrivals cluster in time (Bacry, Mastromatteo & Muzy 2015). Under Hawkes fills the AS strategy's inventory-control advantage becomes more important, because clustered fills generate larger inventory excursions.

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Strategies compared

Strategy	Inventory-aware?	Vol-aware?	Description
AvellanedaStoikovMM	yes	yes	Closed-form HJB-optimal (the paper)
SymmetricMM	no	yes	AS spread but no inventory skew
ConstantSpreadMM	no	no	Fixed half-spread around mid

All implement the same quote(s, q, t, T, sigma) -> (bid, ask) interface, so the simulator runs them through an identical sequence of mid-price innovations and fill events.

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Results

Head-to-head Monte Carlo (200 paths each)

Same fill model, same RNG seeds, same parameters across strategies.

| Strategy | Poisson Sharpe | Hawkes Sharpe | Poisson <|q|> | Hawkes <|q|> | |----------------------|---------------:|--------------:|----------------:|---------------:| | AvellanedaStoikov| 9.80 | 17.43 | 1.03 | 0.78 | | Symmetric (q-blind) | 5.61 | 7.54 | 4.21 | 10.12 | | Constant spread | 5.15 | 8.90 | 4.37 | 9.56 |

Reading the table: AS more than doubles the risk-adjusted return by constraining inventory variance to 1-2 shares instead of 4-10 shares.

γ-sensitivity (the hump)

examples/sensitivity.py sweeps γ over four orders of magnitude:

gamma=0.005   Sharpe= 6.5    <|q|>=3.3      (too risk-neutral, inventory swings)
gamma=0.089   Sharpe=10.1    <|q|>=1.2      <-- optimum on this fill model
gamma=5.000   Sharpe= 2.9    <|q|>=0.04     (too risk-averse, quotes too wide, miss fills)

The Sharpe-vs-γ curve has a clean interior optimum. Too little risk aversion → unconstrained inventory; too much → quotes so skewed the maker misses fills. The sweet spot lives at γ ≈ 0.09 for this fill model.

Replicated paper figure

examples/plot_paths.py reproduces the AS Figure 2 layout: a single sample path of mid + (bid, ask) quotes + inventory + cumulative P&L.

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Run it

pip install -r requirements.txt

pytest tests/                       # 15 passed in 0.48s
python examples/run_sim.py          # head-to-head Monte Carlo
python examples/plot_paths.py       # AS paper Figure 2 replication
python examples/sensitivity.py      # γ sweep with optimum localization

Layout

as_mm/
  model.py            - reservation_price, optimal_spread, optimal_quotes
  simulator.py        - Simulator + PoissonFillModel + HawkesFillModel
  strategies.py       - AS / Symmetric / ConstantSpread interfaces
  analytics.py        - summarize() across MC paths
tests/
  test_model.py       - 15 tests for closed-form limit behavior
examples/
  run_sim.py          - head-to-head comparison
  plot_paths.py       - single-path visualization
  sensitivity.py      - γ-Sharpe curve

Tests

pytest tests/ -> 15/15 passing. Coverage of the closed-form formulas:

• reservation at q=0 equals mid
• long inventory lowers reservation, short inventory raises it
• skew scales linearly with inventory (eq. 9)
• skew shrinks to zero at terminal
• spread decomposes additively (diffusion + liquidity)
• spread monotonic in σ, monotonic-inverse in k
• invalid inputs raise ValueError
• quotes symmetric about reservation (the high-frequency approximation)

Roadmap

• Inventory penalty extension — Cartea-Jaimungal-Penalva 2015 add a running quadratic penalty -φ · q² to the running utility; closed form exists.
• Stochastic intensity calibration — fit A, k to actual LOB market-by-order data (Bouchaud-Farmer-Lillo Empirical Market Microstructure).
• Multi-asset — joint hedging across correlated names; the HJB becomes vector-valued but stays tractable.
• Discrete-tick variant — quotes restricted to the tick grid; small but nonzero impact on the optimal-spread formula.

References

• Avellaneda, M. & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance 8(3): 217-224.
• Cartea, A., Jaimungal, S. & Penalva, J. (2015). Algorithmic and High-Frequency Trading. Cambridge UP. Ch. 10 expands AS with inventory penalty + alpha signals.
• Bacry, E., Mastromatteo, I. & Muzy, J.-F. (2015). Hawkes processes in finance. Market Microstructure and Liquidity 1(1).
• Guéant, O., Lehalle, C.-A. & Fernandez-Tapia, J. (2013). Dealing with the inventory risk: a solution to the market making problem. — finite-horizon refinement of AS.