Modeling Joint Temporary Price Impact with Hawkes Processes

# Reference for Aggregated Temporary Price Impact


# Reference for Aggregated Temporary Price Impact












I am wondering if someone knows relevant literature on the joint temporary price impact. The temporary price impact here refers to the difference between the best ask/bid price and transaction price at a specific time. For example, a large order may consume layers of liquidity within the limit order book.

Here are more details. Let us consider the Almgren-Chriss model. Denote by $(P_t)_{t\in[0,T]}$ the fundamental price of an asset. The market price $(S_t)_{t\in[0,T]}$ follows

$$S_t = P_t + \int_0^t \frac{1}{N} \sum_{i=1}^N v_s^i \, ds,$$

where $v_t^i$ is the trading rate of agent $i$ at time $t$. The integral term is a popular linear example of the joint permanent price impact. The non-linear version can be generalised. For the transaction price $\bar{S}_t^i$ of agent $i$, much of the literature uses "single-agent" temporary price impact as

$$\bar{S}_t^i = S_t + \lambda \, v_t^i$$

for some $\lambda > 0$. This approach neglects the actions from the others. In addition to the following linear example from Schied and Zhang:

$$\bar{S}_t^i = S_t + \frac{\lambda}{N} \sum_{i=1}^N v_t^i,$$

I am not able to find enough literature on the joint temporary impact. Examples of questions I am interested in include:

(1) if there are lots of buying market orders at certain time, how does it affect the temporary impact on the ask side?

(2) if there are lots of buying market orders at certain time, how does it affect the temporary impact on the bid side?

Any theoretical and empirical references are highly appreciated. Thank you!

## Answer by lehalle (score 1)

https://quant.stackexchange.com/a/78060

I think what you are looking for is usually captured by Hawkes models. The reference paper is certainly Bacry, Emmanuel, Sylvain Delattre, Marc Hoffmann, and Jean-François Muzy. "Some limit theorems for Hawkes processes and application to financial statistics" Stochastic Processes and their Applications 123, no. 7 (2013): 2475-2499.

First off all you have to be aware that to account for the kind of effects you have in mind, you need to use Point Processes that are marking events. Typical events are insert, cancel, and transaction. Let's focus on the first limit for simplicity: $dN^{B/A}_{i/c/t}$ is the occurence of such events on any of the sides. It is parametrised by its intensity $$\lambda^{B/A}_{i/c/t} = \lim_{dt\rightarrow 0} \mathbb{E} \left(\frac{dN^{B/A}_{I/c/t}}{dt}\right).$$

They you want to encode in the model the fact that the probability of any of these events to occur is conditioned by the past occurence of others (i.e. your example of "a lot of buying transactions influencing the ask"). In Hawkes models it is done via a kernel that capture this: $$\lambda_t = \mu_t + \sum_{s,e}\int_{\tau<t} \phi^s_e (t-\tau) \;dN^{s}_e(\tau).$$ Thanks to that the models can capture mean reverting effects and cascades of trades.

Here are typical dependencies if you fit them of real data:

If you want to go further you can use the Queue Reactive Model; instead of looking at the sequence of recent events, it looks at the state of the (full) orderbook to predict the intensities of the 2x3 types of events. The seminal reference is Huang, Weibing, Charles-Albert Lehalle, and Mathieu Rosenbaum. "Simulating and analyzing order book data: The queue-reactive model" Journal of the American Statistical Association 110, no. 509 (2015): 107-122.