Order Flow Imbalance and Short-Horizon Price Impact

Scenario

In high-frequency quantitative market making, the instantaneous order flow imbalance (OFI) measures net supply-demand pressures inside the Limit Order Book (LOB) across successive book states.

Your Task

1. Given best bid price $P_t^b$, best bid depth $q_t^b$, best ask price $P_t^a$, and best ask depth $q_t^a$, formulate the event-by-event bid flow contribution $e_t^b$ and ask flow contribution $e_t^a$.
2. Define the total order flow imbalance $OFI_t = e_t^b - e_t^a$.
3. Derive the linear price impact regression parameter $\beta = \frac{\text{Cov}(\Delta P_t, OFI_t)}{\text{Var}(OFI_t)}$ for mid-price changes $\Delta P_t = P_t^{\text{mid}} - P_{t-1}^{\text{mid}}$.
4. Simulate a series of 10 LOB events, compute the resulting OFI time series, and calculate the empirical price impact coefficient $\beta$.

Constraints

• Level 1 top-of-book quotes.
• Standard Cont-Kukanov-Stoikov (2014) OFI definitions.

Examples When a buyer consumes the ask queue (price increases or ask size decreases), $e_t^a < 0$, making $OFI_t > 0$, predicting an upward mid-price movement.