# Modeling liquidity in corporate bond markets: applications to price
adjustments

Philippe Bergault111Université Paris Dauphine-PSL, Ceremade, 75116 Paris,
France<EMAIL_ADDRESS>Olivier Guéant222Université Paris 1
Panthéon-Sorbonne, Centre d’Economie de la Sorbonne, 106 Boulevard de
l’Hôpital, 75642 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract

To assign a value to a portfolio, it is common to use Mark-to-Market prices.
But how to proceed when the securities are illiquid? When transaction prices
are scarce, how to use other available real-time information? In this article
dedicated to corporate bonds, we address these questions using an extension of
the concept of micro-price recently introduced for assets exchanged on limit
order books in the market microstructure literature and ideas coming from the
recent literature on OTC market making. To account for liquidity imbalances in
OTC markets, we use a novel approach based on Markov-modulated Poisson
processes. Beyond an extension to corporate bonds of the concept of micro-
price, we coin the new concept of Fair Transfer Price that can be used to
value or transfer securities in a fair manner even when the market is illiquid
and/or tends to be one-sided.

Key words: fair price, imbalance, liquidity, market making, corporate bonds.

##  1 Introduction

We are all used to see the real-time prices of stocks scrolling on TV or
blinking on the screens of our computers and cellphones. However, we seldom
ask ourselves what these prices we read are or should be. Do they correspond
to the prices of the last recorded trades? Are they some form of mid-prices?
From which exchange(s) or venue(s)? In fact, the very notion of real-time
prices raises a lot of questions.

For liquid securities traded in limit order books (LOBs), a wide variety of
concepts of real-time prices have been proposed under different names such as
mid-price, efficient price, fair price, micro-price, etc. Of course, with
different desired or undesired properties.

The first notion that naturally arises in the case of LOBs is that of the mid-
price $\frac{S^{b}+S^{a}}{2}$, where $S^{b}$ is the best bid price and $S^{a}$
the best offer or ask price. This notion is simple but suffers from several
limitations. If we think that a good notion of price should be the proceed of
a nowcasting procedure, the above notion of mid-price does not use all the
information available in the LOB, in particular the volumes available. Also,
it evolves in a discontinuous manner and may suffer from a form of bid-ask
bounce when limits are depleted by trades (though a less severe form of bid-
ask bounce than in the case of last trade prices). Moreover, if an asset can
be traded on several venues, the mid-price ceases to be defined in an
unambiguous manner: it could be defined, for instance, as the mid-price on the
main venue or as the average between the best bid prices across venues and the
best ask prices across venues. Questions also arise when prices are not
reliable because orders are not firm due to last look practices (a typical
feature in foreign echange markets, see [29]). In spite of these problems,
mid-prices are widely used and they are good enough for a large bunch of
applications.

The most famous extension of mid-price is that of the weighted mid-price (also
called imbalance-based mid-price) defined as
$\frac{V^{a}}{V^{b}+V^{a}}S^{b}+\frac{V^{b}}{V^{b}+V^{a}}S^{a}$ where $V^{b}$
and $V^{a}$ are the volumes available in the LOB at the best bid and best ask
prices respectively. This weighted mid-price is related to the saying “the
price is where the volume is not” (see [14]) that has inspired a lot of the
approaches discussed below. Although it suffers from numerous flaws
(discontinuity, counterintuitive sensitivity to price improvement in some
cases, excessive noise, etc.) this weighted mid-price is widely used. It is
indeed attractive since the imbalance between the volumes posted at the best
bid and at the best ask is known to be a good predictor of the price of the
next trade or of the next (mid-)price move. One can cite [19] for an empirical
study, [12] for a simple expression of the probability of an upward move
conditional on these volumes in a simple Markovian model for the dynamics of a
limit order book, and [9] for an example of use of volume imbalance in trading
strategies, to only cite a few.

Measures of imbalance based on $V^{b}$ and $V^{a}$ just have to be monotone in
$V^{b}/V^{a}$ and they can therefore take a variety of forms. In an attempt to
generalize the price formation model of [27] to large-tick assets, Bonart and
Lillo proposed in [8] an extension of the above weighted mid-price in which
they replaced volumes at the best limits by their squares, i.e.
$\frac{{V^{a}}^{2}}{{V^{b}}^{2}+{V^{a}}^{2}}S^{b}+\frac{{V^{b}}^{2}}{{V^{b}}^{2}+{V^{a}}^{2}}S^{a}$.333To
account for make-take fees on some platforms, they also propose to replace bid
and ask prices in the formulas by rebate-adjusted prices. They argue, based on
theoretical and empirical grounds that the quadratic version is preferable to
the linear one, especially for assets with bid-ask spreads (almost always)
equal to one tick – so-called large-tick assets.

Many other notions of mid-price can be proposed along the above lines. One can
indeed easily extend the above definitions beyond top-of-book prices and
volumes, or consider several venues. Another commonly seen method consists in
regressing signed cumulated volumes in the LOB on prices and to define an
extended mid-price as the intersection between the regression line and the
price axis. In all cases these notions are only heuristics and deserve a
micro-foundation.

In the specific case of large-tick assets, different notions have also emerged
in the academic literature. Delattre et al. introduced in [14] an interesting
approach in which they assume that there exists an unobservable “efficient”
price and deduce the location of that price through the order flow at the bid.
More precisely, they consider limit orders sent with a probability that is a
monotone function of the distance between that unobserved efficient price and
the (observed) bid. Using historical data, they estimate in a nonparametric
way that function and then deduce from the current (in fact recent) order flow
an estimation of the efficient price. Two-sided extensions of this approach
where one uses both the bid and the ask sides could be imagined and would
share a lot with the modelling approach of the trading flow typically used in
market making models. Robert and Rosenbaum proposed in [30] another route to
estimate an efficient price for large-tick assets that does not rely on the
volumes at the best limits in the LOB but rather on transaction prices only.
The main idea underlying the approach is that if a transaction occurs and
changes one of the best limits in the LOB, then the efficient price must be
close enough to the transaction price. Their paper is one of the applications
of the concept of uncertainty zones that has also been used for the optimal
choice of tick sizes (see [13] and [2] for a recent paper).

In an attempt to provide a general framework for defining notions of real-time
prices, Stoikov proposed in [32] the concept of micro-price.444See [33] for a
recent multi-asset extension. This micro-price is defined as the long-term
expectation of the (classical) mid-price conditional on all the information
currently available. In other words, it relies on a long-term limit to
eliminate microstructural noise.555A vast literature exists regarding the
filtering of microstructural noise. However, the aim of that literature is
more that of estimating volatility at the high-frequency level rather than
effectively constructing a denoised price. Close ideas were present in the
paper [26] by Lehalle and Mounjid who, however, restricted the conditioning to
the value of the current mid-price and imbalance.666In fact, the idea could be
traced back to [25]. The general framework proposed by Stoikov leads to
various notions of prices depending on the assumptions made regarding the
random variables at stake. In particular, the notions of mid-price and
weighted mid-price are outcomes of the approach for simple models of the LOB
dynamics. An important advantage of this approach, beyond its versatility, is
that the micro-price is always, by definition, a martingale.

Many concepts have been introduced in the case of markets organized around
LOBs and these concepts are commonly used by practitioners in the equity
world. The case of over-the-counter (OTC) markets, however, has always
attracted less research. In fact, several questions arise naturally when it
comes to them, especially regarding the available information.

On some markets, post-trade transparency is enforced and both dealers and
clients777In this paper, we use the word client to designate a liquidity-
taker, i.e. any market participant who is not a dealer (as in the expression
“dealer-to-client segment”). It is of course not the client of a specific
dealer, although we are going to use a dataset of RFQs sent to a specific
dealer. can have access – at least theoretically – to a consolidated tape of
transactions. This is the case in the US corporate bond markets with TRACE
data (see [15, 16] for relevant statistical methods to exploit TRACE data) but
the situation is different in the European market in spite of recent efforts.
The problem is in fact the fragmented nature of information and, as in all OTC
markets, the lag in reporting.

Beyond transaction prices and volumes, clients usually have access to the
prices streamed by dealers on electronic platforms. In the case of the
European market for corporate bonds, which is the market of interest for this
paper, the main (multi-dealer-to-client) platforms are those of Bloomberg,
MarketAxess and Tradeweb. However, streamed prices are only indicative and for
one given size.

As far as dealers are concerned, the information available to them depends on
the market. In the case of corporate bonds, dealers do not have access to the
prices streamed by competitors but they have access to composite prices
provided by multi-dealer-to-client electronic platforms (CBBT for Bloomberg,
CP+ for MarketAxess, etc.) or can make their own composite from multiple
sources. These prices have a lot of drawbacks but they often constitute a
first useful estimate. Beyond indicative prices, dealers have in fact access
to a lot of information through their customer flows. In the case of corporate
bond markets, requests for quotes (RFQs) probably constitute for a market
maker with a decent market share the main source of information beyond
composite prices. The information content of client flows is indeed very
important: (i) the side / sign of RFQs (i.e. the willingness to buy or to
sell) clearly indicates the sentiment of clients on each bond or more
generally on the bonds with similar characteristics (sector of the issuer,
maturity, etc.), and (ii) client decisions to trade or not to trade at the
price answered by the dealer inform about the demand curve of clients and
therefore about the current (unobservable) price or its distribution.888One
limitation is that some requests are sent without the intention to trade (for
instance to value a portfolio). However, thanks to multi-dealer-to-client
platforms, dealers know whether the requests they answered led to a
transaction with a competitor.

The use of RFQ data to estimate a real-time price in corporate bond markets is
not new in the literature. A multivariate approach based on particle filtering
has been proposed in [21] that exploited information from a proprietary
database of RFQs sent to a dealer and trades in the dealer-to-dealer segment
of the market. This particle filtering approach is interesting in that it is
Bayesian and provides therefore a distribution for real-time prices.

In this paper, we propose two new ideas that both rely on a new approach to
model the flow of RFQs and its complex dynamics. In classical market making
models, RFQs are modelled by Poisson processes: requests arrive randomly and
the probability of occurrence of an RFQ is the same at all time – we call
intensity the infinitesimal probability of an RFQ occurrence per unit of time.
To model varying liquidity, we assume in this paper that RFQs arrive randomly
with an intensity that is itself a stochastic process: a simple continuous-
time Markov chain with only a few states. In technical terms, we model the
flow of RFQs at the bid and at the ask by a bidimensional Markov-modulated
Poisson process (MMPP).999See [17] for an overview about MMPPs and their
historical applications in telecommunication.

