Security Bid/Ask Dynamics with Discreteness and Clustering

Joel Hasbrouck in his paper, “Security Bid/Ask Dynamics with Discreteness and Clustering” , uses Gibbs sampling for estimating the parameters of a stylized market microstructure model. For any model, there are many ways to estimate parameters. One of the common methods is the likelihood approach. Even though this approach makes sense intuitively, the computational complexity explodes as the number of parameters increase. The curse of dimensionality kicks in and hence parameters become notoriously unstable. On the other hand, estimation methods based on MCMC scale linearly with parameters. Thus MCMC becomes an important technique for dealing with curse of dimensionality.

The paper deals with modeling key microstructure effects : a stochastic cost of market-making, discreteness and clustering. Hasbrouck describes the model as,

The permanent price component common to the observed bid and ask quotes follow a random walk. The dealer posting a bid or ask quote is subject to a stochastic trading cost that encompasses clearing, asymmetric information and inventory costs specific to the next trade and some allocation of fixed costs. These costs imply ask and bid prices which map in turn on to the posted discrete quotes via a stochastic rounding process.

The model has three latent state variables denoted by set s ( the permanent price component, the cost of trading and the implicit tick size). A state equation for each of these latent variables is assumed by the author. These latent variables are driven by a set of parameters Θ. The observed quotes, i.e. bid and ask prices are mapped on to the latent state variables in a non linear way. Given the state equation, parameters driving the state equation and the observed equation, the paper then goes on to use Gibbs sampling procedure to estimate the parameters.

State equation :

$\begin{align*} m_t &= m_{t-1} + u_t , \quad u_t \sim N( 0, \sigma_u^2)\\ c_t &\sim N( \mu_c, \sigma_c^2)\\ K_t &= \begin{cases} 1, \text { with probability } k \\ 5, \text { with probability } 1-k \\ \end{cases} \end{align*}$

Observation equation :

$\begin{align*} \text{bid}_t = \text{ Floor }( e^{m_t} - e^{c_t}, K_t) \\ \text{ask}_t = \text{ Ceiling }( e^{m_t} + e^{c_t}, K_t) \end{align*}$

All the priors for the parameters in the model are taken as non-informative. They are chosen so that the posterior distributions fall under convenient closed form distributions.

Gibbs sampling is run at multiple levels and MCMC chain output is used to derive estimates about the parameters. The paper ends by suggesting extensions to the above model to capture more types of microstructure effects such as

Time varying volatility
Incorporate time varying and serially correlated trading costs
Asymmetric bid and ask quotes
Alternate clustering methods

This paper is simple enough that any reader can easily get an idea about the application of MCMC methods to market microstructure modeling.