# The cost of misspecifying price impact

Natascha Hey1, Jean-Philippe Bouchaud1,2,3,
Iacopo Mastromatteo2, Johannes Muhle-Karbe4,
Kevin Webster4 1Chair of Econophysics and Complex Systems, École Polytechnique
2Capital Fund Management
3Académie des Sciences
4Department of Mathematics, Imperial College London
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Portfolio managers’ orders trade off return and trading cost predictions.
Return predictions rely on alpha models, whereas price impact models quantify
trading costs. This paper studies what happens when trades are based on an
incorrect price impact model, so that the portfolio either over- or under-
trades its alpha signal. We derive tractable formulas for these
misspecification costs and illustrate them on proprietary trading data. The
misspecification costs are naturally asymmetric: underestimating impact
concavity or impact decay shrinks profits, but overestimating concavity or
impact decay can even turn profits into losses.

##  1\. Introduction

Price impact refers to price movements induced purely by trading flows,
independently of their information content. For large investors, such adverse
price moves caused by their own trades are the main source of transaction
costs.111In this regime, other sources of costs such as proportional bid-ask
spreads are of secondary concern and hence will be disregarded in the
following; the interested reader is referred to Martin and Schöneborn (2011)
for a detailed discussion of the role of such linear costs for the design of
trading strategies that operate on a smaller scale. Price impact models are
thus essential tools in algorithmic trading, allowing investment teams to
estimate the effect of their trades on asset prices and thereby design, size
and deploy systematic strategies. For instance, (capacity constrained)
statistical arbitrage portfolios seek to achieve the best trade-off between
price predictions and trading costs.222If transaction costs do no constrain
the size of the portfolio tightly enough, it may also be risk constrained,
cf., e.g., Gârleanu and Pedersen (2013) and the references therein.

The price forecast is commonly called an _alpha signal_. Turning the latter
into optimal trades in turn requires an appropriate price impact model. Many
empirical studies find a concave _nonlinear_ relationship between order size
and the price impact of sizeable trades, see, e.g., Bouchaud, Bonart, Donier,
and Gould (2018); Webster (2023) and the references therein for an overview.
More specifically, two essential parameters emerge when determining the
evolution of price impact: _concavity_ and _impact decay_.

_Concavity_ describes how the marginal build-up of price impact slows beyond a
certain order size: larger orders are cheaper than a linear model suggests.
_Impact decay_ describes how long trade-induced price moves linger in the
market and affects how quickly portfolios can turn over.

This paper quantifies how incorrect calibration of these price impact
parameters leads to significant misspecification costs for traders and
portfolio managers. This allows to go beyond purely statistical parameter
estimates by establishing P&L-driven error bounds for price impact parameters,
which are crucial to any team estimating the actual trading capacity of their
alpha signals.

To carry out this analysis, we derive closed-form expressions for
misspecification costs in a nonlinear but nevertheless tractable price impact
model introduced by Alfonsi, Fruth, and Schied (2010). This model was
originally proposed and studied for optimal execution problems, but in fact
admits explicit formulas for optimal trades with general alpha signals.

These in turn allow one to quantify how wrong model parameters can be before
turning a profitable trading strategy for a given alpha signal into an
unprofitable one. For example, linear price impact models as in Obizhaeva and
Wang (2013) overestimate the transaction costs of large trades and thereby
imply overly conservative trading strategies. However, they rarely turn a
winning strategy into a losing one. Conversely, even relatively small
overestimates of impact concavity can lead to overly aggressive trading that
leads to losses despite an accurate alpha signal. Similarly, underestimating
impact decay leads to conservative strategies that trade slower than optimal,
which sacrifices some trading opportunities but does not lead to dramatic
losses. In contrast, overly aggressive trades based on overestimates of impact
decay can quickly turn a portfolio’s P&L negative.

Figure 1. Left panel: Symmetric statistical significance ratio
$R^{2}(\hat{c})/R^{2}(c)$ for point estimate $c=0.48$ plotted against
misspecified impact concavity $\hat{c}$. Right panel: Asymmetric profit ratio
$U(\hat{c})/U(c)$ for point estimate $c=0.48$ plotted against misspecified
impact concavity $\hat{c}$. The dark and light shaded areas are bootstrap
confidence intervals around $c$.

These fundamental asymmetries are illustrated in Figure 1, which contrasts the
symmetric shape of the statistical model fit with the highly asymmetric nature
of the corresponding P&Ls.

The remainder of the paper is organized in three parts:

  * •

_Optimal Trading Strategy:_ Section 2 derives closed-form formulas for trading
general alpha signals with non-linear price impact. These can be implemented
into automated trading algorithms or used in Transaction Cost Analysis (TCA).