Our first idea consists in defining a micro-price à la Stoikov using the
information contained in the flow imbalance. More precisely, we assume that
the price process drifts proportionally to the difference between the
intensity at the ask and the intensity at the bid. When intensities at the bid
and at the ask are the same, the micro-price is nothing but the current price.
However, imbalance leads to a micro-price above or below the current price
depending on the side of the imbalance. The exact value of the micro-price
depends of course on the proportionality factor and on the joint dynamics of
intensities.

Our second idea is inspired by the recent literature on OTC market making (see
the reference books [10, 22] for an overview of the recent market making
literature). When two agents want to agree on a price, they can resort to a
neutral third party. However, if the seller requests a price to a market
maker, then they will get the bid price quoted by that market maker. If,
instead, the buyer requests a price to a market maker, then they will get the
ask price quoted by that market maker. If we assume that this third party is
aware of the flow imbalances in the market, it is therefore natural to regard
the average between these two prices as a fair price, all the more when the
market maker has zero inventory.

In market making models à la Avellaneda-Stoikov [1] (see also [6, 7, 22, 23]
for a presentation which is more consistent with OTC markets), trading flows
depend on the distance of the dealer’s quotes to an exogenous reference price.
If trading flows (or intensities in mathematical models) at the bid and at the
ask are the same,101010We assume here implicitly that inventory costs are also
the same. then the optimal bid and ask prices of a market maker with no
inventory will be symmetric around the reference price, which is therefore a
fair transfer price. What happens, however, when a market maker is aware of
asymmetries in the trading flows is that they skew their quotes even in the
absence of inventory. As a consequence, the average between the optimal bid
and ask quotes ceases to coincide with the reference price. However, it is a
fair transfer price given the current context in terms of liquidity. We
propose therefore an extension of existing market making models to incorporate
MMPPs and obtain a new model in which the average between the bid and ask
quotes (in the absence of inventory) defines a fair transfer price that can be
used to value or transfer securities even when the market is illiquid and/or
tends to be one-sided.

In Section 2 we introduce the modelling framework for the flow of RFQs and
present the statistical estimation of the model parameters. We then deal with
two important extensions that are used throughout the paper. The first one is
linked to an exchangeability assumption between the intensities at the bid and
at the ask. Our assumption means that there is no structural asymmetry between
the bid and the ask. Liquidity can of course be asymmetric from time to time
with a higher intensity on one side but this is only transitory and the same
could have happened on the other side with the same probability. The second
extension allows to go multi-bond and fit the model on a set of bonds rather
than on single instruments. In Section 3, we present a notion of micro-price
inspired by that of Stoikov, but rooted into our model for the flow of RFQs,
and introduce our notion of Fair Transfer Price. Section 4 discusses numerical
methods, presents many numerical examples and discusses them.

##  2 A modelling framework for the flow of RFQs

###  2.1 Introduction and notation

Corporate bond markets, like many markets, see liquidity cycles. The number of
RFQs received by a dealer can be high or low. It can also be high on one side
and low on the other, hence the very important role of dealers who hold
inventory and bridge the gap between different phases.

To model the dynamics of liquidity, the basic idea is to regard the number of
RFQs received by a dealer on a given bond at the bid and at the ask as two
point processes. Of course, Poisson processes are not sufficient: the
intensities $\left(\lambda^{b}_{t}\right)_{t}$ (for the bid) and
$\left(\lambda^{a}_{t}\right)_{t}$ (for the ask) must be stochastic processes.
In quantitative finance, the most commonly used extensions of Poisson
processes are Hawkes processes. Hawkes processes are indeed very good at
modelling events that may happen in clusters. However, they are self-excited
processes and, in a market with limited post-trade transparency, we argue that
it is odd to assume that an RFQ sent by a client is the consequence of an RFQ
sent by another client. Instead of using Hawkes processes, we assume that
intensities are continuous-time Markov chains with values in a finite set and
use the concept of Markov-modulated Poisson process. Because liquidity shocks
can sometimes be symmetric and sometimes be asymmetric, we consider more
precisely a bidimensional MMPP: the intensity111111Throughout the paper, we
call this process an intensity process in spite of it being bidimensional.
process $(\lambda_{t})_{t}=(\lambda^{b}_{t},\lambda^{a}_{t})_{t}$ is a
continuous-time Markov chain taking values in
$\\{\lambda^{1,b},\ldots,\lambda^{m_{b},b}\\}\times\\{\lambda^{1,a},\ldots,\lambda^{m_{a},a}\\}$
with transition (or rate) matrix $Q\in M_{m_{b}m_{a}}$.121212In what follows,
we shall order the states in lexicographic order, i.e.
$(\lambda^{1,b},\lambda^{1,a}),\ldots,(\lambda^{1,b},\lambda^{m_{a},a}),\ldots,(\lambda^{m_{b},b},\lambda^{1,a}),\ldots,(\lambda^{m_{b},b},\lambda^{m_{a},a}).$
The case in which the two intensity processes are considered in an independent
manner is a specific one and corresponds, for the chosen order, to
$Q=Q^{b}\otimes I_{m_{a}}+I_{m_{b}}\otimes Q^{a}$ where $Q^{b}$ and $Q^{a}$
are the transition matrices associated with $\left(\lambda^{b}_{t}\right)_{t}$
and $\left(\lambda^{a}_{t}\right)_{t}$ respectively and $\otimes$ denotes
tensor (or Kronecker) product.

In what follows, we focus on the estimation of the intensities
$\lambda^{1,b},\ldots,\lambda^{m_{b},b}$ and
$\lambda^{1,a},\ldots,\lambda^{m_{a},a}$ and the coefficients of $Q$. The
method we propose is inspired by the EM algorithm proposed in [31] but
generalized to the case of a bidimensional MMPP.

###  2.2 Estimation of the parameters

####  2.2.1 Likelihood of a sample

Our goal in the next paragraphs is to compute the likelihood of a sequence of
RFQ times $t_{1}<\ldots<t_{N}$ with sides
$\mathfrak{s}_{1},\ldots,\mathfrak{s}_{N}$, where the sides are encoded as
elements of $\\{b,a\\}$ for bid and ask.

Let us denote by $(N^{\text{RFQ},b}_{t})_{t}$ and $(N^{\text{RFQ},a}_{t})_{t}$
the processes counting the number of RFQs at the bid and at the ask
respectively, and let us consider the function

$\mathcal{G}:t\mapsto(\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t))_{1\leq
j_{b},k_{b}\leq m_{b},1\leq j_{a},k_{a}\leq m_{a}}$

where

$\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)=\mathbb{P}(N^{RFQ,b}_{t}=0,N^{RFQ,a}_{t}=0,\lambda_{t}=(\lambda^{k_{b},b},\lambda^{k_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a})).$

We have for $h>0$, $1\leq j_{b},k_{b}\leq m_{b}$ and $1\leq j_{a},k_{a}\leq
m_{a}$:

$\displaystyle\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t+h)$
$\displaystyle=$
$\displaystyle\mathbb{P}(N^{RFQ,b}_{t+h}=0,N^{RFQ,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a}))$
$\displaystyle=$
$\displaystyle\sum_{l_{b}=1}^{m_{b}}\sum_{l_{a}=1}^{m_{a}}\mathbb{P}(N^{RFQ,b}_{t+h}=0,N^{RFQ,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a}),\lambda_{t}=(\lambda^{l_{b},b},\lambda^{l_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a}))$
$\displaystyle=$
$\displaystyle\sum_{l_{b}=1}^{m_{b}}\sum_{l_{a}=1}^{m_{a}}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)\mathbb{P}\left(N^{RFQ,b}_{t+h}=0,N^{RFQ,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.|N^{RFQ,b}_{t}=0,N^{RFQ,a}_{t}=0,\lambda_{t}=(\lambda^{l_{b},b},\lambda^{l_{a},a})\right)$
$\displaystyle=$
$\displaystyle\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)\left(1+Q_{(k_{b}-1)m_{a}+k_{a},(k_{b}-1)m_{a}+k_{a}}h+o(h)\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left(1-\lambda^{k_{b},b}h+o(h)\right)\left(1-\lambda^{k_{a},a}h+o(h)\right)$
$\displaystyle+\sum_{1\leq l_{b}\leq m_{b},1\leq l_{a}\leq
m_{a},(l_{b},l_{a})\neq(k_{b},k_{a})}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)\left(Q_{(l_{b}-1)m_{a}+l_{a},(k_{b}-1)m_{a}+k_{a}}h+o(h)\right).$

This leads to the following differential equation:

$\frac{d\
}{dt}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)=\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)\left(Q_{(k_{b}-1)m_{a}+k_{a},(k_{b}-1)m_{a}+k_{a}}-\lambda^{k_{b},b}-\lambda^{k_{a},a}\right)$
$+\sum_{1\leq l_{b}\leq m_{b},1\leq l_{a}\leq
m_{a},(l_{b},l_{a})\neq(k_{b},k_{a})}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)Q_{(l_{b}-1)m_{a}+l_{a},(k_{b}-1)m_{a}+k_{a}}$

which, in matrix form, writes

$\mathcal{G}^{\prime}(t)=\mathcal{G}(t)\left(Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a}\right)$

where $\Lambda^{b}=\text{diag}(\lambda^{1,b},\ldots,\lambda^{m_{b},b})$,
$\Lambda^{a}=\text{diag}(\lambda^{1,a},\ldots,\lambda^{m_{a},a})$,
$\tilde{\Lambda}^{b}=\Lambda^{b}\otimes I_{m_{a}}$ and
$\tilde{\Lambda}^{a}=I_{m_{b}}\otimes\Lambda^{a}$.

As $\mathcal{G}(0)$ is the identity matrix $I_{m_{b}m_{a}}$, we conclude that

$\mathcal{G}(t)=\exp\left(\left(Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a}\right)t\right).$

By Markov property, for $s\geq 0$, if we assume that $\lambda_{s}$ is
distributed according to a distribution represented by a column vector
$\pi_{s}$ (in $\mathbb{R}^{m_{b}m_{a}}$), then for $1\leq j_{b}\leq
m_{b},1\leq j_{a}\leq m_{a}$,
$\pi_{s}^{\prime}\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})t)e^{(j_{b}-1)m_{b}+j_{a}}$
is the probability that there was no RFQ between time $s$ and time $s+t$ and
the intensity process is equal to $(\lambda^{j_{b},b},\lambda^{j_{a},a})$ at
time $s+t$.131313$(e^{1},\ldots,e^{m_{b}m_{a}})$ is the canonical basis of
$\mathbb{R}^{m_{b}m_{a}}$.

