Intraday Liquidity & Optimal Execution (Work-Trial Task)

This repo builds an intraday liquidity curve from quote data, calibrates a piecewise temporary impact model (flat cost up to depth; concave power-law tail), and solves for the optimal minute-by-minute execution schedule for a fixed parent order using KKT conditions with a bisection on the Lagrange multiplier.

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What’s in here

• Modeling note — estimating the flat cost $c$, intraday depth $D_t$, and tail exponent $p$, with plots and rationale.
• Allocation note — KKT-based algorithm with a bisection on the multiplier to hit the target $S$.
• Notebook / code — end-to-end data loading, P-spline smoothing, power-law fit, and final allocation.

Key figures:

Penalised cubic B-spline fit to minute-level average depth $D_t$.

Filtered average premium $g(x)$ with tail $x^{p}$ (log–log).

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Method (at a glance)

1. Depth curve $D_t$
   For each trading minute $t\in{0,\dots,389}$, compute the average of the first non-zero ask size per quote event for each symbol–day, then average across panels to get $D_t$. Smooth with a penalized cubic B-spline (GCV selects $\lambda$).

2. Flat cost $c$
   Take the median half-spread per symbol during RTH, then the median across symbols → global $c$.

3. Impact tail exponent $p$
   Build empirical $g(x)$ by averaging the premium paid above best-ask as one walks up the ask ladder; fit
   $\log g(x)=\log a + p\log x$ on filtered buckets → $p\approx 0.45$.

4. Piecewise temporary impact $g_t(x)$

$$g_t(x)= \begin{cases} c, & 0\le x\le D_t\\\ a_t\,x^{\,p}, & x>D_t \end{cases} \qquad a_t=\dfrac{c}{D_t^{\,p}},\quad 0<p<1.$$