5.8.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and 03 · Transaction-Cost Models.

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1. Intuition & Practical Objective

The four extensions below are what separate a textbook rebalance from an institutional trading program:

1. The aim portfolio (multi-period). You do not trade to the current target; you trade toward a long-horizon aim that averages over where you expect the target to go - and you only close a fraction of the gap. With quadratic impact the fraction is a matrix (cross-asset coupling); with proportional costs the policy is closed-form (Gârleanu & Pedersen 2013).
2. Fixed costs. Paying a charge whenever you touch a name (custody, funding, minimum ticket, tax lot) makes the cost function non-convex. The standard trick is a convex relaxation that trades the whole position in a name or nothing - the bang-bang pattern you actually see at well-run desks.
3. Non-convex (square-root) impact. The empirical $|\Delta w{|}^{3/2}$ law is concave, so it must be convexified by successive convex approximation (SCA) - majorize it by a tangent quadratic at each iteration, which is exactly how production optimizers handle it.
4. Cross-impact. Real cost matrices $\Lambda$ have off-diagonal entries: selling eight correlated names is one market-wide trade. The additive per-name model understates cost, and this is the mechanism behind “we can’t be the only ones doing this.”

The one-sentence essence. “The static, one-period, additive, convex problem is a tractable idealization; the advanced extensions are the four honest corrections to it - a multi-period aim, fixed charges, concave impact, and a cost matrix with off-diagonals.”

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2. Mathematical Ground Truth & Derivations

2.1 The aim portfolio and the matrix partial adjustment

Minimize tracking error plus quadratic impact over one step:

$$\underset{w}{\min }\frac{\delta }{2}(w-w^{\ast }{)}^{\top }\Sigma (w-w^{\ast })+\frac{\kappa }{2}(w-w_0{)}^{\top }\Lambda (w-w_0).$$

Setting the gradient to zero, $\delta \Sigma (w-w^{\ast })+\kappa \Lambda (w-w_0)=0$, gives the aim/target update in closed form:

$$w^{\star }=w_0+\underset{F}{\underbrace{(\delta \Sigma +\kappa \Lambda {)}^{-1}\delta \Sigma }}(w^{\ast }-w_0),\boxed{F=(\delta \Sigma +\kappa \Lambda {)}^{-1}\delta \Sigma }$$

Three properties:

• $F$ is not a scalar. Because $\Sigma$ is not diagonal, a signal in one name moves the weights of all names. Only in the special case $\Lambda =\lambda I$ and $\Sigma$ diagonal does $F$ reduce to a scalar fraction $\delta {\sigma }_i^2/(\delta {\sigma }_i^2+\kappa \lambda )$.
• The eigenvalues of $F$ are $\delta {\lambda }_i/(\delta {\lambda }_i+\kappa {\eta }_i)\in (0,1)$. Cheap, low-risk directions are traded almost fully; expensive, high-risk directions are barely touched. The matrix anisotropically discounts the trade by exactly where the cost lives.
• Gârleanu–Pedersen (2013) generalise this to many periods with proportional cost and predictable returns, obtaining $x_t=x_{t-1}+(I+\kappa \Sigma {)}^{-1}({\text{aim}}_t-x_{t-1})$: trade a constant fraction of the gap to the aim every period, never the whole thing.

2.2 Fixed costs and the convex relaxation

A fixed charge $K_i$ for touching name $i$ (plus a linear part $c_i$) gives

$$C(\Delta w)=\sum _{i=1}^N[K_{i}1\{\Delta w_i\ne 0\}+c_i|\Delta w_i|],$$

which is discontinuous at zero and therefore non-convex. Lobo–Fazel–Boyd (2007) relax $1\{\Delta w_i\ne 0\}$ by a variable $u_i\in [0,1]$ with $|\Delta w_i|\le M_i{u}_i$, so the cost becomes ${\sum }_i(K_i{u}_i+c_i|\Delta w_i|)$ - a linear program inside the QP, solvable exactly, whose solution is bang-bang in $u_i$: either you trade name $i$ fully (up to the position limit $M_i$) or you do not touch it at all. That is the mathematical origin of the “we rebalance a handful of names and leave the rest alone” behaviour in §3(B).

2.3 Concave impact and successive convex approximation

The empirical cost ${\kappa }_i|\Delta w_i{|}^{3/2}$ is concave, so its KKT points are not guaranteed global. The standard remedy: at iterate $\Delta w^{(k)}$, majorize by the tangent quadratic (a valid convex upper bound),

$$|\Delta w_i{|}^{3/2}\le \frac{3}{2}(\Delta w_i^{(k)}{)}^{1/2}\Delta w_i^2-\frac{1}{2}(\Delta w_i^{(k)}{)}^{3/2},$$

and iterate. Each subproblem is a QP; the sequence converges to a local optimum, and the majorant is tight at the iterate - the same SCA machinery used for ${\ell }_p$ regularisation in robust optimisation.