If we assume that $\lambda_{0}$ is distributed according to a distribution
represented by a column vector $\pi_{0}$, then the likelihood of the whole
sample writes

$\displaystyle\mathcal{L}(Q,\Lambda^{b},\Lambda^{a}|t_{1},\ldots,t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N})$
$\displaystyle=$
$\displaystyle\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp\left(\left(Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a}\right)(t_{n}-t_{n-1})\right)\tilde{\Lambda}^{\mathfrak{s}_{n}}\right)e$

where $t_{0}=0$ and
$e=\sum_{j_{b}=1}^{m_{b}}\sum_{j_{a}=1}^{m_{a}}e^{(j_{b}-1)m_{a}+j_{a}}=(1,\ldots,1)^{\prime}$.

Maximizing the above likelihood expression is not straightforward. Instead, we
propose in the next paragraph an EM algorithm in which the hidden variable is
the trajectory of the unobservable intensity process.

####  2.2.2 An EM algorithm

Let us consider as hidden variables a sequence of times
$0=\tau_{0}<\ldots<\tau_{P}(\leq t_{N})$ corresponding to transitions of the
process $(\lambda_{t})_{t}$ and a sequence of couples
$(s^{b}_{0},s^{a}_{0}),\ldots,(s^{b}_{P},s^{a}_{P})$ in
$\\{1,\ldots,m_{b}\\}\times\\{1,\ldots,m_{a}\\}$ such that
$(\lambda^{b}_{t},\lambda^{a}_{t})=(\lambda^{s^{b}_{p},b},\lambda^{s^{a}_{p},a})$
over $(\tau_{p},\tau_{p+1}]$ (where, by convention $\tau_{P+1}=t_{N}$).

The likelihood of $t_{1}<\ldots<t_{N}$,
$\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}$, $\tau_{1}<\ldots<\tau_{P}$,
$s^{b}_{0},\ldots,s^{b}_{P}$ and $s^{a}_{0},\ldots,s^{a}_{P}$ is

$\displaystyle\mathcal{L}(Q,\Lambda^{b},\Lambda^{a}|t_{1},\ldots,t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N},\tau_{1},\ldots,\tau_{P+1},s^{b}_{0},\ldots,s^{b}_{P},s^{a}_{0},\ldots,s^{a}_{P})$
$\displaystyle=$
$\displaystyle(\pi_{0})_{(s^{b}_{0}-1)m_{a}+s^{a}_{0}}\left(\prod_{p=0}^{P-1}Q_{(s^{b}_{p}-1)m_{a}+s^{a}_{p},(s^{b}_{p+1}-1)m_{a}+s^{a}_{p+1}}\exp\left({Q_{(s^{b}_{p}-1)m_{a}+s^{a}_{p},(s^{b}_{p}-1)m_{a}+s^{a}_{p}}(\tau_{p+1}-\tau_{p})}\right)\right)$
$\displaystyle\times\exp\left({Q_{(s^{b}_{p}-1)m_{a}+s^{a}_{p},(s^{b}_{p}-1)m_{a}+s^{a}_{p}}(\tau_{P+1}-\tau_{P})}\right)\left(\prod_{p=0}^{P}\left(\lambda^{s^{b}_{p},b}\right)^{c^{b}_{p}}\exp\left({-\lambda^{s^{b}_{p},b}(\tau_{p+1}-\tau_{p})}\right)\right)$
$\displaystyle\times\left(\prod_{p=0}^{P}\left(\lambda^{s^{a}_{p},a}\right)^{c^{a}_{p}}\exp\left({-\lambda^{s^{a}_{p},a}(\tau_{p+1}-\tau_{p})}\right)\right)$

where
$c^{b}_{p}=\text{Card}(\\{n|t_{n}\in(\tau_{p},\tau_{p+1}],\mathfrak{s}_{n}=b\\})$
and
$c^{a}_{p}=\text{Card}(\\{n|t_{n}\in(\tau_{p},\tau_{p+1}],\mathfrak{s}_{n}=a\\})$.

The associated log-likelihood writes

$\displaystyle\log\left((\pi_{0})_{(s^{b}_{0}-1)m_{a}+s^{a}_{0}}\right)+\sum_{p=0}^{P-1}\log\left(Q_{(s^{b}_{p}-1)m_{a}+s^{a}_{p},(s^{b}_{p+1}-1)m_{a}+s^{a}_{p+1}}\right)$
(1)
$\displaystyle+\sum_{p=0}^{P}\left(Q_{(s^{b}_{p}-1)m_{a}+s^{a}_{p},(s^{b}_{p}-1)m_{a}+s^{a}_{p}}-\lambda^{s^{b}_{p},b}-\lambda^{s^{a}_{p},a}\right)(\tau_{p+1}-\tau_{p})+\sum_{p=0}^{P}c^{b}_{p}\log(\lambda^{s^{b}_{p},b})+\sum_{p=0}^{P}c^{a}_{p}\log(\lambda^{s^{a}_{p},b})$
$\displaystyle=$
$\displaystyle\log\left((\pi_{0})_{(s^{b}_{0}-1)m_{a}+s^{a}_{0}}\right)+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m_{b}\\\ 1\leq j_{a}\leq
m_{a}\end{subarray}}\sum_{\begin{subarray}{c}1\leq k_{b}\leq m_{b}\\\ 1\leq
k_{a}\leq m_{a}\\\
(j_{b},j_{a})\neq(k_{b},k_{a})\end{subarray}}\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\log(Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}})$
$\displaystyle+\sum_{\begin{subarray}{c}1\leq j_{b}\leq m_{b}\\\ 1\leq
j_{a}\leq
m_{a}\end{subarray}}\tilde{T}^{(j_{b},j_{a})}\left(Q_{(j_{b}-1)m_{a}+j_{a},(j_{b}-1)m_{a}+j_{a}}-\lambda^{j_{b},b}-\lambda^{j_{a},a}\right)$
$\displaystyle+\sum_{\begin{subarray}{c}1\leq j_{b}\leq m_{b}\\\ 1\leq
j_{a}\leq
m_{a}\end{subarray}}\tilde{n}^{b}_{(j_{b},j_{a})}\log(\lambda^{j_{b},b})+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m_{b}\\\ 1\leq j_{a}\leq
m_{a}\end{subarray}}\tilde{n}^{a}_{(j_{b},j_{a})}\log(\lambda^{j_{a},a})$
$\displaystyle=$
$\displaystyle\log\left((\pi_{0})_{(s^{b}_{0}-1)m_{a}+s^{a}_{0}}\right)+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m_{b}\\\ 1\leq j_{a}\leq
m_{a}\end{subarray}}\sum_{\begin{subarray}{c}1\leq k_{b}\leq m_{b}\\\ 1\leq
k_{a}\leq m_{a}\\\
(k_{b},k_{a})\neq(j_{b},j_{a})\end{subarray}}\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\log(Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}})$
$\displaystyle-\sum_{\begin{subarray}{c}1\leq j_{b}\leq m_{b}\\\ 1\leq
j_{a}\leq m_{a}\end{subarray}}\left(\left(\sum_{\begin{subarray}{c}1\leq
k_{b}\leq m_{b}\\\ 1\leq k_{a}\leq m_{a}\\\
(k_{b},k_{a})\neq(j_{b},j_{a})\end{subarray}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\right)+\lambda^{j_{b},b}+\lambda^{j_{a},a}\right)\tilde{T}^{(j_{b},j_{a})}$
$\displaystyle+\sum_{\begin{subarray}{c}1\leq j_{b}\leq m_{b}\\\ 1\leq
j_{a}\leq
m_{a}\end{subarray}}\tilde{n}^{b}_{(j_{b},j_{a})}\log(\lambda^{j_{b},b})+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m_{b}\\\ 1\leq j_{a}\leq
m_{a}\end{subarray}}\tilde{n}^{a}_{(j_{b},j_{a})}\log(\lambda^{j_{a},a})$

where:

  * •

for $1\leq j_{b},k_{b}\leq m_{b}$ and $1\leq j_{a},k_{a}\leq m_{a}$ with
$(j_{b},j_{a})\neq(k_{b},k_{a})$, $\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}$ is
the number of transitions of the intensity process $(\lambda_{t})_{t}$ from
$(\lambda^{j_{b},b},\lambda^{j_{a},a})$ to
$(\lambda^{k_{b},b},\lambda^{k_{a},a})$ over the time interval $[0,t_{N}]$,

  * •

for $1\leq j_{b}\leq m_{b}$ and $1\leq j_{a}\leq m_{a}$,
$\tilde{T}^{(j_{b},j_{a})}$ is the total time spent by the intensity process
$(\lambda_{t})_{t}$ in $(\lambda^{j_{b},b},\lambda^{j_{a},a})$ over the time
interval $[0,t_{N}]$,

  * •

for $1\leq j_{b}\leq m_{b}$ and $1\leq j_{a}\leq m_{a}$,
$\tilde{n}_{(j_{b},j_{a})}^{b}$ and $\tilde{n}_{(j_{b},j_{a})}^{a}$ are the
number of RFQs at the bid and at the ask respectively over the time interval
$[0,t_{N}]$ while the intensity process $(\lambda_{t})_{t}$ is in
$(\lambda^{j_{b},b},\lambda^{j_{a},a})$.

The EM algorithm consists in iteratively computing the expectation of the log-
likelihood expression (1) conditionally on the real observables
$t_{1}<\ldots<t_{N}$ and $\mathfrak{s}_{1},\ldots,\mathfrak{s}_{N}$ under the
assumption that the unobservable variables are distributed according to the
model with given values $\widehat{\Lambda^{b}}$, $\widehat{\Lambda^{a}}$ and
$\widehat{Q}$ of $\Lambda^{b}$, $\Lambda^{a}$ and $Q$, and, then, carrying out
a maximization of the resulting expression over the diagonal coefficients of
$\Lambda^{b}$ and $\Lambda^{a}$ and the non-diagonal coefficients of $Q$ to
update the values of $\widehat{\Lambda^{b}}$, $\widehat{\Lambda^{a}}$ and
$\widehat{Q}$.

Ignoring the first term which contributes almost nothing, we easily see that
the EM algorithm boils down to the following updates:

$\widehat{\Lambda^{b}}_{j_{b},j_{b}}\leftarrow\frac{\sum_{j_{a}=1}^{m_{a}}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(j_{b},j_{a})}^{b}\right]}{\sum_{j_{a}=1}^{m_{a}}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]}\quad\text{
for }1\leq j_{b}\leq m_{b},$
$\widehat{\Lambda^{a}}_{j_{a},j_{a}}\leftarrow\frac{\sum_{j_{b}=1}^{m_{b}}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(j_{b},j_{a})}^{a}\right]}{\sum_{j_{b}=1}^{m_{b}}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]}\quad\text{
for }1\leq j_{a}\leq m_{a},$

and, for $1\leq j_{b},k_{b}\leq m_{b}$ and $1\leq j_{a},k_{a}\leq m_{a}$ with
$(j_{b},j_{a})\neq(k_{b},k_{a})$,

$\widehat{Q}_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\leftarrow\frac{\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\right]}{\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]}.$

Assuming that the initial intensity is distributed according to a distribution
represented by a column vector $\pi_{0}$, we get that the conditional
expectation of the number of RFQs at the bid while the intensity process is
equal to $(\lambda^{j_{b},b},\lambda^{j_{a},a})$ is141414We write
$\widetilde{\widehat{\Lambda^{b}}}=\widehat{\Lambda^{b}}\otimes I_{m_{a}}$ and
$\widetilde{\widehat{\Lambda^{a}}}=I_{m_{b}}\otimes\widehat{\Lambda^{a}}$.

