Using State Transitions to Forecast the Limit Order Book Micro-Price

# Stoikov Micro Price Absorbing States


# Stoikov Micro Price Absorbing States












Digging into Stoikov's Micro Price paper, I'm having some trouble understanding the intuition behind the second of the two different absorbing states and it's implication for the calculation of G*.

To find the first order adjustment, G1, we need to find R1 as part of the calculation:

$$ R1_{xk} = Prob(M_{t+1} - M_{t} = k | X_{t} = x) $$

This I understand as the transition probabilities of the absorbing states. So the probabilities of the mid price changing by some value k in [-0.01, -0.005, 0.005, 0.01] from period t to t+1, conditional on our state variables X = (I, S) starting and remaining in some combination, x, for t to t+1.

So then for R2 (or T, as noted in the paper):

$$ R2_{xy} = Prob(M_{t+1} - M_{t} \neq 0 ∧ X_{t+1} = y | X_{t} = x) $$

Is this telling us the probability that both the mid price and the combination of state variables change from t to t+1? As an observation, it looks like Q:

$$ Q_{xy} = Prob(M_{t+1} - M_{t} = 0 ∧ X_{t+1} = y | X_{t} = x) $$

Which is probability of changing from transient state x to transient state y. What does this mean B represents and why successively multiply it with G1?

## Answer by lehalle (score 1, accepted)

https://quant.stackexchange.com/a/79964

Sorry but I am not sure to understand your question (changing the notations of the paper you ask a question on is not very nice with the reader). I have to confess that Sasha's paper is not extraordinary clear neither, and having attended to some of the conference where he explained his approach helped me. So let me recap the framework of this micro-price paper, that is in reality very useful.

The micro-price has to be seen as the expected price given some information that is influencing, according to you, its moves. Consider the bid-ask spread and the imbalance as Sasha's favorite ones, but you can consider the shape of the orderbook (that is more detailed) if you prefer, like in Huang, Weibing, C-A L, 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.

The natural notation is then $$M_t:=\mathbb{E}(P_{t+1}|(I_t,S_t)=(i,s))$$ for the microprice at $t$.

What is important to keep in mind is that in the paper you consider only the set of information, $(I,S)=$(Imbalance, BA-Spread), that you use. It means that you have to keep track of their expected values too: $$(\hat I_{t+1}, \hat S_{t+1})=\mathbb{E}((I_{t+1}, S_{t+1})|(I_t,S_t)=(i,s)).$$ Another important observation in the paper is that there is no hope that the micro-price (likewise the price) is stationary, but its increments are.

Last but not least, events on the orderbook are occurring at stopping times $\tau_k,\tau_{k-1},\ldots$ (the Queue Reactive model tells us that they follow a Poisson process once conditioned by the shape of the orderbook; since this shape contains the bid-ask spread and the imbalance, the micro-price model is a special case of the QR model) and not at $t,t-1,\ldots$, the notations have to be complicated.

Let me nevertheless keep the notation $t,t-1,\ldots$ for simplicity.

Based on that, the correct state space is ${\cal F}=(I,S)$, whereas you want to keep track of $M_t-M_{t-1}$ The Markov chain you have to consider has transition of interest: $${\cal T}_{t+1}:=\mathbb{E}\left[ \begin{pmatrix}M_{t+1}-M_t\\ I_{t+1}\\ S_{t+1}\end{pmatrix} \middle| \begin{pmatrix} I_{t}\\ S_{t}\end{pmatrix} \right].$$ And use the notation $${\cal T}_{t+1}|_{IS}:=\mathbb{E}\left[ \begin{pmatrix}I_{t+1}\\ S_{t+1}\end{pmatrix} \middle| \begin{pmatrix} I_{t}\\ S_{t}\end{pmatrix} \right]$$ For the transition of the Markov chain of interest, and $${\cal T}_{t+1}|_{\Delta M}:=\mathbb{E}\left[ M_{t+1} - M_t \middle| \begin{pmatrix} I_{t}\\ S_{t}\end{pmatrix} \right]$$ for the change in micro-price given the state space.

This is in the paper the first transition. The $t$-th order transition is named $i$-th transition in Sahsa's paper; that is misleading because it may let you think that $i$ stands for imbalance as it stands for the $i$-th stopping time. With my notation $I_t=i$ replaces $I_{\tau_i}=\iota$ that is easier to read.

We need to last notations

- $\frak T$ for the transition matrix, a big matrix with all the possible states as rows and columns

- $V_t$ that is a vector with zeros everywhere except the state that you observe at $t$.

This is standard for Markov chains.

Anyway, the good way to keep track of this model is to compose the transitions: $$\mathbb{E}(M_{t+i+1}-M_{t+i}|{\cal F}_{t+i}) = {\cal T}_{t+i+1}|_{\Delta M} \circ \underbrace{{\cal T}_{t+i}|_{IS} \circ {\cal T}_{t+i-1}|_{IS} \circ \ldots \circ {\cal T}_{t+1}|_{IS}}_{{\frak T}^i \cdot V_t}.$$

Now, to recover the expected price in $i$ steps, one can write $$M_{t+i+1} = M_t + \sum_{n=1}^{i} {\cal T}_{t+n+1}|_{\Delta M} \circ {\frak T}^n \cdot V_t.$$

This is a long answer because the notations have to be complex, but it is not complicated, thus I hope it helps.