Price impact and optimal execution

A large order moves the price against itself: the more you buy, the higher the price you pay, and the more you sell, the lower. This lesson separates the two ways this happens, gives the empirical rule of thumb for how big the move is, and derives how to split an order over time. It builds on "The limit order book", where a market order walks through the queue of resting orders, and on "Informed trading and Kyle's lambda", where trades move the price because they carry information.

Prices are measured as fractions of the stock's price and quantities as multiples of the order itself, so a cost of $0.0025$ means 25 basis points (bp) of the order's value.

Temporary and permanent impact

Temporary impact is what you pay for demanding liquidity quickly. A buy market order eats through the asks in the book, so its average price is above the quote before the trade. A few moments later new orders refill the book and the quote returns to roughly where it was. The cost is real, but it does not last.

Permanent impact is the part of the move that stays. If the market learns something from the trade, the price should move for good, which is exactly the linear price response $\lambda$ of Kyle's model. In a clean data set both are read off prices before the order, during it and some time after: the move that has disappeared after a while was temporary, the move that is still there is permanent.

Permanent impact is the same for every way of executing the order, because it depends only on the total traded. Temporary impact is not, and that is what execution is about: trading slowly keeps it small.

The square-root rule

Empirical studies of large equity orders find that the price move is not proportional to the order size but grows roughly with its square root:

Here $\sigma$ is the daily volatility, $Q$ the number of shares in the order, $V$ the average daily volume, and $Y$ a constant of order 1. A stock with 2% daily volatility and an order of 5% of daily volume moves by about $2\% \times \sqrt{0.05} \approx 0.447\%$, or 44.7 bp, with $Y = 1$.

Concavity has a consequence: doubling the order does not double the impact per share, it only multiplies it by $\sqrt 2$. The total cost of $Q$ shares grows as $Q^{3/2}$, still faster than linearly, so splitting a big order over several days or hours lowers the cost per share.

Splitting an order

The rule says what an order costs when done at once; the Almgren–Chriss model says how to spread it out. A trader must sell $X$ shares within a horizon $T$ (one trading day below). Time is cut into $N$ intervals of length $\tau = T/N$, with times $t_j = j\tau$. Let $x_j$ be the number of shares still held at $t_j$, so $x_0 = X$ and $x_N = 0$, and let $n_j = x_{j-1} - x_j$ be the number sold in interval $j$.

Between trades the price follows a random walk with volatility $\sigma$ per square-root day, and both impacts are assumed linear. Selling $n_j$ lowers the price for good by $\gamma\,n_j$ (permanent impact), and the trade itself is executed $\eta\,n_j/\tau$ below the price before it (temporary impact, which depends on the trading rate $n_j/\tau$). A fixed fee per share is left out, since it is the same for every schedule.

Expected cost and risk

The cost of the liquidation is the initial value of the position minus the money collected. Adding up the two impacts gives two pieces, plus a random one from the price moves. Permanent impact costs $\gamma\,n_j$ on every share still held after trade $j$, and temporary impact costs $\eta\,n_j^2/\tau$. Since $x_{j-1}^2 - x_j^2 = 2x_j n_j + n_j^2$, adding over $j$ gives $\sum_j x_j n_j = \tfrac12\left(X^2 - \sum_j n_j^2\right)$, so the expected cost is

The first term does not depend on the schedule. The second is smallest when the squares are small, which happens when the trades are equal: slow, even selling minimises temporary impact. The random part of the cost comes from the price moving while shares are still held: in interval $j$ the position $x_j$ is exposed to a price change of variance $\sigma^2\tau$, so

The variance is smallest when the shares are sold early, because the holdings $x_j$ are then small. The two goals pull in opposite directions: cost wants an even schedule, risk wants a fast one. This is the same kind of conflict as between return and risk in "Risk aversion and the optimal complete portfolio".

The optimal schedule

A trader who dislikes risk minimises $E + \lambda_{\text{risk}}V$, where $\lambda_{\text{risk}}$ is the risk aversion. (This is not Kyle's $\lambda$, the price response to order flow; the subscript keeps them apart.) Setting the derivative with respect to each interior holding $x_j$ to zero gives

Trying $x_j \propto e^{\pm\kappa t_j}$ turns this into $2\cosh(\kappa\tau) - 2 = \tilde\kappa^2\tau^2$. With the boundary values $x_0 = X$ and $x_N = 0$ the solution is

The parameter $\kappa$ sets how fast the holding runs down. When $\lambda_{\text{risk}} \to 0$, $\kappa \to 0$ and the formula becomes the straight line $x_j = X(T - t_j)/T$, the even schedule. The larger $\lambda_{\text{risk}}$ is, or the larger the volatility, or the smaller the temporary impact $\eta$, the larger $\kappa$ is and the more of the order is sold at the start.

The efficient frontier of execution

Each value of $\lambda_{\text{risk}}$ gives one schedule, with its own expected cost and standard deviation $\sqrt V$. Plotting them gives a curve like the efficient frontier of a portfolio: no schedule has both a lower cost and a lower risk than a point on the curve, and the trader picks the point that suits the risk aversion. The even schedule is the end with the lowest cost and the highest risk; selling everything at once is the end with the highest cost and no risk from price moves.

The figure sells the order in 20 slices over one day. The left chart shows how much is still held at each time, for the chosen risk aversion (solid) and for the even schedule (dashed). The right chart is the frontier; drag the dot along it to change the risk aversion. At the defaults ($\lambda_{\text{risk}} = 100$, 2% daily volatility, $\eta = 20$ bp) the expected cost is 49.3 bp with a standard deviation of 59.3 bp, and 67.5% of the order is sold in the first quarter of the day. The even schedule costs about 25 bp with a standard deviation of 111 bp, so the extra 24 bp of cost buys a cut in risk of almost half. Lower the risk aversion to see the schedule straighten, and raise the volatility to see it bend again.

A worked example

First the rule of thumb, then the smallest possible schedule that shows the trade-off. In the toy case, $X = 2$ shares are sold in $N = 2$ intervals of length $\tau = 1$, with $\sigma = 1$, $\tilde\eta = 1$ and no permanent impact.

A trader with $\lambda_{\text{risk}} < 2$ prefers to sell evenly, and one with $\lambda_{\text{risk}} > 2$ prefers to sell at once. The sinh formula is the same decision with many intervals, where the best answer lies in between.

Summary

• Temporary impact fades after the trade and depends on how fast you trade; permanent impact stays and depends only on the total traded.
• Empirically the price move of a large order is about $Y\sigma\sqrt{Q/V}$, concave in the order size.
• With linear impact the expected cost is $\tfrac12\gamma X^2 + (\tilde\eta/\tau)\sum n_j^2$ and the variance is $\sigma^2\tau\sum x_j^2$; the first prefers even trading, the second fast trading.
• Minimising cost plus $\lambda_{\text{risk}}$ times variance gives $x_j = X\sinh(\kappa(T - t_j))/\sinh(\kappa T)$: a straight line for a risk-neutral trader and more front-loading as risk aversion or volatility grows.
• The (risk, cost) pairs of all schedules form an efficient frontier, like the one for portfolios.