$\displaystyle\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(j_{b},j_{a})}^{b}\right]$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{N}1_{\mathfrak{s}_{r}=b}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[1_{\lambda^{b}_{t_{r}}=\lambda^{j_{b},b}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\sum_{r=1}^{N}1_{\mathfrak{s}_{r}=b}\pi_{0}^{\prime}\left(\prod_{n=1}^{r}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e^{(j_{b}-1)m_{a}+j_{a}}$
$\displaystyle\quad\times{e^{(j_{b}-1)m_{a}+j_{a}}}^{\prime}\left(\prod_{n=r+1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e.$

Similarly, we have

$\displaystyle\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(j_{b},j_{a})}^{a}\right]$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{N}1_{\mathfrak{s}_{r}=a}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[1_{\lambda^{b}_{t_{r}}=\lambda^{j_{b},b}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\sum_{r=1}^{N}1_{\mathfrak{s}_{r}=a}\pi_{0}^{\prime}\left(\prod_{n=1}^{r}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e^{(j_{b}-1)m_{a}+j_{a}}$
$\displaystyle\quad\times{e^{(j_{b}-1)m_{a}+j_{a}}}^{\prime}\left(\prod_{n=r+1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e.$

Regarding the time spent in $(\lambda^{j_{b},b},\lambda^{j_{a},a})$, we get

$\displaystyle\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{t_{N}}\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[1_{(\lambda^{b}_{t},\lambda^{a}_{t})=(\lambda^{j_{b},b},\lambda^{j_{a},a})}\right]dt$
$\displaystyle=$
$\displaystyle\frac{1}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\int_{0}^{t_{N}}\pi_{0}^{\prime}\left(\prod_{n=1}^{n(t)}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)$
$\displaystyle\quad\quad\quad\times\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t-t_{n(t)}))e^{(j_{b}-1)m_{a}+j_{a}}{e^{(j_{b}-1)m_{a}+j_{a}}}^{\prime}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n(t)+1}-t))$
$\displaystyle\quad\quad\quad\times\left(\prod_{n=n(t)+2}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e\
dt$

where $n(t)=\max\\{n,t_{n}<t\\}$.

This also writes

$\displaystyle\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\sum_{r=1}^{N}\left(\pi_{0}^{\prime}\left(\prod_{n=1}^{r-1}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)\right.$
$\displaystyle\quad\quad\quad\times\int_{t_{r-1}}^{t_{r}}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t-t_{r-1}))e^{(j_{b}-1)m_{a}+j_{a}}{e^{(j_{b}-1)m_{a}+j_{a}}}^{\prime}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{r}-t))dt$
$\displaystyle\quad\quad\quad\times\left.\left(\prod_{n=r+1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e\right).$

Using a similar reasoning, we have

$\displaystyle\mathbb{E}_{\widehat{\Lambda^{b}},\widehat{\Lambda^{a}},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\right]$
$\displaystyle=$
$\displaystyle\frac{\widehat{Q}_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\int_{0}^{t_{N}}\pi_{0}^{\prime}\left(\prod_{n=1}^{n(t)}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)$
$\displaystyle\quad\quad\quad\times\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t-t_{n(t)}))e^{(j_{b}-1)m_{a}+j_{a}}{e^{(k_{b}-1)m_{a}+k_{a}}}^{\prime}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n(t)+1}-t))$
$\displaystyle\quad\quad\quad\times\left(\prod_{n=n(t)+2}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e\
dt$ $\displaystyle=$
$\displaystyle\frac{\widehat{Q}_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}}{\pi_{0}^{\prime}\left(\prod_{n=1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e}$
$\displaystyle\quad\times\sum_{r=1}^{N}\left(\pi_{0}^{\prime}\left(\prod_{n=1}^{r-1}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)\right.$
$\displaystyle\quad\quad\quad\times\int_{t_{r-1}}^{t_{r}}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t-t_{r-1}))e^{(j_{b}-1)m_{a}+j_{a}}{e^{(k_{b}-1)m_{a}+k_{a}}}^{\prime}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{r}-t))dt$
$\displaystyle\quad\quad\quad\times\left.\left(\prod_{n=r+1}^{N}\exp((\widehat{Q}-\widetilde{\widehat{\Lambda^{b}}}-\widetilde{\widehat{\Lambda^{a}}})(t_{n}-t_{n-1}))\widetilde{\widehat{\Lambda^{\mathfrak{s}_{n}}}}\right)e\right).$

These quantities can be computed iteratively and it is noteworthy that we do
not need to compute the denominators as they cancel out when we update
$\widehat{\Lambda^{b}}$, $\widehat{\Lambda^{a}}$ and $\widehat{Q}$. It must
also be noted that the only computational difficulty lies in finding a scaling
factor to avoid ending up with very low or very high values.

####  2.2.3 Estimating the current state

Once an estimation of the parameters has been carried out, it is possible to
estimate the state of the intensity processes at any point in time $t$. If
indeed we consider a prior probability distribution for the initial value
$\lambda_{0}=(\lambda^{b}_{0},\lambda^{a}_{0})$ of intensity process
represented by a column vector $\pi_{0}$, then, given a sequence of observed
RFQs times $0=t_{0}<t_{1}<\ldots<t_{n}(\leq t)$ prior to time $t$ along with
their associated sides $\mathfrak{s}_{1},\ldots\mathfrak{s}_{n}$, the a
posteriori distribution $\pi_{t}$ of $(\lambda^{b}_{t},\lambda^{a}_{t})$
writes

$(\pi_{t})_{(j_{b}-1)m_{a}+j_{a}}\propto\pi_{0}^{\prime}\left(\prod_{r=1}^{n}\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t_{r}-t_{r-1}))\tilde{\Lambda}^{\mathfrak{s}_{r}}\right)\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t-t_{n}))e^{(j_{b}-1)m_{a}+j_{a}}$

i.e.

$\pi_{t}=\frac{\pi_{0}^{\prime}\left(\prod_{r=1}^{n}\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t_{r}-t_{r-1}))\tilde{\Lambda}^{\mathfrak{s}_{r}}\right)\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t-t_{n}))e^{(j_{b}-1)m_{a}+j_{a}}}{\pi_{0}^{\prime}\left(\prod_{r=1}^{n}\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t_{r}-t_{r-1}))\tilde{\Lambda}^{\mathfrak{s}_{r}}\right)\exp((Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a})(t-t_{n}))e}.$
(2)

###  2.3 Two important extensions

####  2.3.1 An exchangeability assumption to impose symmetry in the
asymmetries

In the above estimation procedure, we considered two sets
$\\{\lambda^{1,b},\ldots,\lambda^{m_{b},b}\\}$ and
$\\{\lambda^{1,a},\ldots,\lambda^{m_{a},a}\\}$: one for the bid and one for
the ask. Even if one considers $m^{b}=m^{a}=m$, when one estimates intensities
on real data, there is no chance that the estimated parameters coincide
between the bid and the ask. However, if parameters are close and/or if there
is no reason to believe that there is a structural asymmetry between the bid
and the ask, it makes sense to impose that both sides share a unique set of
intensities $\\{\lambda^{1},\ldots,\lambda^{m}\\}$. In what follows, we shall
furthermore assume, when indicated, some form of symmetry in liquidity
asymmetries: there may be periods when liquidity is higher on one side than
the other but the exact opposite could have happened with the same
probability. In mathematical terms, this corresponds to a point-in-time
exchangeability assumption. In our Markovian setup, this means that the
transition matrix $Q$ of the Markov chain
$(\lambda^{b}_{t},\lambda^{a}_{t})_{t}$ is also that of the Markov chain
$(\lambda^{a}_{t},\lambda^{b}_{t})_{t}$. These assumptions are essential to
build a model where prices are driven by imbalances. In particular, they
guarantee that the price process does not drift indefinitely in such a model.

A natural question is of course that of estimating the intensities (or
equivalently the diagonal matrix
$\Lambda=\textrm{diag}(\lambda^{1},\ldots,\lambda^{m})$) and the transition
matrix $Q\in M_{m^{2}}$ using a set of RFQs at the bid and at the ask.

The likelihood computed in Section 2.2.1 is of course valid in the specific
case we consider here with $\Lambda^{b}=\Lambda^{a}=\Lambda$, but the EM
algorithm has to be adapted to take into account the constraints imposed by
the symmetry assumptions. The log-likelihood (1) now writes

$\displaystyle\mathcal{L}(Q,\Lambda|t_{1},\ldots,t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N},\tau_{1},\ldots,\tau_{P+1},s^{b}_{0},\ldots,s^{b}_{P},s^{a}_{0},\ldots,s^{a}_{P})$
$\displaystyle=$
$\displaystyle\log\left((\pi_{0})_{(s^{b}_{0}-1)m+s^{a}_{0}}\right)+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m\\\ 1\leq j_{a}\leq m\end{subarray}}\sum_{\begin{subarray}{c}1\leq
k_{b}\leq m\\\ 1\leq k_{a}\leq m\\\
(k_{b},k_{a})\neq(j_{b},j_{a})\end{subarray}}\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\log(Q_{(j_{b}-1)m+j_{a},(k_{b}-1)m+k_{a}})$
$\displaystyle-\sum_{\begin{subarray}{c}1\leq j_{b}\leq m\\\ 1\leq j_{a}\leq
m\end{subarray}}\left(\left(\sum_{\begin{subarray}{c}1\leq k_{b}\leq m\\\
1\leq k_{a}\leq m\\\
(k_{b},k_{a})\neq(j_{b},j_{a})\end{subarray}}Q_{(j_{b}-1)m+j_{a},(k_{b}-1)m+k_{a}}\right)+\lambda^{j_{b}}+\lambda^{j_{a}}\right)\tilde{T}^{(j_{b},j_{a})}$
$\displaystyle+\sum_{\begin{subarray}{c}1\leq j_{b}\leq m\\\ 1\leq j_{a}\leq
m\end{subarray}}\tilde{n}^{b}_{(j_{b},j_{a})}\log(\lambda^{j_{b}})+\sum_{\begin{subarray}{c}1\leq
j_{b}\leq m\\\ 1\leq j_{a}\leq
m\end{subarray}}\tilde{n}^{a}_{(j_{b},j_{a})}\log(\lambda^{j_{a}})$

and the matrix $Q$ verifies
$Q_{(j_{b}-1)m+j_{a},(k_{b}-1)m+k_{a}}=Q_{(j_{a}-1)m+j_{b},(k_{a}-1)m+k_{b}}$
for $1\leq j_{b},k_{b}\leq m$ and $1\leq j_{a},k_{a}\leq m$.

