Are there references about liquidation, transaction, market impact costs in portfolio optimization

Question

I am looking for some references treating of what I would call

1. liquidation cost
2. market impact cost
3. transaction cost(*)

in the usual "portfolio optimization problem under linear constraints".

Let me define the terms here. $$ \text{Find } w^*=\underset{w}{\text{argmax}} \ \ r^Tw - \lambda w^{T} \Sigma w\\ \text{uc.} \ \ l_b \leq Aw \leq u_b $$ where $w \in \mathbb{R}^n$ is the final position of a portfolio of $n$ assets. $r \in \mathbb{R}^n$ is the vector of expected returns over a time period (say $[0,T]$), $\Sigma$ is the $\mathbb{M}_{n,n}(\mathbb{R})$ covariance matrix of the asset returns.

One can use a factor model $r=Xf+u$, where returns are modeled as a linear combination of the weighted factor returns $f$ plus an idiosyncratic part $u$. By assuming independence between $f$ and $u$, and between the individual components of $u$ one gets $\Sigma = X^T F X + D$.

So we rewrite $$ \text{Find } w^*=\underset{w}{\text{argmax}} \ \ r^Tw - \lambda w^{T} (X^T F X + D) w\\ \text{uc.} \ \ l_b \leq Aw \leq u_b $$

Finally $A \in \mathbb{M}_{p,n}(\mathbb{R})$ is a matrix that allows constraints (initial positions, delta). If one wants turnover, bookvalue constraints the usual linearisation trick $n \rightarrow 5n$ allows to keep the exact same formulation, only $A$ would change.

This brings my question 1:

It is well known from Grinhold and Kahn that

market impact should increase as the square root of the amount traded. This agrees remarkably well with the empirical work of Loeb (1983). Because the total trading cost depends on the cost per share times the number of shares traded, it increases as the 3/2 power of the amount traded.

As such my view is that whatever portfolio achieved should take into account the cost it will take to liquidate the portfolio.

Same thing for question 2:

Solving the problem does not take into account the impact that working out the solution portfolio will have on the expected returns.

I think question 3 brings non-continuous cost on the table so, I'll leave it for now (tell me if I am wrong)

As such I think I want to solve

$$ \text{Find } w^*=\underset{w}{\text{argmax}} \ \ (r-I(t))^Tw - \lambda w^{T} (X^T F X + D) w - \Theta |w|^{\frac{3}{2}}\\ \text{uc.} \ \ l_b \leq Aw \leq u_b $$ where $t=|w-w_0|$

Can you give me references on adding those additional terms, the more applied the better.

Answer 1

The state of the art is Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach by Jiatu Cai, Mathieu Rosenbaum, Peter Tankov. Hence the references of this paper are the ones to read.

In the paper, they explain how you have to consider the prefactors (your $\lambda$ and $\Theta$) to be able to stay close to a target portfolio trajectory.

They also explain how most control problems can be (in this scope) redefined as a target following one. It means you can define an equivalent ideal portfolio, and try to follow it according to their results.

[EDIT] Following a comment, I agree this paper is reasonably complex. You can go back to Dynamic Trading with Predictable Returns and Transaction Costs, by Nicolae Garleanu and Lasse Pedersen. They show a simple version of this very generic problem. In terms of rewriting the problem, you will read that an important step is to express the maximization not in terms of weights $w$ but in terms of trades $\Delta w$ that will lead to a transition from $w_0$ to $w_0 + \Delta w$, then you can write quite easily the transaction costs (related to $\Delta w$ and not to $w$).

For you first example $\max_w r^T w - \lambda w^T \Sigma w$ u.c. $l_b\leq A w\leq u_b$ should for instance (following Garleanu-Pederson approach) be rewritten as

$$\max_{\Delta w} r^T (w_0+\Delta w) - T_\mbox{costs}(\Delta w)- \lambda (w_0+\Delta w)^T \Sigma (w_0+\Delta w),$$ $$\mbox{u.c.}\quad l_b - A w_0\leq A \Delta w\leq u_b - A w_0$$

When the ''transaction costs'' term is non linear (i.e. when it includes a market impact term), this is a different optimization program.