Transaction Timing and Temporary Impact in Almgren–Chriss Execution Models
Summary
The document examines why Almgren and Chriss define a trade's transaction price using the market price from the preceding time step, plus a temporary-impact term. This setup appears in optimal liquidation models that seek to control execution costs while selling or buying a portfolio over a fixed horizon. The transaction price, rather than the market price alone, determines the proceeds from each trade.
The explanation distinguishes temporary impact, paid in the trade, from permanent impact, which arises after the transaction and is reflected in the next market price. It also notes that using the contemporaneous market price would make the market-to-transaction price difference equal to the impact term; when that term is deterministic, the spread would be predictable. The discussion supplies a modeling interpretation rather than empirical evidence or a full derivation, and the source model's assumptions about impact and timing matter to the conclusion.
Key ideas
- The model calculates transaction prices from the prior period's market price and a temporary-impact term.
- Temporary impact affects the trade price, while permanent impact is reflected in the subsequent market price.
- Using the current market price changes how impact costs are assigned across time.
- A deterministic impact term would make the spread predictable under the alternative definition.
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Full text
# Time lag in the definition of transaction price in Almgren & Chriss (2001) and Almgren (2003)
# Time lag in the definition of transaction price in Almgren & Chriss (2001) and Almgren (2003)
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.
## Answer by Daneel Olivaw (score 1)
https://quant.stackexchange.com/a/84000
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.