Liquidity risk and closeout horizons

Definition Market liquidity risk is the risk that closing out a position realizes less than its mark, because the trades pay the bid-ask spread, move the execution price, and take time during which the price can change. It is distinct from funding liquidity risk, the risk of being unable to meet payments when they fall due, which is not treated here. In this lesson time $t$ is measured in trading days, and a long position of $q > 0$ shares marked at the mid price $S_0$ is sold over $[0, h]$ along a deterministic schedule. The inventory $x(t)$ has $x(0) = q$ and $x(h) = 0$, and the selling rate is $v(t) = -x'(t) \ge 0$ shares per day. The model assumes that the unaffected mid price is $S_t = S_0 + \sigma W_t$, with $W$ a standard Brownian motion under $\mathbb{P}$, zero drift and $\sigma > 0$ in currency units per share per square root of a day, and that a share sold at time $t$ fetches $S_t - b - \eta v(t)$ for constants $b \ge 0$ and $\eta > 0$. The liquidation cost, measured against the mark, is

Since $x$ is deterministic with $x(h) = 0$ and $W_0 = 0$, the product rule gives $\int_0^h v(t) W_t\,dt = \int_0^h x(t)\,dW_t$, and so

The cost is positive when the sale fetches less than the mark, and a rising price lowers it.

Definition The half-spread $b \ge 0$, in currency units per share, is the distance from the mid price to the bid, where a seller trades, or to the ask, where a buyer trades. Selling $q$ shares at the bid instead of the mid costs $bq$ relative to a mid mark, whatever the schedule, because $\int_0^h v(t)\,dt = q$. In this model $b$ is constant, so the spread cost does not depend on the horizon, and every schedule has $\mathbb{E}[C] \ge bq$. If the full spread is quoted as a fraction $s$ of the mid, then $b = sS_0/2$. A short position bought back pays the same amount at the ask. A spread that widens in a stress is a separate scenario input, not something the model produces.

Definition Under linear temporary impact, a share sold at time $t$ fetches $\eta v(t)$ less than $S_t - b$, where $v(t)$ is the selling rate in shares per day and $\eta > 0$ is measured in currency unit days per share squared. The impact affects only the trades made at that moment and does not move later prices. Permanent impact is outside this model. The impact cost is $\eta \int_0^h v(t)^2\,dt$, a deterministic amount. By the Cauchy-Schwarz inequality,

with equality only for a constant rate, so for a fixed horizon the constant rate $v = q/h$ has the smallest impact cost, $\eta q^2/h$. The expected cost of the constant-rate schedule is

It decreases toward the spread floor $bq$ as $h$ grows and becomes unbounded as $h \to 0$, because linear impact makes very fast trading arbitrarily expensive.

Definition In the model, the random part of the cost is $-\sigma \int_0^h x(t)\,dW_t$, an Itô integral of a deterministic integrand. By the Itô isometry of The Ito integral and isometry, its variance is

and since the integrand is deterministic, $C$ is normally distributed. For the constant-rate schedule the inventory is $x(t) = q(1 - t/h)$, and the substitution $u = t/h$ gives

so $\operatorname{Var}(C(h)) = \sigma^2 q^2 h/3$, in currency units squared. It grows linearly with the horizon. Holding all $q$ shares for the time $h$ and selling them at the end would give the price variance $\sigma^2 q^2 h$, three times as much, because the constant-rate schedule reduces its exposure as it goes. The units of $\sigma$ and $h$ have to match, here a square root of a day and days.

Definition The liquidation horizon $h > 0$ is the time over which the position is closed out. In the constant-rate model with a risk aversion $\lambda \ge 0$, in inverse currency units, the horizon is chosen to minimize the mean-variance objective

For $\eta$, $\lambda$ and $\sigma$ all positive, $J'(h) = -\eta q^2/h^2 + \lambda \sigma^2 q^2/3$ and $J''(h) = 2\eta q^2/h^3 > 0$, so $J$ is strictly convex on $(0, \infty)$ with the unique minimizer

which does not depend on $q$ in this model. Multiplying $J'(h^{*}) = 0$ by $h^{*}$ shows that at the optimum the impact cost equals the risk penalty, so $J(h^{*}) = bq + 2q^2\sqrt{\eta \lambda \sigma^2/3}$. The objective decreases before $h^{*}$ and increases after it, so if the feasible horizons form an interval, the best feasible horizon is $h^{*}$ moved to the nearer end of the interval when it lies outside. If $\lambda = 0$ or $\sigma = 0$, the objective decreases in $h$ and the formula for $h^{*}$ does not apply. The longest feasible horizon is then best, and without an upper bound no horizon is optimal, since the objective only approaches the spread floor $bq$. If $\eta = 0$, the objective increases and the shortest feasible horizon is best. The optimum is over constant-rate schedules only. With $\lambda$ and $\sigma$ positive, the best of all deterministic schedules for a given horizon sells faster at the start, which lies outside this restricted model.

Definition A liquidity adjustment changes a mark-to-market value or a risk figure to account for the cost and the time of closing out. In this model the expected liquidation proceeds are

so the adjustment to the mid mark is $-\mathbb{E}[C(h)]$ for the chosen horizon. A risk-sensitive version uses the objective $J(h) = \mathbb{E}[C] + \lambda \operatorname{Var}(C)$, which is a decision criterion whose size depends on the chosen $\lambda$, not a quantile. Since $C$ is normal in this model, its quantile at a confidence level $\alpha$ is $\mathbb{E}[C] + \Phi^{-1}(\alpha)\sqrt{\operatorname{Var}(C)}$, a different number. When a liquidity amount is added to a market risk figure, both have to be checked for the price risk of the inventory, which is counted twice if the market figure already covers price moves over the liquidation horizon and the liquidity amount includes the inventory variance. Jumps, changing depth or spreads, stochastic participation, price trends and permanent impact all change the distribution of the cost and lie outside this model.