  * •

_Empirical Estimation:_ Section 3 estimates the impact decay and concavity
parameters of the non-linear price impact model using proprietary trading
data.

  * •

_Sensitivity Analysis:_ Section 4 combines the results from the previous two
sections. Using the explicit trading strategies from Section 2, we evaluate
the P&L implications of the parameter estimates from Section 3.

Both optimal trading and parameter estimation in models with nonlinear price
impact are studied in greater detail in the technical companion paper of the
present study (Hey, Mastromatteo, Muhle-Karbe, and Webster, 2023), to which we
also refer for the derivations of the results presented here.

##  2\. Optimal Trading Strategy

This section takes a price impact model as given and focuses on deriving the
trading strategy that maximizes an alpha signal’s value net of price impact
costs. We write $Q_{t}$ for a portfolio’s position at time $t$. Therefore,
$dQ_{t}$ represents the trade at time $t$. Let $S$ be an Itô process
describing the midprice in the absence of trading, also called the
“unperturbed” or “fundamental” price. The observed midprice at time $t$ is

$P_{t}=S_{t}+I_{t}(Q),$

where $I_{t}(Q)$ is the impact caused by the trades $(Q_{s})_{s\leq t}$ up to
time $t$.

###  2.1. The AFS model

We focus on the price impact model introduced by Alfonsi et al. (2010)
(henceforth AFS) and studied further by Gatheral et al. (2012), which captures
the nonlinear nature of price impact but nevertheless remains remarkably
tractable.

In the AFS model, price impact is a nonlinear function of a moving average of
current and past order flow:333To simplify the exposition, this paper focuses
on price impact functions of power form and leaves the general treatment of
the AFS model to the technical companion paper (Hey et al., 2023). One can
also incorporate time-dependent and even stochastic liquidity by modelling
$\tau,\lambda$ as processes instead of constants. Such a model still leads to
closed-form formulas, reported in the companion paper.

(2.1) $I_{t}=\lambda\ \text{sign}(J_{t})|J_{t}|^{c},$

where $J_{t}$ is the exponentially weighted moving average

(2.2) $dJ_{t}=-\frac{1}{\tau}J_{t}dt+dQ_{t},\quad J_{0}=0.$

Here, $\lambda>0$ describes the magnitude of price impact, $c\in(0,1]$
measures its concavity, and $\tau>0$ describes the time scale over which
impact decays.444Note that this specification implies that impact $I_{t}$
relaxes all the way to zero over long time horizons, i.e., trades have no
_permanent_ impact. But see Gabaix and Koijen (2021); Bouchaud (2022) for a
detailed discussion of this point.

For $c=1$, price impact is linear and one recovers the model of Obizhaeva and
Wang (2013). $c=0.5$ corresponds to the “square-root law” practitioners
commonly use in Transaction Cost Analysis (TCA).

###  2.2. The optimization problem

Given a model for the unperturbed price $S$, a trading strategy $Q$ and an
impact model $I(Q)$, a portfolio’s P&L $Y$ has the dynamics555This
representation holds exactly if the trading strategy is smooth. If the trading
strategy includes larger transactions such as bulk trades or diffusive
purchases and salers, then there are some additional correction terms detailed
in the technical companion paper.

$dY_{t}=Q_{t}dS_{t}-I_{t}dQ_{t}.$

For many statistical arbitrage strategies, price impact rather than risk is
the main capacity constraint. In this regime, one maximizes the expected P&L,
given future return predictions. Such predictions take the form of an alpha
signal:

$\alpha_{t}=\mathbb{E}_{t}\left[S_{t+h}-S_{t}\right]\quad\mbox{for some
prediction horizon $h>T$.}$

In addition to the _level_ $\alpha_{t}$ of the current alpha prediction,
another crucial statistic in this context is its _decay_ , captured by its
drift rate $\mu^{\alpha}_{t}$. For smooth alphas, this is simply the
derivative $\mu^{\alpha}_{t}=\alpha^{\prime}_{t}$.

For a given alpha signal, a risk-neutral statistical arbitrage portfolio
maximizes

(2.3)
$\mathbb{E}\left[Y_{T}\right]=\mathbb{E}\left[\int_{0}^{T}(\alpha_{t}-I_{t})dQ_{t}\right].$

That is, each trade captures the expected alpha but pays price impact since –
as noted above – the latter does eventually disappear.