Subsequently, the $M$-step (i.e. the update) is modified and becomes

$\widehat{\Lambda}_{j,j}\leftarrow\frac{\sum_{k=1}^{m}\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(j,k)}^{b}\right]+\sum_{k=1}^{m}\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}_{(k,j)}^{a}\right]}{\sum_{k=1}^{m}\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j,k)}\right]+\sum_{k=1}^{m}\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(k,j)}\right]}\quad\text{
for }1\leq j\leq m,$

and, for $1\leq j_{b},k_{b}\leq m$ and $1\leq j_{a},k_{a}\leq m$ with
$(j_{b},j_{a})\neq(k_{b},k_{a})$,

$\widehat{Q}_{(j_{b}-1)m+j_{a},(k_{b}-1)m+k_{a}}\leftarrow\frac{\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}^{(j_{b},j_{a}),(k_{b},k_{a})}\right]\\!+\\!\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{n}^{(j_{a},j_{b}),(k_{a},k_{b})}\right]}{\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{b},j_{a})}\right]+\mathbb{E}_{\widehat{\Lambda},\widehat{Q},t_{1},\ldots
t_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N}}\left[\tilde{T}^{(j_{a},j_{b})}\right]}$

where the expectations are the same as in Section 2.2.2 with
$\widehat{\Lambda^{b}}=\widehat{\Lambda^{a}}=\widehat{\Lambda}$.

####  2.3.2 A multi-bond extension

What determines liquidity in corporate bond markets? Depending on the risk
aversion of market participants, the level of equity prices, those of yield
curves and those of credit spreads, credit risk can be attractive or
unattractive to investors. Some sectors or some ranges of maturity can also be
more attractive while others may be less. In all cases, it is reasonable to
believe that the interest of clients for bonds is shared across bonds having
similar characteristics.

In what follows, we consider a set of $d$ bonds and propose a one-factor
liquidity model that echoes in some sense the CAPM. More precisely, we
consider a Markov chain similar to the one used above, but we assume that the
intensity process of bond $i$ is given by
$(\lambda^{i}_{t})_{t}=(\lambda^{i,b}_{t},\lambda^{i,a}_{t})_{t}=(\beta^{i,b}\lambda^{b}_{t},\beta^{i,a}\lambda^{a}_{t})_{t}$.
In other words, $(\lambda_{t})_{t}=(\lambda^{b}_{t},\lambda^{a}_{t})_{t}$
represents an aggregate, while bond-level sensitivities to this aggregate are
represented by coefficients $(\beta^{i,b})_{i}$ and
$(\beta^{i,a})_{i}$.151515For identifiability reasons, we consider the
normalization  $\sum_{i=1}^{d}\beta^{i,b}=\sum_{i=1}^{d}\beta^{i,a}=1.$

To compute the likelihood of a sequence of RFQ times $t_{1}<\ldots<t_{N}$
corresponding to RFQs in bonds $i_{1},\ldots,i_{N}$ and sides
$\mathfrak{s}_{1},\ldots,\mathfrak{s}_{N}$ where the sides are encoded as
elements of $\\{b,a\\}$ as above, let us introduce two counting processes
$(N^{RFQ,i,b}_{t})_{t}$ and $(N^{RFQ,i,a}_{t})_{t}$ for each bond $i$, and the
function

$\mathcal{G}:t\mapsto(\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t))_{1\leq
j_{b},k_{b}\leq m_{b},1\leq j_{a},k_{a}\leq m_{a}}$

where

$\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)=\mathbb{P}(\forall
i,N^{RFQ,i,b}_{t}=0,N^{RFQ,i,a}_{t}=0,\lambda_{t}=(\lambda^{k_{b},b},\lambda^{k_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a}))$

Using the same reasoning as above, we obtain for $h>0$, $1\leq j_{b},k_{b}\leq
m_{b}$ and $1\leq j_{a},k_{a}\leq m_{a}$:

$\displaystyle\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t+h)$
$\displaystyle=$ $\displaystyle\mathbb{P}(\forall
i,N^{RFQ,i,b}_{t+h}=0,N^{RFQ,i,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a}))$
$\displaystyle=$
$\displaystyle\sum_{l_{b}=1}^{m_{b}}\sum_{l_{a}=1}^{m_{a}}\mathbb{P}(\forall
i,N^{RFQ,i,b}_{t+h}=0,N^{RFQ,i,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a}),\lambda_{t}=(\lambda^{l_{b},b},\lambda^{l_{a},a})|\lambda_{0}=(\lambda^{j_{b},b},\lambda^{j_{a},a}))$
$\displaystyle=$
$\displaystyle\sum_{l_{b}=1}^{m_{b}}\sum_{l_{a}=1}^{m_{a}}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)\mathbb{P}\left(\forall
i,N^{RFQ,i,b}_{t+h}=0,N^{RFQ,i,a}_{t+h}=0,\lambda_{t+h}=(\lambda^{k_{b},b},\lambda^{k_{a},a})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\Big{|}\forall
i,N^{RFQ,i,b}_{t}=0,N^{RFQ,i,a}_{t}=0,\lambda_{t}=(\lambda^{l_{b},b},\lambda^{l_{a},a})\right)$
$\displaystyle=$
$\displaystyle\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)\left(1+Q_{(k_{b}-1)m_{a}+k_{a},(k_{b}-1)m_{a}+k_{a}}h+o(h)\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\prod_{i=1}^{d}\left(1-\beta^{i,b}\lambda^{k_{b},b}h+o(h)\right)\left(1-\beta^{i,a}\lambda^{k_{a},a}h+o(h)\right)$
$\displaystyle+\sum_{1\leq l_{b}\leq m_{b},1\leq l_{a}\leq
m_{a},(l_{b},l_{a})\neq(k_{b},k_{a})}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)\left(Q_{(l_{b}-1)m_{a}+l_{a},(k_{b}-1)m_{a}+k_{a}}h+o(h)\right).$

This leads to the following differential equation:

$\displaystyle\frac{d\
}{dt}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)$
$\displaystyle=$
$\displaystyle\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}(t)\left(Q_{(k_{b}-1)m_{a}+k_{a},(k_{b}-1)m_{a}+k_{a}}-\sum_{i=1}^{d}\beta^{i,b}\lambda^{k_{b},b}-\sum_{i=1}^{d}\beta^{i,b}\lambda^{k_{a},a}\right)$
$\displaystyle+\sum_{1\leq l_{b}\leq m_{b},1\leq l_{a}\leq
m_{a},(l_{b},l_{a})\neq(k_{b},k_{a})}\mathcal{G}^{(j_{b}-1)m_{a}+j_{a},(l_{b}-1)m_{a}+l_{a}}(t)Q_{(l_{b}-1)m_{a}+l_{a},(k_{b}-1)m_{a}+k_{a}}$

which, in matrix form, writes

$\mathcal{G}^{\prime}(t)=\mathcal{G}(t)\left(Q-\sum_{i=1}^{d}\beta^{i,b}\Lambda^{b}\otimes
I_{m_{a}}-\sum_{i=1}^{d}\beta^{i,a}I_{m_{b}}\otimes\Lambda^{a}\right).$

As $\mathcal{G}(0)=I_{m_{b}m_{a}}$, we conclude that

$\mathcal{G}(t)=\exp\left(\left(Q-\sum_{i=1}^{d}\beta^{i,b}\Lambda^{b}\otimes
I_{m_{a}}-\sum_{i=1}^{d}\beta^{i,a}I_{m_{b}}\otimes\Lambda^{a}\right)t\right)=\exp\left(\left(Q-\Lambda^{b}\otimes
I_{m_{a}}-I_{m_{b}}\otimes\Lambda^{a}\right)t\right)$

thanks to the normalization choice.

If we assume that $\lambda_{0}$ is distributed according to $\pi_{0}$, then,
using the same reasoning as above, the likelihood writes

$\displaystyle\mathcal{L}(Q,\Lambda^{b},\Lambda^{a}|t_{1},\ldots,t_{N},i_{1},\ldots
i_{N},\mathfrak{s}_{1},\ldots\mathfrak{s}_{N})$ $\displaystyle=$
$\displaystyle\pi^{\prime}\left(\prod_{n=1}^{N}\exp\left(\left(Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a}\right)(t_{n}-t_{n-1})\right)\beta^{i_{n},\mathfrak{s}_{n}}\tilde{\Lambda}^{\mathfrak{s}_{n}}\right)e$
$\displaystyle=$
$\displaystyle\left(\prod_{i=1}^{d}(\beta^{i,b})^{K^{i,b}}\right)\left(\prod_{i=1}^{d}(\beta^{i,a})^{K^{i,a}}\right)\pi^{\prime}\left(\prod_{n=1}^{N}\exp\left(\left(Q-\tilde{\Lambda}^{b}-\tilde{\Lambda}^{a}\right)(t_{n}-t_{n-1})\right)\tilde{\Lambda}^{\mathfrak{s}_{n}}\right)e$

where $K^{i,b}=\text{Card}(\\{n,i_{n}=i,\mathfrak{s}_{n}=b\\})$ and
$K^{i,a}=\text{Card}(\\{n,i_{n}=i,\mathfrak{s}_{n}=a\\})$.

From this expression we deduce that (i) we can merge RFQs at the bid across
bonds and RFQs at the ask across bonds to estimate the parameters of $Q$,
$\Lambda^{b}$ and $\Lambda^{a}$ using the EM algorithm of Section 2.2 or that
of Section 2.3.1, and (ii) we can separately estimate the $\beta$
coefficients. Regarding the former, the EM algorithm presented above can be
used on merged data. The latter (the estimation of the sensitivities) is
trivial: maximizing $\prod_{i=1}^{d}(\beta^{i,b})^{K^{i,b}}$ subject to
$\sum_{i=1}^{d}\beta^{i,b}=1$ indeed boils down to setting $\beta^{i,b}$
proportional to $K^{i,b}$, i.e.
$\beta^{i,b}=\frac{K^{i,b}}{\sum_{j=1}^{d}K^{j,b}}$ – and similarly we obtain
$\beta^{i,a}=\frac{K^{i,a}}{\sum_{j=1}^{d}K^{j,a}}$.

