Poisson intensity of limit order execution, calibration of parameters A and k using level 1 tick data

Description

A limit order placed at a price S_t ± δ, has the instantaneous probability of execution λ(δ)dt where the intensity λ(δ) is given by:

λ(δ) = A e ^-kδ

λ - Poisson order execution intensity
δ - spread (distance from mid price S_t)
A - parameter, positively related to trading intensity
k - parameter, positively related to market depth

Package Execution Intensity Estimator (EIE) contains single and multi threaded calibration procedure of A and k parameters. Methods that calculate intensity λ(δ, A, k) and spread δ(λ, A, k) are provided as well. Algorithm operates on level 1 tick data, therefore it is suitable in a setting where liquidity is not fully observable (i.e. dark pools). Calibration is two step procedure performed separately for buy and sell limit orders.

Steps:

• For each spread δ_k of N predefined spreads (δ₀ , δ₁ , δ₂ , ... δ_N-1) estimate execution intensity λ(δ_k) using "waiting time" approach described in [1] 4.4.2.. Result of this step is set of N points (δ_k , λ(δ_k)) on empirical Spread Intensity Curve (SIC)
• Estimate A and k based on N points from previous step. This can be achieved by various approaches. Code implements two approaches described in [2] 3.2:
  • LOG_REGRESSION performs OLS regression of log(λ_k) on δ_k. Finally k = -slope and A = e ^intercept
  • MULTI_CURVE from set of N points creates N_s = (N*(N-1))/2 unique pairs fo points ((δ_x , λ_x) , (δ_y , λ_y)). For each set of points solves the following set of equations for A' and k' :
    λ_x = A' e ^-k'δ_x
    λ_y = A' e ^-k'δ_y
    Final estimates are A = mean(A'₁ , A'₂ , ... A'_Ns) and k = mean(k'₁ , k'₂ , ... k'_Ns)

Once A and k are calibrated, depending on context of usage, user can specify:

• spread δ to obtain corresponding intensity λ(δ)
• intensity λ to obtain corresponding spread δ(λ)
  
  
  

References

• [1] Fernandez-Tapia, Joaquin. (2015). Modeling, optimization and estimation for the on-line control of trading algorithms in limit-order markets. 10.13140/RG.2.1.1490.5684.

• [2] Sophie Laruelle, Faisabilité de l’apprentissage des paramètres d’un algorithme de trading sur des données réelles