###  2.3. Solving the problem in impact space

Maybe surprisingly, the statistical arbitrage problem (2.3) has a
straightforward, closed-form solution for arbitrary alpha signals, even for
non-linear price impact described by the AFS model. The result hinges on a
simple observation, proven in the companion paper using a technique introduced
by Fruth, Schöneborn, and Urusov (2013, 2019). The key insight is that, for
any choice of position $Q$ or, _equivalently_ , the corresponding impact
variable $J$ (respectively $I$), one has

$\mathbb{E}\left[Y_{T}\right]=\mathbb{E}\left[\frac{1}{\tau}\int_{0}^{T}(\alpha_{t}J_{t}-\lambda|J_{t}|^{1+c})dt-\int_{0}^{T}J_{t}d\alpha_{t}+\alpha_{T}J_{T}-\frac{1}{1+c}|J_{T}|^{1+c}\right].$

Whence, by switching the control variables from positions $Q$ to the
corresponding impact $J$ (“passing to _impact space_ ”), the complex control
problem (2.3) becomes a simple pointwise maximization. The optimal impact in
turn is666The companion paper extends this result to general non-linear price
impact functions as well as stochastic liquidity.

(2.4)
$I^{*}_{t}=\lambda\left(J^{*}_{t}\right)^{c}=\begin{cases}\frac{1}{1+c}\left(\alpha_{t}-\tau\mu^{\alpha}_{t}\right),&t\in(0,T)\\\
\alpha_{T},&t=T.\end{cases}$

The corresponding optimal positions can be recovered via

(2.5) $Q^{*}_{t}=J^{*}_{t}+\frac{1}{\tau}\int_{0}^{t}J^{*}_{s}ds.$

Whence, even though the optimal trading strategy $Q^{*}$ can become rather
complex, switching to impact space allows one to retain a straightforward
linear relationship between the optimal policy as well as the current alpha
signal and its decay. Up to a bulk trade at the terminal time that exhausts
all remaining alpha, it is optimal to keep impact equal to a constant fraction
of the alpha signal, adjusted for alpha decay relative to impact decay. The
intuition for this adjustment is that one has to trade more aggressively and
accept higher impact costs for signals that decay quickly compared to impact,
as already highlighted by Gârleanu and Pedersen (2013) for linear impact
models:

> “The alpha decay is important because it determines how long time the
> investor can enjoy high expected returns and, therefore, affects the trade-
> off between returns and transactions costs.”

The fraction of the adjusted alpha that is optimally paid in impact depends on
the concavity parameter $c$ of the price impact function. For linear price
impact ($c=1$), the optimal impact is one half of the adjusted alpha, compare,
e.g., (Isichenko, 2021, p. 193). For strictly concave price impact functions,
this ratio increases and reaches two thirds for $c=1/2$ compatible with
square-root impact. As a result, linear models prescribe to trade less
aggressively than is optimal in their strictly concave counterparts. The
impact scale $\lambda$ does not appear in the formula for the optimal impact –
it only determines how the corresponding optimal trades need to be chosen to
attain the optimal impact state.

###  2.4. Implications for Trading

In trading applications, it is common to normalize the scaling factor
$\lambda$ as

(2.6) $\lambda=\frac{\sigma}{V^{c}}\,g(c,\tau),$

where $V$ is the average daily volume (ADV); compare, e.g., Almgren et al.
(2005). This normalization expresses the quantities of interest in trader-
friendly units: return predictions compare to the volatility $\sigma$ of price
changes and trading volumes are expressed as fractions of the average daily
volume $V$. With this normalization, the price impact coefficient also becomes
comparable across different contracts; our notation $g(c,\tau)$ highlights
that this estimate will depend on the corresponding impact concavity $c$ and
decay time scale $\tau$.

#### Long-term alpha signal

We first illustrate the application of the general optimal trading result
(2.4) for the simplest case where $\alpha_{t}=\alpha$ is constant. This means
that the signal predicts a return that happens in the distant future, e.g., an
event at the end of the month when focusing on today’s trading. For square-
root impact ($c=1/2$), Equation (2.4) then simplifies to

(2.7) $I^{*}=\frac{2}{3}\alpha.$

That is, the optimal trading strategy (for sizeable orders) pays two thirds of
the constant alpha in impact. The corresponding optimal order size is

(2.8)
$\frac{Q_{T}}{V}=\Lambda^{-2}\cdot\text{sign}(\alpha)\left(\frac{\alpha}{\sigma}\right)^{2},\quad\mbox{where
}\quad\Lambda=\frac{g(1/2,\tau)}{\sqrt{1+\frac{4}{9}\frac{T}{\tau}}}.$

Conversely, one implies a constant alpha from a long-term order of size $Q$
via the formula

(2.9) $\frac{\alpha}{\sigma}=\Lambda\cdot\text{sign}(Q)\sqrt{\frac{|Q|}{V}}.$

The order size formula 2.8 helps portfolio managers size trades considering
price impact’s square-root law. Then, traders use the implied alpha formula
2.9 to agree on a baseline alpha with the portfolio team. After establishing a
baseline alpha, the trading team can linearly add their own short-term signals
into the trading strategy.