##  3 New notions of price

###  3.1 A micro-price for corporate bond markets

####  3.1.1 Definition

In [32], Stoikov introduced the notion of micro-price for an asset traded
through a limit order book. It is defined as the asymptotic value of the
expected mid-price, given all the information available (in the limit order
book).

It is reasonable to extend the ideas introduced in [32] to corporate bonds
through the use of our model for RFQ arrival. If we consider a reference price
process $(S_{t})_{t}$ that can be CBBT, CP+ or another composite, it is
commonplace to assume a Brownian dynamics $dS_{t}=\sigma dW_{t}$. However, if
we know the current state of liquidity, it makes more sense to consider a
dynamics of the form

$dS_{t}=\sigma dW_{t}+\kappa(\lambda^{a}_{t}-\lambda^{b}_{t})dt$

where $\kappa$ is a nonnegative constant. Then, the micro-price at time $t$ is
naturally defined by

$S^{\text{micro}}_{t}=\lim_{T\to+\infty}\mathbb{E}[S_{T}|S_{t},\lambda^{b}_{t},\lambda^{a}_{t}]=S_{t}+\kappa\lim_{T\to+\infty}\mathbb{E}\left[\left.\int_{t}^{T}(\lambda^{a}_{s}-\lambda^{b}_{s})ds\right|\lambda^{b}_{t},\lambda^{a}_{t}\right]$

if that limit exists.161616For bonds, Brownian dynamics can only be valid in
the short run. Nevertheless, we consider a long-term limit to define the
micro-price. This may seem problematic at first sight but, in our model, the
long-term limit corresponds to a time horizon equivalent to that of the return
to a symmetric state of liquidity, which is typically short.

####  3.1.2 Mathematical analysis

To study this notion, let us define

$v_{T}(t,\lambda^{b},\lambda^{a})=\mathbb{E}\left[\left.\int_{t}^{T}(\lambda^{a}_{s}-\lambda^{b}_{s})ds\right|\lambda^{b}_{t}=\lambda^{b},\lambda^{a}_{t}=\lambda^{a}\right].$

If we write $v_{T}(t)$ the vector with coordinates
$(v_{T}(t,\lambda^{b,j_{b}},\lambda^{a,j_{a}}))_{1\leq j_{b}\leq m_{b},1\leq
j_{a}\leq m_{a}}$ in lexicographic order, then we have that $v_{T}$ solves

$\frac{d}{dt}v_{T}(t)+I_{m_{b}}\otimes\lambda_{\text{vec}}^{a}-\lambda_{\text{vec}}^{b}\otimes
I_{m_{a}}+Qv_{T}(t)=0\quad\text{and}\quad v_{T}(T)=0$

where
$\lambda_{\text{vec}}^{b}=(\lambda^{b,1},\ldots,\lambda^{b,m_{b}})^{\prime}$
and
$\lambda_{\text{vec}}^{a}=(\lambda^{a,1},\ldots,\lambda^{a,m_{a}})^{\prime}$,
i.e.

$v_{T}(t)=\int_{t}^{T}\exp(Q(s-t))(I_{m_{b}}\otimes\lambda_{\text{vec}}^{a}-\lambda_{\text{vec}}^{b}\otimes
I_{m_{a}})ds.$

Let us now assume as in Section 2.3.1 that
$(\lambda^{b}_{s},\lambda^{a}_{s})_{s}$ and
$(\lambda^{a}_{s},\lambda^{b}_{s})_{s}$ have the same distribution. Then, it
is straightforward to see that $v_{T}(t,\lambda,\lambda)=0$ for all
$\lambda\in\\{\lambda^{1},\ldots,\lambda^{m}\\}$. Moreover, writing
$\lambda_{\text{vec}}=(\lambda^{1},\ldots,\lambda^{m})^{\prime}$, we have that
$I_{m_{b}}\otimes\lambda_{\text{vec}}^{a}-\lambda_{\text{vec}}^{b}\otimes
I_{m_{a}}=I_{m}\otimes\lambda_{\text{vec}}-\lambda_{\text{vec}}\otimes I_{m}$
has all coordinates corresponding to symmetric states equal to $0$. Therefore,
if we denote by the superscript ${}^{\text{ns}}$ vectors and matrices where
all symmetric states have been dropped, we have:

$v^{\text{ns}}_{T}(t)=\int_{t}^{T}\exp(Q^{\text{ns}}(s-t))(I_{m}\otimes\lambda_{\text{vec}}-\lambda_{\text{vec}}\otimes
I_{m})^{\text{ns}}ds.$

If we add the assumption that the matrix $Q$ is such that, in each asymmetric
state, the intensity associated with returning to at least one symmetric state
is positive, then $Q^{\text{ns}}$ is a strictly diagonally-dominant matrix
with negative diagonal terms and we have therefore, from Gershgorin circle
theorem, that $Q^{\text{ns}}$ is invertible and that
$\lim_{T\to+\infty}\exp(Q^{\text{ns}}T)=0$. We conclude that

$\displaystyle v^{\text{ns}}_{T}(t)$ $\displaystyle=$
$\displaystyle(Q^{\text{ns}})^{-1}\exp(Q^{\text{ns}}(T-t))(I_{m}\otimes\lambda_{\text{vec}}-\lambda_{\text{vec}}\otimes
I_{m})^{\text{ns}}-(Q^{\text{ns}})^{-1}(I_{m}\otimes\lambda_{\text{vec}}-\lambda_{\text{vec}}\otimes
I_{m})^{\text{ns}}$ (3)
$\displaystyle\to_{T\to+\infty}-(Q^{\text{ns}})^{-1}(I_{m}\otimes\lambda_{\text{vec}}-\lambda_{\text{vec}}\otimes
I_{m})^{\text{ns}}.$

Under the above assumptions, we can therefore define
$v(\lambda^{b},\lambda^{a})=\lim_{T\to+\infty}v_{T}(t,\lambda^{b},\lambda^{a})$
and write

$S^{\text{micro}}_{t}=S_{t}+\kappa v(\lambda^{b}_{t},\lambda^{a}_{t}).$

Of course, in practice, one never knows the current state of liquidity and
rather uses a probability distribution $\pi$ over the states. Then, one gets
at time $t$ a micro-price with mean

$\bar{S}^{\text{micro}}_{t}=S_{t}+\kappa\sum_{1\leq j_{b}\leq m,1\leq
j_{a}\leq m}\pi^{j_{b},j_{a}}v(\lambda^{j_{b}},\lambda^{j_{a}})$ (4)

and standard deviation171717This standard deviation only quantifies the
uncertainty linked to the estimation of the current market liquidity state for
given values of the model parameters. given by

$\kappa\sqrt{\sum_{1\leq j_{b}\leq m,1\leq j_{a}\leq
m}\pi^{j_{b},j_{a}}v(\lambda^{j_{b}},\lambda^{j_{a}})^{2}-\left(\sum_{1\leq
j_{b}\leq m,1\leq j_{a}\leq
m}\pi^{j_{b},j_{a}}v(\lambda^{j_{b}},\lambda^{j_{a}})\right)^{2}}.$

####  3.1.3 Main remarks on our assumptions to obtain a micro-price

To be able to define the notion of micro-price and ensure that the limit
exists, we imposed three structural assumptions to our model for the flow of
RFQs. We indeed impose that:

  * •

the set of possible intensities is shared across the bid and the ask;

  * •

the transition matrix $Q$ is both that of $(\lambda^{b}_{t},\lambda^{a}_{t})$
and $(\lambda^{a}_{t},\lambda^{b}_{t})$ – in the case of two liquidity states,
high and low, this means that the chance of any transition leading to or from
an unbalanced state does not depend on the side of the imbalance;

  * •

any unbalanced state has a chance to be followed by at least one balanced
state.

These assumptions are quite light and natural. What is more questionable is
the linearity assumption in the drift. It is however important to have in mind
that intensities are not observed but that only the probability to be in the
different states can be estimated. This results in noisy estimates of $\kappa$
as we shall see. Parsimony clearly guided our modelling choice.

###  3.2 A fair transfer price

####  3.2.1 From a market making model to a fair transfer price

We now want to go beyond the notion of micro-price and use ideas coming from
the market making literature to define a fair transfer price.

We consider a theoretical market maker receiving RFQs to buy and sell a bond
(for a given unique size $z$ in this model). We model the number of RFQs at
the bid and at the ask by MMPPs as above and do not impose here, for the sake
of generality, our symmetry assumptions that guaranteed the well-posedness of
the definition of micro-price.

We consider a reference price process $(S_{t})_{t}$ and we assume that

$dS_{t}=\sigma dW_{t}+\kappa(\lambda^{a}_{t}-\lambda^{b}_{t})dt.$

Upon receiving at time $t$ an RFQ at the bid (resp. at the ask), the market
maker answers a price $S_{t}^{b}=S_{t}-\delta_{t}^{b}$ (resp.
$S_{t}^{a}=S_{t}+\delta_{t}^{a}$) and this leads to a trade with probability
$f^{b}(S_{t}-S_{t}^{b})=f^{b}(\delta_{t}^{b})$ (resp.
$f^{a}(S_{t}^{a}-S_{t})=f^{a}(\delta_{t}^{a})$) where $f^{b}$ (resp. $f^{a}$)
is a decreasing function from $1$ to $0$ (sometimes called S-function or
S-curve). The inventory process $(q_{t})_{t}$ of the market maker evolves
subsequently as

$dq_{t}=zdN^{b}_{t}-zdN^{a}_{t}.$

The cash process $(X_{t})_{t}$ evolves therefore as

$dX_{t}=zS^{a}_{t}dN^{a}_{t}-zS^{b}_{t}dN^{b}_{t}=-S_{t}dq_{t}+z\delta_{t}^{b}dN_{t}^{b}+z\delta_{t}^{a}dN_{t}^{a}$

and the PnL process $(\text{PnL}_{t})_{t}=(X_{t}+q_{t}S_{t})_{t}$ as

$d\text{PnL}_{t}=z\delta_{t}^{b}dN_{t}^{b}+z\delta_{t}^{a}dN_{t}^{a}+\sigma
q_{t}dW_{t}+\kappa(\lambda_{t}^{a}-\lambda_{t}^{b})q_{t}dt.$

A market maker wishing to capture the bid-ask spread while mitigating the risk
(see [10, 11, 22]) typically maximizes, over the set of predictable processes
$(\delta^{b}_{t})_{t}$ and $(\delta^{a}_{t})_{t}$, the objective function

$\displaystyle\mathbb{E}\left[\text{PnL}_{T}-\frac{\gamma}{2}\langle\text{PnL}\rangle_{T}\right]=\mathbb{E}\left[\int_{0}^{T}\left(z\lambda_{t}^{b}\delta_{t}^{b}f^{b}(\delta_{t}^{b})+z\lambda_{t}^{a}\delta_{t}^{a}f^{a}(\delta_{t}^{a})+\kappa(\lambda_{t}^{a}-\lambda_{t}^{b})q_{t}-\frac{\gamma}{2}\sigma^{2}q_{t}^{2}\right)dt\right]$

for a given risk aversion parameter $\gamma>0$, where
$\langle\cdot\rangle_{T}$ stands for the quadratic variation.

