Kuan Yang

Kuan Yang

Postgraduate in Mathematical and Computational Finance @ Oxford

A Parsimonious Stochastic Simulation for LOB Dynamics

The Problem

What we care about is the following: \(E(X_t | \mathcal{F}_{t-20})\) From the perspective of statistics, we are finding a regression or any other model, that characterizes the relation by learning from previous observations.

From the aspect of probability, we have a coarse mindset, that what kind of the model would be. Based on observations, we are calibrating our model parameters.

Adjustment to the data

We adjust the LOB:

  • Group by Time Tick

  • Group Bid Ask / Trade into different Classes

Reconstructed LOB{#fig:LOB}

Statistical Approach - Bid-Ask Pressure

Here are two factors that is intended to reflect the pressure between buy and sell:

  • $F_1 = a1 - b1$ (The difference between best ask and best bid)

  • $F_2 = a1/b1$ (The ratio between best ask and best bid)

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Both Factors have good approximation for approximate window.

Statistical Approach - Willingness to be Executed

Consider the gap between the volume (price) between the second rank Bid and Ask.

  • $F_3$ = a2 - b2

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Statistical Approach - Market Situation

How many orders remain is also a good indicator for the market dynamics.

  • $F_4 = \Sigma a_i - \Sigma b_i$

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Long / Short Memory

A question is whether or not the price would be affected by the historical event that has happened long time ago? We considered the moving average of bid and ask.

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Probabilistic Approach - How many orders arrived in unit time?

The Mid Price is the average of best bid and best ask.

  • Poisson Process describes the arrival of events in unit time

  • We have to consider the randomness of the $\lambda$ and the discrete property of Poisson Distribution - Cox Process

Volume of $a1${#fig:a width=”\textwidth”}

Volume of $b1${#fig:b width=”\textwidth”}

Relation between Mid-Price and LOs

\(dS_t = (v + \alpha_t) dt + \sigma dW_t\) where \(d \alpha_t = -\xi \alpha_t dt + \sigma_{\alpha} dB_t + \epsilon^+ dL_t^+ - \epsilon^- dL_t^-\) While we simplify to the following numerical approach: \(dS_t = w \cdot dX_t\) \(X_t|_{\lambda_t} \sim Pois(\lambda_t)\) where \(d \lambda_t = \mu_t dt + \sigma_t dW_t\)

Calibrating the Model

  • $w$: measuring the impact of orders, OLS / WLS

  • $\mu$: measuring the drift of order arrivals during unit time - mean of difference

  • $\sigma$: measuring the volatility of order arrivals - variance of LO amount

Numerical Results

100 trajectories of simulated path{#fig:a width=”\textwidth”}

Return by Probabilistic Method{#fig:b width=”\textwidth”}

Reflections on Probabilistic Model

  • Slow decay when the prediction interval is growing (Robustness)

  • Seamless connection with Optimal Execution

  • However, it takes longer time to run the Monte Carlo simulation