#### Mean-reverting alpha signal:

Another standard specification assumes that the alpha signal $\alpha_{t}$ is
an Ornstein-Uhlenbeck process with relaxation time $\theta$:

$d\alpha_{t}=-\frac{1}{\theta}\alpha_{t}dt+\sigma dW_{t}.$

Then, $\mu^{\alpha}_{t}=-\theta^{-1}\alpha_{t}$, i.e., the alpha decays
exponentially in the absence of new information. In this model, a statistical
arbitrage portfolio continuously updates its signals based on a steady
information stream. The optimal impact in this context is

(2.10) $I^{*}=\frac{2}{3}\left(1+\frac{\tau}{\theta}\right)\alpha.$

Whence, mean-reverting alphas should always be traded more aggressively than
constant ones, and the size of this adjustment depends on the relative
magnitudes of impact decay $\tau$ and alpha decay $\theta$.

With decaying alpha, the portfolio’s turnover increases compared to a constant
alpha scenario, as the mean-reverting alpha case leads to higher impact
states. For instance, with square-root impact, the corresponding portfolio
(2.5) trades faster over $(0,T)$ by a factor $(1+\tau/\theta)^{2}$ due to the
alpha’s decay $\theta$.

###  2.5. Implications for Transaction Cost Analysis (TCA)

Practitioners can directly use Equation (2.4) for TCA. Indeed, this optimality
condition massively simplifies trade monitoring for statistical arbitrage
portfolios. For instance, instead of scrutinizing each trade, portfolio
managers can compare overall transaction costs with alpha signal
characteristics. Verifying best execution then boils down to checking that
alpha, impact, and alpha decay remain within reasonable bounds from the linear
formula in Equation (2.4).

Let us illustrate with some examples how portfolio managers and traders can
deploy this analysis for specific portfolios. Suppose the price impact model
is of square root form ($c=1/2$) and first consider sizeable orders with
constant long-term alpha. Then, in view of (2.7), one simply compares an
order’s transaction costs to the alpha that triggered it.

For mean-reverting alphas the optimality equation (2.10) implies that one
compares alpha capture to transaction costs considering the alpha’s decay rate
$\theta$ and the impact model’s timescale $\tau$. The quicker the alpha’s
decay, the more impact the strategy has to create by trading faster in order
to maximize profits.

The benefit to this impact-first approach to TCA is that, unlike in Section
2.4, one does not need to share order or portfolio data to verify a trading
algorithm’s behavior: the entire analysis is done comparing alpha, impact, and
alpha decay at the aggregate level.777(Webster, 2023, Chapter 3) details this
TCA method for linear price impact models.

##  3\. Empirical Estimation

###  3.1. Description of the data

The empirical analysis uses proprietary Capital Fund Management (CFM) trading
data comprised of roughly $1.9\cdot 10^{5}$ meta-orders of future contracts
traded over 2012-2022. The time at the start and the end of each meta-order is
indicated, as well as the mid-price and the number of child-orders. All meta-
orders were executed through at least three child-orders and accounted for a
fraction between 0.01% and 10% of ADV; the average order size was of the order
of 0.1%. No meta-order was traded longer than one day and the average
execution time is 2.5h.

###  3.2. Fitting methodology

Price returns are fitted against the increments of a power-law price impact
function of the form (2.1). Since i) no meta-order in the sample lasts more
than one trading day, and ii) each of them is executed with a profile close to
a TWAP, the volume impact $J_{t}$ is computed under the assumption that trades
are executed uniformly during the execution time $\Delta t$ of each meta-
order, i.e., $Q_{t}=\frac{Q}{\Delta t}t$. Note that the alpha signals driving
the decision are typically realized over a time window much larger than one
day, implying that it is appropriate to attribute to impact, rather than
alpha, the average price variation occurring during the execution of meta-
orders in the sample.

Given $J_{t}$ the three parameters of the AFS model are calibrated:

  1. (i)

The concavity coefficient $c$;

  2. (ii)

The impact decay timescale $\tau$;

  3. (iii)

The scaling factor $\lambda$, normalized as in (2.6).

###  3.3. Estimating impact concavity

As an initial visualization of price impact concavity, Figure 2 plots the
average return against order size without impact decay. The log-log plot is
linear, indicating a power-law relationship in the tails. The slope of the
log-log fit measures the exponent of the power law for sizable orders. Here,
this analysis leads to an estimate of $c=0.48$ , consistent with the commonly
observed square-root law (see e.g. Bouchaud et al. (2018)). The bootstrap
confidence interval from Figure 1 in the introduction is computed with the
standard deviation of this parameter estimate, $\textrm{std}(c)=2.5\cdot
10^{-2}$.