Assuming that the theoretical market maker is able to identify in which state
$(\lambda^{j_{b},b},\lambda^{j_{a},a})$ the market is at any point in time,
then the value functions $(\theta^{j_{b},j_{a}})_{1\leq j_{b}\leq m_{b},1\leq
j_{a}\leq m_{a}}$ satisfy the following system of Hamilton-Jacobi-Bellman
(HJB) equations:

$\displaystyle\partial_{t}\theta^{j_{b},j_{a}}(t,q)+\kappa(\lambda^{j_{a},a}-\lambda^{j_{b},b})q-\frac{1}{2}\gamma\sigma^{2}q^{2}+\sum_{1\leq
k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\theta^{k_{b},k_{a}}(t,q)$
$\displaystyle+z\lambda^{j_{b},b}H^{b}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q+z)}{z}\right)+z\lambda^{j_{a},a}H^{a}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q-z)}{z}\right)=0$

with terminal condition $\theta^{j_{b},j_{a}}(T,q)=0$, where
$H^{b/a}(p)=\underset{\delta\in\mathbb{R}}{\sup}f^{b/a}(\delta)(\delta-p).$

Under mild assumptions on the functions $f^{b}$ and $f^{a}$ (see for instance
[7]), the optimal bid and ask quotes of the market maker if the current state
of the market is $(\lambda^{j_{b},b},\lambda^{j_{a},a})$ write

$\displaystyle\delta^{b,i,\star}_{t}$
$\displaystyle=\bar{\delta}^{b}\left(\frac{\theta^{j_{b},j_{a}}(t,q_{t-})-\theta^{j_{b},j_{a}}(t,q_{t-}+z)}{z}\right)$
$\displaystyle\delta^{a,i,\star}_{t}$
$\displaystyle=\bar{\delta}^{a}\left(\frac{\theta^{j_{b},j_{a}}(t,q_{t-})-\theta^{j_{b},j_{a}}(t,q_{t-}-z)}{z}\right).$

If the matrix $Q$ is irreducible, an ergodic limit exists, i.e. there exists a
constant $c$ such that we have
$\lim_{T\to\infty}\theta^{j_{b},j_{a}}(t,q)-c(T-t)=\theta^{j_{b},j_{a}}_{\infty}(q)$
(see [24] for a general framework), and we can define a time-independent
notion of skew by

$\text{skew}^{j_{b},j_{a}}_{\infty}=\bar{\delta}^{a}\left(\frac{\theta^{j_{b},j_{a}}_{\infty}(0)-\theta^{j_{b},j_{a}}_{\infty}(-z)}{z}\right)-\bar{\delta}^{b}\left(\frac{\theta^{j_{b},j_{a}}_{\infty}(0)-\theta^{j_{b},j_{a}}_{\infty}(z)}{z}\right).$

This skew “projects” the asymmetry of the market liquidity into the price
space because of the market maker’s need to quote asymmetrically in order to
account for that asymmetric liquidity (even in the absence of inventory).

The notion of Fair Transfer Price we propose (from now on FTP) is then defined
as the mid-price of a market maker with infinite horizon and no inventory,
i.e. it is defined at time $t$ by:

$S^{\text{FTP}}_{t}=S_{t}+\frac{1}{2}\text{skew}_{\infty}.$

It corresponds to the average between the price answered to a buyer and the
price answered to a seller by a theoretical market maker with no inventory, if
they were requested – hence the dimension of fairness.

Of course, in practice, one never knows the current state of liquidity and
rather uses a probability distribution $\pi$ over the states. One gets at time
$t$ an FTP with mean

$\displaystyle\bar{S}^{\text{FTP}}_{t}=S_{t}+\frac{1}{2}\sum_{1\leq j_{b}\leq
m_{b},1\leq j_{a}\leq
m_{a}}\pi^{j_{b},j_{a}}\text{skew}^{j_{b},j_{a}}_{\infty}$ (5)

and standard deviation given by

$\frac{1}{2}\sqrt{\sum_{1\leq j_{b}\leq m_{b},1\leq j_{a}\leq
m_{a}}\pi^{j_{b},j_{a}}\left(\text{skew}^{j_{b},j_{a}}_{\infty}\right)^{2}-\left(\sum_{1\leq
j_{b}\leq m_{b},1\leq j_{a}\leq
m_{a}}\pi^{j_{b},j_{a}}\text{skew}^{j_{b},j_{a}}_{\infty}\right)^{2}}.$

####  3.2.2 Main remarks on FTP

It is a priori hard to relate our concept of FTP to those used in the case of
LOBs. However, since top-of-book volumes in LOBs are inversely related to the
appetite of liquidity takers (the clients in a dealer market), FTP shares a
lot of good characteristics with the weighted mid-prices of LOBs. Bid/ask
imbalances in LOBs are indeed comparable to bid/ask asymmetries in client
flows, though in the opposite direction. A very high volume at the bid
(relatively to the ask) in an LOB is similar to a situation, in a dealer
market, where the clients are more willing to buy than to sell, and in this
context a market maker has to skew their quotes towards the right, pushing the
FTP upwards. This is in line with a weighted mid-price above the mid-price in
an LOB.

At first sight, the notion of FTP depends strongly on the reference price
chosen to build the market making model. This dependence is real but it is not
the serious caveat it might seem. If indeed we replace $(S_{t})_{t}$ by
$(S_{t}+\xi)_{t}$, then the functions $f^{b}$ and $f^{a}$ should be shifted
accordingly in the estimation procedure if we assume that trading decisions
depend on the absolute values (as opposed to relative) of proposed prices.
Subsequently, value functions should be translated by a term $\xi q$ and it is
easy to see that the FTP would be unchanged since the functions
$\bar{\delta}^{b}$ and $\bar{\delta}^{a}$ are themselves translated by
$\pm\xi$. Of course, this invariance is limited to constant shifts, but,
still, it shows that differences between (relevant) reference prices should be
partially or entirely compensated by their impact on the definition /
estimation of $f^{b}$ and $f^{a}$.

In the definition of FTP, the transaction size $z$ inputted in the market
making model plays a role although it might seem arbitrary. This transaction
size $z$ is typically the reference size for which market makers stream prices
in corporate bond markets. One can naturally generalize the concept to
consider other sizes.

Another important point is that the FTP depends on the risk aversion parameter
$\gamma$ inputted into the objective function. This seems to be a problem
since the parameter $\gamma$ can be chosen arbitrarily. However, the choice of
$\gamma$ leaves one degree of freedom, and this is rather an opportunity. In
Section 4, we use $\gamma$ to calibrate the model to bid-ask spreads.

Many improvements can be made to the market making model in line with what
exists in the literature, like considering several transaction sizes (see
[7]), taking account of client tiering (see [4]), using the possibility to
externalize part of the flow (see [4, 3]), replacing the quadratic running
penalty – linked to the mean/quadratic variation objective function proposed
in [11] – by a more complicated one, etc.181818The introduction of asymmetric
inventory costs, linked to repo rates for instance, is not recommended if the
goal is to define a fair price between two parties that would apply
symmetrically. One can also decide to use a multi-bond market making model
instead of a single-bond one (see for instance [5], [6] and [23]). In fact,
the notion of FTP is versatile and one can choose the market making model they
want. It must be noted that the role of the market making model is, here, not
to provide quotes that will be used inside a trading algorithm, but only that
of projecting information regarding liquidity levels, liquidity imbalances and
volatility into the price space. In particular, there is no real problem in
keeping the market making model relatively simple (as soon as liquidity
dynamics is taken into account), all the more since one can rely on the degree
of freedom provided by the risk aversion parameter $\gamma$ to match a desired
target (see Section 5).

##  4 From theory to practice

###  4.1 Introduction

In the above sections, we have extended the notion of micro-price to our OTC
framework and defined the new concept of Fair Transfer Price. To be able to
use these notions in practice we need to estimate several parameters.

First, we need to estimate the parameters of the MMPPs. We detailed in Section
2 some estimation procedures based on EM algorithms. Then, to compute the
micro-price and/or to estimate the dynamics of the reference price in the
market making model, we need to estimate the constant $\kappa$ and this is
typically done with a linear regression of price moves on terms of the form
$\sum_{1\leq j_{b}\leq m,1\leq j_{a}\leq
m}\pi^{j_{b},j_{a}}v(\lambda^{j_{b}},\lambda^{j_{a}})$, following Eq. (4).
When it comes to using FTP, the volatility parameter $\sigma$ is also
necessary and classical estimators can be used for that matter. One also needs
to estimate the parameters used in the modelling of the conversion of an RFQ
into a trade, i.e. the parameters of $f^{b}$ and $f^{a}$ once a parametric
functional form has been chosen – $f^{b}$ and $f^{a}$ are typically chosen
logistic and the estimation procedure boils down to a logistic regression. In
addition to the estimation of parameters, the use of FTP requires to choose a
risk aversion parameter and solve an HJB equation to get optimal quotes.

In what follows, we illustrate our approach and the different concepts of
prices on corporate bond data. For that purpose, we use an anonymized dataset
of RFQs on high-yield corporate bonds kindly provided by J.P. Morgan. It
contains, for each RFQ, the date and time of the request, the bond requested,
the direction of the request (buy or sell), the notional (odd lots have been
removed from the dataset), the price answered to the client, current market
(in fact composite) prices at the bid and at the ask, and the status – i.e.
whether the price was accepted by the client or not. Because some requests are
only sent to gather information, we focused on RFQs that led to a trade with
J.P. Morgan or with another dealer (this piece of information, but of course
not the identity of the other dealer, is known as we focus on RFQs sent
through multi-dealer-to-client platforms). Our dataset contains bonds from 4
different sectors.191919For confidentiality reasons, we do not give the id of
the bonds nor that of the sectors. It covers more than half a year of trading,
over the post-covid period.202020For confidentiality reasons, we do not
document the exact period of time. Throughout the paper, the unit for times is
in days since the beginning of the period, excluding weekends. Nights have
also been excluded so that the beginning of the next trading day follows the
end of the current one – trading hours have been set from 7am to 5pm.

###  4.2 Estimation of the Markov-modulated Poisson process

For the estimation of the parameters of the bidimensional Markov-modulated
Poisson process, we consider the multi-bond extension (see Section 2.3.2) to
carry out the process at the sector level. To be able to illustrate our notion
of micro-price, we also rely on the exchangeability assumption of Section
2.3.1.

