Time lag in the definition of transaction price in Almgren & Chriss (2001) and Almgren (2003)

External question — Quantitative Finance Stack Exchange Author: Daneel Olivaw Original post: https://quant.stackexchange.com/questions/83892 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. In Almgren & Chriss (2001) and Almgren (2003) the authors study the problem of optimally liquidating a share portfolio over a time period of length $T$ while minimizing market impact. The final proceeds from portfolio liquidation depend, not on the share's market price, but instead on the transaction price which incorporates any temporary impact. Introducing a time grid $0=t_0,\dots,t_n=T$ with $t_k-t_{k-1}=\tau$ , the transaction price $\tilde{S}$ is defined as follows in these papers: $$\tilde{S}_k=S_{k-1}+f(v_k,\varepsilon_k)\tag{1}$$ where $S$ is the market price; $v$ is the instantaneous trading rate that is the number of shares sold (or bought) between $t_{k-1}$ and $t_k$ divided by $\tau$ ; $\varepsilon$ a random variable with zero mean and unit variance; and $f$ some function. In this model, the transaction price depends on the market price from the previous period . Is there any fundamental reason, or modelling constraint, that justifies this choice? Could we not simply define the transaction price as: $$\tilde{S}_k=S_\color{blue}{k}+f(v_k,\varepsilon_k)\ ?\tag{2}$$ References R. Almgren & N. Chriss (2001). "Optimal execution of portfolio transactions", Journal of Risk , 3 (2), 5-39. R. Almgren (2003). "Optimal execution with nonlinear impact functions and trading-enhanced risk", Applied Mathematical Finance , 10 (1), 1-18.

External answer — Quantitative Finance Stack Exchange Author: Daneel Olivaw Original post: https://quant.stackexchange.com/a/84000 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. The difference between Equations $(1)$ and $(2)$ is that in the former, permanent impact arising at $k$ is not incurred by the dealer when trading at $k$ , i.e. the only cost incurred is the temporary impact $f(v_k,\varepsilon_k)$ . In other words, permanent impact occurs after the transaction and will be reflected in $S_{k}$ and not $S_{k-1}$ . Alternatively, as explained by @markleeds, choice $(1)$ makes the spread between market and transaction prices uncertain, otherwise: $$\tilde{S}_k-S_k=f(v_k,\varepsilon_k)$$ In the original paper by Almgren and Chriss, $\varepsilon\equiv0$ hence that would have made the spread predictable.