Figure 2. Expected signed return plotted against the volume fraction
$|Q|/\text{ADV}$ in log-log scale. The relation is concave and follows a
power-law with $c=0.48$.

The estimation of the price impact parameters with impact decay is more
subtle. To this end, we consider a grid of impact decays $\hat{\tau}$ and
concavity parameters $\hat{c}$. For each pair, we then first compute the
volume impact $J_{t}$ corresponding to the impact timescale $\hat{\tau}$
according to (2.2); then the price impact $I_{t}$ is computed for the
corresponding concavity parameter $\hat{c}$. With the impact variable at hand,
we then run a linear regression of price changes against impact to determine
the pre-factor of the price impact model.

Figure 3 shows the model’s statistical sensitivity $R^{2}(\hat{c},\hat{\tau})$
and pre-factor $g(\hat{c},\hat{\tau})$ across a broad range of concavity and
decay parameters. The heatmap of $R^{2}$ in Subfigure 3(a) shows an ellipsoid
around the point estimated parameters $(c,\tau)$. Figure 3(c) plots the
_statistical_ sensitivity of the model to the concavity parameter $c$ together
with the pre-factor. Here, the point estimate of the decay parameter
$\tau=0.2$ days remains fixed, i.e., $R^{2}(\hat{c},\tau)$ and
$g(\hat{c},\tau)$ are displayed.

As the concavity parameter $c$ varies, the model’s $R^{2}$ peaks at $c=0.48$
and decreases markedly around this point estimate.888The magnitude of the
model’s $R^{2}$ is consistent with results published in papers using the
public trading tape. Indeed, when one runs the price impact model on the full
trading tape, the $R^{2}$ varies between 10% and 20%. However, in this study,
only CFM’s metaorders are used. Therefore, the $R^{2}$ drops in line with the
fraction of the public tape CFM’s orders represent. The pre-factor
$g(\hat{c},\tau)$ increases with increasing concavity parameter. This increase
is because the brunt of the data consists of smaller orders, leading to the
linear model under-estimating impact compared to a concave model.

(a) $R^{2}(\hat{c},\hat{\tau})$.

(b) $g(\hat{c},\hat{\tau})$.

(c) $R^{2}(\hat{c},\tau)$ and $g(\hat{c},\tau)$.

(d) $R^{2}(c,\hat{\tau})$ and $g(c,\hat{\tau})$.

Figure 3. Calibration results. Panel (A) shows the statistical sensitivity;
Panel (B) depicts the model’s pre-factor across $\hat{c},\hat{\tau}$. The
sensitivity analysis yields an ellipsoid around the best-fit parameters. On
the other hand, the model’s pre-factor remains roughly constant along the
$\hat{\tau}$-axis and increases with concavity $\hat{c}$. Panel (C) shows the
sensitivity across $\hat{c}$ for $\tau=0.2$ days in green where $R^{2}$ peaks
at the point estimate of $c=0.48$ in line with the square-root law. The fitted
pre-factor $g(\hat{c},\tau)$ is displayed in blue. Panel (D) displays the
sensitivity and the model’s pre-factor across $\hat{\tau}$ for the point
estimate $c$.

###  3.4. Estimating impact decay

To investigate the statistical sensitivity with respect to the impact decay,
the concavity parameter $c$ is fixed at the point estimate $c=0.48$ and the
impact decay $\hat{\tau}$ varies. Figure 3(d) displays $R^{2}(c,\hat{\tau})$
and $g(c,\hat{\tau})$. There is a peak at $\tau=0.2$ days, indicated that one
measures the short-time scale of impact decay. The model’s pre-factor remains
relatively constant for timescales larger than 0.1 days.

The primary take-away is that the _statistical_ loss is significantly more
sensitive to impact concavity than decay. For example, misspecifying $c=1$
instead of the point estimate $c=0.48$ reduces $R^{2}$ by 50%. Conversely,
getting $\tau$ wrong by a factor of ten reduces $R^{2}$ by less than
2%.999This is not a new observation: studies across many datasets and asset
classes agree on price impact’s concavity and general order of magnitude but
disagree on decay. The reason is that different orders decay at different
timescales, which are better captured by a multi-exponential or power law
decay kernel. With linear price impact, there are some results for such decay
kernels, cf. Bouchaud et al. (2004); Gatheral et al. (2012); Abi Jaber and
Neuman (2022) and the references therein. Extending this analysis to models
with nonlinear impact _and_ impact decay is an important direction for future
research

These empirical results do _not_ imply that calibrating impact decay correctly
is unimportant. Indeed, a central result in Section 4 is that a portfolio’s
P&L can be _highly_ sensitive to $\tau$, even if _statistically_ , $R^{2}$ is
not as sensitive to $\tau$.

