Lab | Liquidity value adjustment

56a.4 Liquidity value adjustment

Key points

The liquidity value adjustment (56a.59) accounts for the adverse market impact of liquidating a portfolio on the realized value.

Suppose that we want to liquidate our portfolio. In this case, the theoretical value $v_{t_{n o w}}$ (56a.9) is not achievable. Long (short) positions should be aggregated at their bid (ask) quote, when these are available, and more generally we should model the negative market impact effect of a liquidation, as we discuss extensively in the Execution step (Chapter 61).

Let us consider the current position $h_{t_{n o w}}$ in a single instrument. Suppose that we want to liquidate the position in the period $[t_{s t a r t}, t_{e n d})$ , where $t_{s t a r t} = t_{n o w}$ is the current time.

We model market impact as in Section 61.2.5, considering the price dynamics in clock time rather than volume time and a power law temporary impact $g (u) \equiv \pm η | u |^{α}$ with $α = 0.5$ . If we implement the quasi-optimal execution strategy (61.29) in the liquidation case (i.e. $h_{t_{e n d}} = 0$ and $Δ h^{p a r e n t} = - h_{t_{n o w}}$ ), then the expected P&L (61.31) reads

E {{ˆ Π}_{h (\cdot), t_{n o w} \to t_{e n d}}} = - \frac{γ}{2} h_{t_{n o w}}^{2}      permanent impact - \frac{η ξ}{\sqrt{t_{e n d} - t_{n o w}}} | h_{t_{n o w}} |^{3 ∕ 2}      temporary impact,

(56a.57)

where $ξ \equiv β^{3 ∕ 2} ∕ (β + (β - 1) ∕ 2) > 0$ ; and the variance (61.32) reads

V {{ˆ Π}_{h (\cdot), t_{n o w} \to t_{e n d}}} = σ^{2} (t_{e n d} - t_{n o w}) h_{t_{n o w}}^{2} (1 - \frac{2}{β + 1} + \frac{1}{2 β + 1}) .

(56a.58)

Therefore, in a liquidation, the value of each position in the portfolio should be adjusted for liquidity risk

\begin{matrix} v^{l v a} & \approx \sum_{n} (h_{n, t_{n o w}} v_{n, t_{n o w}}^{b i d} + E {{ˆ Π}_{h_{n} (\cdot), t_{n o w} \to t_{e n d}}}) 1_{{h_{n, t_{n o w}} > 0}} & (56a.59) + \sum_{n} (h_{n, t_{n o w}} v_{n, t_{n o w}}^{a s k} + E {{ˆ Π}_{h_{n} (\cdot), t_{n o w} \to t_{h o r}}}) 1_{{h_{n, t_{n o w}} < 0}} + c a s h_{t_{n o w}} . \end{matrix}

Example 56a.13. Liquidity-adjusted value of stock portfolio
Consider a portfolio composed of $¯ n \equiv 2$ stocks, AAPL and ABT, where the holdings are

h \equiv (\begin{matrix} 1, 000 - 5, 000 \end{matrix})

(56a.60)

The current value of the portfolio is determined using the linearity assumption (56a.9) reads

v = $ 281, 350 .

(56a.61)

Suppose we want to liquidate this portfolio over a horizon of $5$ days. In this case we need to consider the liquidity adjusted value (56a.59), where we assume a linear trading strategy

β = 1,

(56a.62)

for both stocks. In practical applications, this parameter should be chosen to optimize the trading trajectory. Then the liquidity-adjusted value (56a.59) is

v^{l v a} = $ 270, 630 .

(56a.63)

\begin{matrix} v^{l v a} & \approx \sum_{n} (h_{n, t_{n o w}} v_{n, t_{n o w}}^{b i d} + E {{ˆ Π}_{h_{n} (\cdot), t_{n o w} \to t_{e n d}}}) 1_{{h_{n, t_{n o w}} > 0}} & (56a.59) + \sum_{n} (h_{n, t_{n o w}} v_{n, t_{n o w}}^{a s k} + E {{ˆ Π}_{h_{n} (\cdot), t_{n o w} \to t_{h o r}}}) 1_{{h_{n, t_{n o w}} < 0}} + c a s h_{t_{n o w}} . \end{matrix}

v_{h, t_{n o w}} = h_{t_{n o w}}^{'} v_{t_{n o w}} + c a s h_{t_{n o w}} .

h_{q} \equiv h_{q_{n o w}} + {(\frac{q - q_{n o w}}{q_{e n d} - q_{n o w}})}^{β} Δ h^{p a r e n t}, for q \in [q_{n o w}, q_{e n d}],

E {{ˆ Π}_{h (\cdot), q_{n o w} \to q_{e n d}}} = \frac{γ}{2} (h_{q_{e n d}}^{2} - h_{q_{n o w}}^{2})      permanent impact - η ξ | Δ h^{p a r e n t} |^{1 + α} {(q_{e n d} - q_{n o w})}_{e n d}^{- α}      temporary impact,

V {{ˆ Π}_{h (\cdot), q_{n o w} \to q_{e n d}}} = σ^{2} (q_{e n d} - q_{n o w}) (h_{q_{n o w}}^{2} + 2 h_{q_{n o w}} \frac{Δ h^{p a r e n t}}{β + 1} + \frac{{(Δ h^{p a r e n t})}^{2}}{2 β + 1}) .

56a.4 Liquidity value adjustment

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