Overview
Kyle proposed the following equation as a model of market impact on the price of a traded asset.
{% \hat{S}_t = S_t + \psi \frac{\Delta \theta}{\Delta t} %}
Here we have
- {% S_t %} is the price of the asset without market impact effects
- {% \hat{S}_t %} is the (average) price of the asset that is obtained with market impact included
- {% \\Delta \theta %} is the number of shares traded
- {% \Delta t %} is the time frame over which the trades occur
- {% \psi %} is an adjustable parameter that represents the illiquidity of the asset being traded. That is, {% \psi %} is larger when the market impact is greater.
Multiple Assets
The model can be extended to trading multiple assets as follows.
{% \vec{\hat{S}}_t = \vec{S}_t = \frac{1}{\Delta t} \psi \Delta \vec{\theta} %}
where now
- {% S_t %} and {% \hat{S}_t %} and {% \theta %} are now vectors
- {% \psi %} is a positive definite matrix. When it is diagonal, the multi asset model is equivalent to a set of single asset models. When it is not diagonal, there are correlations among the assets such that trading one asset may have market impact on another.