##  4\. Sensitivity Analysis

We now combine the theoretical and empirical results from the previous
sections to quantify the performance losses caused by trading based on a
misspecified model.

###  4.1. Quantifying misspecification costs

Recall that, for given estimates $\hat{c},\hat{\tau}$ of impact concavity and
decay, the corresponding fitted pre-factor is

$\lambda(\hat{c},\hat{\tau})=\frac{\sigma}{V^{\hat{c}}}g(\hat{c},\hat{\tau}).$

Therefore, assuming $\hat{c},\hat{\tau}$ are the correct model parameters, a
portfolio trading an alpha signal $\alpha_{t}$ with decay $\mu^{\alpha}_{t}$
will implement

$I_{t}(\hat{c},\hat{\tau})=\frac{\alpha_{t}-\hat{\tau}\mu^{\alpha}_{t}}{1+\hat{c}}\quad\mbox{or,
equivalently,}\quad
J_{t}(\hat{c},\hat{\tau})=\left(\frac{\alpha_{t}-\hat{\tau}\mu^{\alpha}_{t}}{(1+\hat{c})\lambda(\hat{c},\hat{\tau})}\right)^{1/\hat{c}}.$

We call $I(\hat{c},\hat{\tau})$ or, equivalently, $J(\hat{c},\hat{\tau})$, the
_misspecified_ policy. This is the optimal policy when the impact parameters
$\hat{c},\hat{\tau}$ correctly describe price impact, but not if these
parameters differ from the price impact parameters $c,\tau$ of the _actual_
data generating model. To ease notation, we omit the corresponding variables
when one of the parameters is held fixed at its actual value, e.g., we use the
shorthand $I(\hat{\tau})=I(c,\hat{\tau})$.

The P&L of the _misspecified_ policy $J(\hat{c},\hat{\tau})$ under the
_actual_ price impact model with parameters $c$, $\tau$ is

$\displaystyle U\Big{(}J(\hat{c},\hat{\tau});c,\tau\Big{)}=$
$\displaystyle\frac{1}{\lambda}\mathbb{E}\Bigg{[}\frac{1}{\tau}\int_{0}^{T}((\alpha_{t}-\tau\mu^{\alpha}_{t})J_{t}(\hat{c},\hat{\tau})-\lambda|J_{t}(\hat{c},\hat{\tau})|^{1+c})dt$
$\displaystyle\qquad+\alpha_{T}J_{T}(\hat{c},\hat{\tau})-\frac{1}{1+c}|J_{T}(\hat{c},\hat{\tau})|^{1+c}\Bigg{]}.$

This general formula is straightforward to implement numerically in a back
test. For instance, statistical arbitrage teams can plug in historical alpha
signals to measure the expected P&L of trading based on $\hat{c},\hat{\tau}$
when the actual parameters are $c,\tau$.

To illustrate the misspecification formula’s implications, Sections 4.2 and
4.3 cover two common cases. By specifying concrete parametric alpha signals,
the cost of misspecifying concavity or decay can be quantified in closed-form,
which reveals the dependence on characteristics of the corresponding alpha
signals without relying on a back test.

###  4.2. Misspecification cost of impact concavity

We first consider the simplest case of a constant alpha signal:
$\alpha_{t}=\alpha$, so that $\mu^{\alpha}_{t}=0$. For example, this
assumption is reasonable when trading intraday based on a signal that predicts
a return past today’s close, such as an event taking place the next day or
week.

As impact decay only plays a minor role in this context,101010Indeed, for
constant alpha, the P&L of the believed strategy only depends on the impact
decay parameter $\hat{\tau}$ through the estimate of the magnitude of price
impact $\lambda(\hat{c},\hat{\tau})$. As this dependence is rather weak,
impact decay only plays a minor role in the absence of alpha decay. we assume
for simplicity that it is correctly specified ($\hat{\tau}=\tau$) and focus on
the misspecification cost of the concavity parameter $\hat{c}\neq c$. In
words:

> How much P&L does the portfolio lose if it trades with the wrong price
> impact concavity?

For example, one canonical case is when $c=0.5$ and $\hat{c}=1$: then, one
estimates the cost of implementing a linear price impact model when actual
impact follows a square-root law.