2.4 Cross-impact and the additivity failure

Replace the diagonal $\Lambda$ by a full matrix $\Lambda =q{q}^{\top }D$ (a common one-factor impact model) so that trading any name moves the price of every correlated name. Then

$$C_{\text{impact}}=\frac{1}{2}\Delta w^{\top }\Lambda \Delta w=\frac{1}{2}\left(q^{\top }\Delta w{)}^{\top }D\right(q^{\top }\Delta w),$$

i.e. the cost depends on the factor-level net trade $q^{\top }\Delta w$, not on the per-name trades. Selling eight names of the same factor is one large trade, not eight small ones - the diagonal model understates it. A SOCP reformulation (introduce $z=D^{1/2}{q}^{\top }\Delta w$ and constrain $\Vert z{\Vert }_2\le t$) keeps the problem tractable at the cost of a second-order cone per factor.

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3. Computational Implementation - the matrix aim and the sparse rebalance

(A) build the aim/target update matrix $F=(\delta \Sigma +\kappa \Lambda {)}^{-1}\delta \Sigma$, show it is non-diagonal, and verify the closed form against direct numerical optimisation. (B) with a large alpha revision, sweep the cost penalty and watch the rebalance shut down - the no-trade region in weight space. numpy + scipy.

Two verified readings:

• (A) The trade is a matrix, and the matrix is where the cost lives. The diagonal of $F$ ranges from $0.1262$ (asset 2, the most expensive to trade, $\eta =0.030$) to $0.3074$ (asset 5, the cheapest, $\eta =0.010$) - the fraction of each gap you close varies by a factor of $2.4$ across assets purely because of liquidity. The off-diagonal mass $\Vert F-\mathrm{diag}{\Vert }_F=0.39122$ is large: a buy in asset 1 of one unit induces buys in assets 2–6 through the covariance structure, because managing risk requires moving correlated names together. The closed form agrees with direct numerical optimisation to $5.5\times 10^{-9}$, confirming the derivation.
• (B) The no-trade region shuts the whole rebalance down. Even a $+500$ bp/yr alpha revision on asset 2 - an enormous signal - produces zero trade once the per-unit cost penalty reaches $\lambda \approx 0.003$ (monthly). Note the sharp discontinuity: the trade does not decay smoothly to zero, it stops ($0.3755\to 0.2135\to 0.0227\to 0$). That is the piecewise character of the ${\ell }_1$ path, and it is exactly the bang-bang behaviour the fixed-cost relaxation predicts. A practitioner who expects “a little less trading” is surprised by a switch.

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4. Failure Modes & First-Principles Breakdowns

1. Ignoring cross-impact. The additive model $\frac{1}{2}{\sum }_i{\eta }_i\Delta w_i^2$ understates cost whenever trades share a factor. Calibrate the off-diagonal entries of $\Lambda$ (or the factor form $q{q}^{\top }D$) from realized executions, or you will systematically over-trade multi-name baskets.
2. Solving concave impact as if it were convex. A naive QP on the $|\Delta w{|}^{3/2}$ surface can converge to a poor local point. Use SCA with the tangent majorant of §2.3 and check that the majorant is tight at the solution.
3. Treating the aim as a target. $F$ has eigenvalues in $(0,1)$: you never close the gap in one step, by design. A desk that reports “we are 35% away from target” may be perfectly on policy, not behind - the gap is a feature of the aim.
4. Choosing $M_i$ carelessly in the fixed-cost relaxation. The big-$M$ constants in $|\Delta w_i|\le M_i{u}_i$ must be valid position limits; too small and the relaxation becomes infeasible, too large and the LP loses numerical meaning. Set $M_i$ from the position caps of §02.
5. Confusing the portfolio problem with the execution problem. This model decides what to trade and roughly how much; it does not decide the intraday schedule. Handing the resulting $\Delta w$ to a naive market-order execution destroys the very impact estimates the model assumed (Almgren–Chriss).

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5. References

• Gârleanu, N. & Pedersen, L. H. (2013), Dynamic Trading with Predictable Returns and Transaction Costs, Journal of Finance 68(6):2309–2340
• Lobo, Fazel & Boyd (2007), Portfolio Optimization with Linear and Fixed Transaction Costs, Annals of OR 152:341–365
• Boyd, Busseti, Diamond, Kahn, Koh, Nystrup & Speth (2017), Multi-Period Trading via Convex Optimization, FnT in Optimization 3(1):1–72
• Almgren & Chriss (2000/01), Journal of Risk 3(2):5–39
• Hasbrouck (2007), Empirical Market Microstructure
• Avellaneda & Stoikov (2008), High-Frequency Trading in a Limit Order Book, Quantitative Finance 8(3):217–224

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · 04 · Turnover & the Multi-Period Trade-Off
• Execution bridge: Optimal Execution (Almgren–Chriss) · POV · Queue Position & Fill Probability
• Market-making bridge (the cost side of the trade): Avellaneda–Stoikov · Inventory Management & Quote Skewing · Spread Decomposition & the Roll Model
• Siblings: Robust Optimization · 06 · Covariance Shrinkage & RMT
• Foundations: Calculus & Optimization · Numerical Methods
• Hub: Index Hub