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)
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Daneel Olivaw · External communityPost link
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/
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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.
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Daneel Olivaw · External communityPost link
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.
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Quoted from Forex.com.bd-Editorial External question — Quantitative Finance Stack Exchange Author: Daneel Olivaw Source score (net votes, not local likes): 1 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.
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