The expected P&L for the _misspecified_ policy is

$\displaystyle U\Big{(}J(\hat{c});c\Big{)}=$
$\displaystyle\frac{\sigma\,V}{g(\hat{c})^{1/\hat{c}}}\Bigg{[}\left(\frac{\alpha}{\sigma}\right)^{1+1/\hat{c}}\left(\frac{T}{\tau(1+\hat{c})^{1/\hat{c}}}+1\right)$
$\displaystyle\qquad\qquad-\frac{g(c)}{g(\hat{c})^{c/\hat{c}}}\left(\frac{\alpha}{\sigma}\right)^{(1+c)/\hat{c}}\left(\frac{T}{\tau(1+\hat{c})^{(1+c)/\hat{c}}}+\frac{1}{1+c}\right)\Bigg{]}.$

The natural comparison for this is the expected P&L of the optimal policy for
the actual concavity parameter $c$:

$U\Big{(}J(c);c\Big{)}=\frac{\sigma\,V}{g(c)^{1/c}}\left(\frac{\alpha}{\sigma}\right)^{1+1/c}\frac{c}{1+c}\left(\frac{T}{\tau(1+c)^{1/c}}+1\right).$

In this setting, the primary alpha characteristic is the signal’s Sharpe
ratio, $\alpha/\sigma$. A core result from Figure 4 is that misspecification
costs are more sensitive to $c$ as a signal’s Sharpe ratio increases.
Therefore, the stronger a team’s alpha signal, the more important it becomes
to correctly estimate the price impact model’s concavity accurately.

Figure 4. Profit ratios between correctly and incorrectly specified strategies
when $c=0.48$. Left panel: Heat map of the profit ratio across $\hat{c}$ and
$\alpha/\sigma$. Right panel: Profit ratio across $\hat{c}$ for an alpha
signal with Sharpe $\alpha/\sigma=1$. The profits are asymmetric: over-
estimating concavity (low $\hat{\delta}$) quickly leads to losses, while
under-estimating concavity only shrinks profits.

In addition to quantifying the empirical sensitivity of P&L to
misspecifications of the concavity parameter $c$, the closed-form formulas
also allow one to compute _how wrong_ a parameter can become before turning
profitable strategies unprofitable. Indeed, for each true parameter $c$, there
exists a critical value $c_{\min}$ such that, for any choice
$\hat{c}<c_{\min}$, the expected P&L becomes a _decreasing_ function of the
alpha signal. Naturally, statistical arbitrage portfolios crucially need to
avoid this critical regime where adding more predictive power leads to worse
outcomes. Figure 4 illustrates the sharpness of the phase transition between
profitable and non-profitable misspecified trading strategies. Because
misspecification costs become more sensitive with higher alpha Sharpe ratios,
the critical value becomes tighter as the Sharpe increases. Again, this
highlights the importance of correctly estimating price impact’s concavity
when trading high Sharpe ratio signals.

A crucial insight is that this misspecification risk is asymmetric: even for
large over-estimates $\hat{c}>c$, the expected P&L typically remains positive
(except for very small Sharpe ratios). The intuition is that overestimating
the concavity parameter $c$ leads the portfolio to submit smaller trades.
While this is suboptimal and sacrifices some profit opportunities, it will
only lead to shrinking profits but not losses. Conversely, underestimates
below the critical value $c_{\min}<c$ can lead to significant losses, as the
corresponding trading strategy submits outsized trades leading to excessive
trading costs.

While these results are derived in a concrete model with a convenient closed-
form solution, we expect them to remain true qualitatively across a much wider
range of models. This leads to a practical takeaway when fitting price impact
models: when considering confidence intervals for a price impact model’s
concavity, the robust solution is to deploy a more conservative estimate from
the upper part of the confidence interval, i.e., use a impact model that is
somewhat less concave than the point estimate.

###  4.3. Misspecification cost of impact decay

We now turn to misspecification of the impact decay parameter $\tau$, while
keeping the concavity parameter $c$ fixed to its correct value. As impact
decay predominantly interacts with the alpha decay, we study this for a mean-
reverting alpha with $\mu^{\alpha}_{t}=-\alpha_{t}/\theta$. To obtain crisp
results, we focus on the steady-state limit
$\lim_{T\to\infty}\frac{1}{T}\mathbb{E}[Y_{T}]$ of the P&L. Unlike the model’s
statistical $R^{2}$, it turns out that the expected P&L is surprisingly
sensitive to misspecifications in $\tau$.

