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- Impact Models

# Impact Models

An impact model is used to predict changes in price due to trading activity.

This expectation of price change may be used to inform execution and ptf construction decisions.

Several well-known models have been proposed.

Almgren and al. use a proprietary data set obtained from Citigroup's equity trading desk in which a trade's direction is known.

For most public data sets, trade direction is not available and has to be estimated by a classification algorithm.

In their model, trades serve as a proxy for trading imbalance.

The authors assume that, some time after the complete execution of a parent order, only permanent impact remains.

This allows them to separate impact into its temporary and permanent components.

The model parameters can then be calculated from a regression, giving the following results:

1) Permanent impact cost is linear in trade size and volatility.

2) Temporary impact cost is linear in volatility and roughly proportional to the square root (Almgren and al. find a power of 3/5) of the fraction of volume represented by one's own trading during the period of execution. Hence, for a given rate of trading, a less volatile stock with large average daily volume has the lowest temporary impact costs.

**The Buy-side Perspective**

Optimal execution algos have less value to a typical ptf manager if analyzed separately from the corresponding returns earned by his trading strategy.

In fact, high transaction costs are not bad *per se* - they could simply prove to be necessary for generating superior returns.

The typical sell-side perspective of algo trading does not take expectation of profits or the client's ptf objectives into account.

Needless to say, this is an important component of execution.

The decisions of the trader and the ptf manager are based on different objectives.

The **trader** decides on the timing of the execution, breaking large parent orders into a series of child orders that, when executed over time, represent the correct trade-off between opportunity cost, market impact, and risk. He sees only the trading assets, whereas the ptf manager sees the entire ptf, which includes both the trading assets and the static nontrading assets.

The **ptf manager**'s task is to construct a ptf by optimizing the trade-off between opportunity cost, market impact, and risk for the full set of trading and nontrading assets. In general, the optimal execution framework described by Almgren and Chriss is not appropriate for the ptf manager.

Engle and Ferstenberg proposed a framework that unites these objectives by combining optimal execution and classical mean-variance optimization models.

In their model, trading takes place at discrete time intervals as the ptf manager rebalances his ptf holding w_{t} at times t = 0, 1, ..., T subject to changing expected returns, µ_{t}, and risk (measured by the covariance matrix of returns), , until he reaches the ptf that reflects his final view: .

The joint dynamic optimization problem has the form:

where is the temporary market impact function. This is a dynamic programming problem that has to be solved by numerical techniques.

Intuitive interpretation:

1) The first term represents the standard mean-variance optimization problem.

2) The second term corresponds to the optimal execution problem.

3) The third term is the covariance between the remaining shares to be traded and the final position.

In the single asset case, this term is positive (negative) for buying (selling) orders, which implies that risk is reduced (increased).