For the _misspecified_ policy, we have

$\displaystyle U\Big{(}J(\hat{\tau});\tau\Big{)}=$
$\displaystyle\frac{\sigma\,V}{\tau}\left(\frac{1+\hat{\tau}/\theta}{g(\hat{\tau})(1+c)}\right)^{1/c}\left(1+\frac{\tau}{\theta}-\frac{g(\tau)(1+\hat{\tau}/\theta)}{g(\hat{\tau})(1+c)}\right)$
$\displaystyle\qquad\times\lim_{T\to\infty}\frac{1}{T}\left[\int_{0}^{T}\left|\frac{\alpha_{t}}{\sigma}\right|^{1+1/c}dt\right].$

The optimal value derived from the policy for the actual impact decay $\tau$
is

$\displaystyle U\Big{(}J(\tau);\tau\Big{)}=$
$\displaystyle\frac{\sigma\,V}{\tau}\frac{c(1+\tau/\theta)^{1+1/c}}{g(\tau)^{1/c}(1+c)^{1+1/c}}\times\lim_{T\to\infty}\frac{1}{T}\left[\int_{0}^{T}\left|\frac{\alpha_{t}}{\sigma}\right|^{1+1/c}dt\right].$

In this setting, the key alpha characteristic is its decay $\theta$. More
specifically, the ratio of performances for the misspecified and optimal
policies mostly depends on the estimated and true impact decays through their
ratios relative to alpha decay, $\hat{\tau}/\theta$ and $\tau/\theta$. Indeed,
as depicted in Figure 3(d), $g(\tau)\approx g(\hat{\tau})$ is a good
approximation for a wide range of decay parameters; with this, the ratio of
performance simplifies to

$\displaystyle\frac{U\Big{(}J(\hat{\tau});\tau\Big{)}}{U\Big{(}J(\tau);\tau\Big{)}}=\frac{1}{c}\left(\frac{1+\hat{\tau}/\theta}{1+\tau/\theta}\right)^{1/c}\left(1+c-\frac{1+\hat{\tau}/\theta}{1+\tau/\theta}\right).$

Figure 5. Profit ratio between correctly and incorrectly specified strategies
when $\tau=0.2$ days. Left panel: Heat map of the profit ratio across
$\hat{\tau}$ and $\theta$ in log-log scale. Right panel: Profit ratio across
$\hat{\tau}$ in log-scale for an alpha signal with a decay timescale of 1 day.
The profits are asymmetric: over-estimating decay (high $\hat{\tau}$) quickly
leads to losses, while under-estimating decay only shrinks profits.

The profit ratio between misspecified and optimal policy is displayed in
Figure 5. There is a sharp boundary between the profitable and unprofitable
regions that depends linearly on the ratios $\hat{\tau}/\theta$ and
$\tau/\theta$. This boundary appears then curved in log-log scale on the left
panel in Figure 5 separating the positive and negative profit regions. A wrong
estimation of impact decay for a long-lasting alpha signal (small $\theta$) is
less costly than for a fast decaying one.

Like for impact concavity, the costs of misspecifying impact decay are highly
asymmetric. Indeed, overestimating impact decay quickly turns profitable alpha
signals unprofitable, as illustrated in the right panel of Figure 5 for a
moderate alpha decay time scale of 1 day. The intuition is that overestimated
$\hat{\tau}$ lead to larger than optimal impacts, i.e., overly aggressive
trading just like underestimates of the concavity parameter.

##  5\. Conclusion

In this paper, we investigate the effect that the misspecification of the
price impact model has on the performance of a trader. This leads to three
broad takeaways:

  1. (i)

For the AFS model, the closed-form formula

$I^{*}_{t}=\frac{1}{1+c}\left(\alpha_{t}-\tau\mu^{\alpha}_{t}\right)$

relates the optimal impact state $I^{*}$ to a trader’s alpha level
$\alpha_{t}$, alpha decay $\mu^{\alpha}_{t}$, impact concavity $c$, and impact
decay $\tau$.

  2. (ii)

Statistical estimates yield $c\approx 0.5$ for sizeable meta-orders, in
consensus with the literature (Almgren et al., 2005; Bouchaud et al., 2018).
However, statistical estimates of $\tau$ are less precise: a broad range of
impact decays $\tau$ statistically fit the model similarly well.

  3. (iii)

The P&L cost of getting an impact parameter wrong can be evaluated using the
AFS model’s tractable _misspecification cost_ formula. This P&L-driven
approach complements a statistical approach and shows that the opportunity
costs of getting model parameters wrong are _asymmetric_ :

     * •

It is better to over- than under-estimate $c$. An excessively concave price
impact model can lead to losses.

     * •

It is better to under- than over-estimate $\tau$. A price impact model with
excessively fast impact decay can lead to losses.

In both cases, overly aggressive trading has a substantially bigger effect
than an overly cautious approach. While we illustrate this with the closed-
form solutions for the AFS model, these insights are expected to remain
qualitatively true for many models. Therefore, a general approach when
considering confidence intervals for price impact model parameters is to
deploy parameters within the error band that lead to more conservative trades.

Our results thereby support the relevance of a robust approach to optimal
trading problems in the presence of trading costs, and supports the idea of
jointly tackling parameter estimation and alpha exploitation.

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