In the EM algorithm of Section 2.3.1, one must choose the number $m$ of
intensities, set their initial values and those of the coefficients of the
transition matrix. To obtain a first naive estimation of the intensities of
the Markov-modulated Poisson processes for each sector, we start by computing
the number of RFQs per day at the bid and at the ask. The results are plotted
in Figures 1, 2, 3 and 4. We clearly see that liquidity is volatile and that
upward or downward bumps may be simultaneous across bid and ask (see for
instance what happens around $t=60$ and $t=90$ in Figure 1), but also
asymmetric with one side seeing a rise or a decrease in liquidity while the
other does not (see for instance what happens around $t=75$ in Figure 2).

For each sector, we decide to consider two intensities ($m=2$), and initialize
them using the average over the bid and ask sides of the quantile of order 10%
(for the low liquidity state) and 90% (for the high liquidity state) of the
distribution of the number of RFQs per day. To initialize $Q$, we use very
naive values corresponding to independence between the intensities at the bid
and at the ask and a transition rate from low to high or high to low equal to
$1$ (per day).

We ran the EM algorithm of Section 2.3.1 over our RFQ database, sector by
sector (using the trick of the multi-bond extension – see Section 2.3.2). To
normalize likelihoods, we used classical regression techniques. We noticed
convergence of the values of $\lambda^{1}$, $\lambda^{2}$ and the coefficients
of the matrix $Q$ after approximately 50 steps. The resulting parameters of
the MMPPs are reported in Table 1. We clearly see that, for the 4 sectors, the
algorithm manages to separate low liquidity from high liquidity. We also see
that the transition matrices are different across sectors: high transition
rates and a relatively high probability to jump from an imbalanced state to
the opposite imbalanced state in the case of Sector 1, a very stable (resp.
unstable) low/low-liquidity (resp. high/high-liquidity) state in the case of
Sector 2, and low transition rates for Sector 4.

Figure 1: Number of RFQs per day at the bid and at the ask for Sector 1.
Figure 2: Number of RFQs per day at the bid and at the ask for Sector 2.
Figure 3: Number of RFQs per day at the bid and at the ask for Sector 3.
Figure 4: Number of RFQs per day at the bid and at the ask for Sector 4. Sector | $\lambda^{1}$ | $\lambda^{2}$ | $Q$
---|---|---|---
Sector 1 | 10.83 | 73.03 | $\begin{pmatrix}-14.01&4.37&4.37&5.27\\\ 19.32&-60.91&12.54&29.05\\\ 19.32&12.54&-60.91&29.05\\\ 23.67&15.00&15.00&-53.67\end{pmatrix}$
Sector 2 | 8.44 | 58.28 | $\begin{pmatrix}-4.55&1.00&1.00&2.55\\\ 18.53&-28.31&0.13&9.65\\\ 18.53&0.13&-28.31&9.65\\\ 14.77&16.73&16.73&-48.23\end{pmatrix}$
Sector 3 | 15.73 | 81.78 | $\begin{pmatrix}-9.98&2.79&2.79&4.40\\\ 20.53&-23.73&0.02&3.18\\\ 20.53&0.02&-23.73&3.18\\\ 9.87&4.17&4.17&-18.21\end{pmatrix}$
Sector 4 | 7.33 | 28.32 | $\begin{pmatrix}-1.67&0.48&0.48&0.71\\\ 1.92&-2.02&0.00&0.10\\\ 1.92&0.00&-2.02&0.10\\\ 0.84&0.11&0.11&-1.06\end{pmatrix}$
Table 1: Estimated parameters of the Markov-modulated Poisson process for the
four sectors.

Once the parameters of the MMPPs have been estimated, we can evaluate at each
point in time the probability to be in each state (see Section 2.2.3). In
Figures 5, 6, 7 and 8, we document the distribution of the values taken by the
probability $\pi^{0,1}$ (resp. $\pi^{1,0}$) to be in a low/high-liquidity
(resp. high/low-liquidity) state. We clearly see that high values of these
probabilities are quite rare: it is hard to be certain that a desquilibrium in
RFQs indeed corresponds to an underlying asymmetric regime.

Figure 5: Distribution of the values taken by $\pi^{0,1}$ and $\pi^{1,0}$ for
Sector 1.
Figure 6: Distribution of the values taken by $\pi^{0,1}$ and $\pi^{1,0}$ for
Sector 2.
Figure 7: Distribution of the values taken by $\pi^{0,1}$ and $\pi^{1,0}$ for
Sector 3.
Figure 8: Distribution of the values taken by $\pi^{0,1}$ and $\pi^{1,0}$ for
Sector 4.

###  4.3 Micro-price

####  4.3.1 Estimation of $\kappa$

In order to illustrate our concept of micro-price, we need first to estimate
the parameter $\kappa$ in the dynamics of bond prices. For each sector, we
consider 4 bonds amongst those available in the database. We first compute the
functions $v$ given by Eq. (3) and then use Eq. (4) to perform a linear
regression and estimate $\kappa$ for each bond. We also compute the arithmetic
volatility $\sigma$ and the weight $\beta$ associated with each bond (see
Section 2.3.2). The results are reported in Table 2.

The estimated values for $\kappa$ are not all significantly different for $0$
(given the standard deviations reported), but it is nevertheless interesting
to notice that the figures are positive for all bonds. This tends to prove
that imbalance in the flow of RFQs has a consistant predictive power on the
variation of the price, hence the interest of the concept of micro-price.

####  4.3.2 Micro-price in practice

In Table 3, we took the last composite mid-price and bid-ask spread in the
dataset for each of the 16 bonds we focus on, and computed the corresponding
micro-price when we are $100\%$ sure that the market is imbalanced, one way or
the other.

Of course, and this is confirmed by the above histograms, one can seldom be
certain to be in any of the two imbalanced states. In practice, the micro-
price must therefore be computed as an expectation over the different possible
states, i.e. as a function of the current estimates of the probabilities to be
in each state. In particular, the micro-prices exhibited in Table 3 correspond
to theoretical bounds for the micro-price that would be used in practice.

Sector | Bond | $\beta$ |  $\kappa$ (stdev) | $\sigma$
---|---|---|---|---
1 | 1 | 0.10 | 2.29 (0.55) | 18.39
| 2 | 0.10 | 0.25 (0.49) | 15.43
| 3 | 0.06 | 2.83 (1.66) | 22.55
| 4 | 0.05 | 0.33 (2.23) | 19.75
2 | 1 | 0.19 | 0.57 (0.19) | 13.75
| 2 | 0.14 | 0.90 (0.22) | 16.05
| 3 | 0.11 | 0.65 (0.16) | 9.80
| 4 | 0.10 | 0.86 (0.68) | 20.36
3 | 1 | 0.11 | 0.61 (0.34) | 9.93
| 2 | 0.09 | 0.05 (0.16) | 18.41
| 3 | 0.06 | 0.11 (0.08) | 12.23
| 4 | 0.05 | 0.08 (0.11) | 18.68
4 | 1 | 0.21 | 0.04 (0.02) | 13.00
| 2 | 0.12 | 0.01 (0.01) | 24.09
| 3 | 0.12 | 0.08 (0.04) | 16.91
| 4 | 0.07 | 0.09 (0.05) | 12.67
Table 2: Estimations of $\beta$, $\kappa$ and $\sigma$. Sector | Bond | Mid-price | Bid price | Ask price | Micro-price $\pi^{1,0}\\!\\!\\!=\\!\\!1$ | Micro-price $\pi^{0,1\\!}\\!\\!=\\!\\!1$
---|---|---|---|---|---|---
1 | 1 | 103.593 | 103.098 | 104.088 | 101.652 | 105.534
| 2 | 97.107 | 96.614 | 97.600 | 96.892 | 97.322
| 3 | 99.146 | 98.631 | 99.661 | 96.752 | 101.541
| 4 | 94.187 | 93.049 | 95.325 | 93.909 | 94.465
2 | 1 | 99.823 | 99.291 | 100.355 | 98.819 | 100.827
| 2 | 99.270 | 98.603 | 99.936 | 97.700 | 100.840
| 3 | 99.649 | 98.815 | 100.483 | 98.513 | 100.784
| 4 | 98.903 | 97.570 | 100.235 | 97.970 | 99.835
3 | 1 | 95.338 | 94.674 | 96.001 | 93.634 | 97.041
| 2 | 92.394 | 91.860 | 92.927 | 92.252 | 92.535
| 3 | 97.137 | 96.484 | 97.790 | 96.819 | 97.455
| 4 | 94.839 | 94.220 | 95.458 | 94.810 | 94.867
4 | 1 | 102.632 | 102.151 | 103.112 | 102.252 | 103.011
| 2 | 104.785 | 104.327 | 105.242 | 104.717 | 104.853
| 3 | 104.824 | 104.293 | 105.355 | 103.994 | 105.654
| 4 | 108.438 | 107.991 | 108.884 | 107.500 | 109.375
Table 3: Micro-price for the different bonds in imbalanced states.

In what follows, we study how the micro-price evolves depending on the
probabilities of the different states of the MMPP, for the first bond of each
sector. Notice that, in our case, the respective values of $\pi^{0,0}$ and
$\pi^{1,1}$ have no impact on the micro-price: only $\pi^{0,1}$ and
$\pi^{1,0}$ matter.

In Figure 9, we plot the micro-price as a function of $\pi^{0,1}$, when
$\pi^{1,0}=0.$ Figure 10 documents similarly the micro-price as a function of
$\pi^{0,1}$, when $\pi^{1,0}=0.3$. To study the impact of the uncertainty on
the parameter $\kappa$, we also plot the micro-prices corresponding to values
of $\kappa$ one standard deviation above and below our estimation. Composite
bid-ask spreads are also reported in the Figures.

Naturally, when $\pi^{0,1}=\pi^{1,0}$, the micro-price is equal to the mid-
price. As expected, we also see that a rise in $\pi^{0,1}$ leads to an
increase in micro-prices. In Figure 9, we see that micro-prices are within the
composite bid-ask spread for moderate values of $\pi^{0,1}$ but beyond for
large values (except for Bond 4.1). Values beyond the bid-ask spread could be
seen as a real trading signal, but it is important to have in mind that the
results obtained for high values of $\pi^{0,1}$ have to be taken with care
because the linear regressions have been carried out with only a few high
values for $\pi^{0,1}$ (see the above histograms). We see in Figure 10 that
when $\pi^{1,0}=0.3$, most values remain inside the bid-ask spread. In fact,
we see in Figures 11, 12, 13 and 14 that micro-prices are significantly
outside of the bid-ask spread for extreme values of $\pi^{0,1}$ and
$\pi^{1,0}$ (in opposite directions) only, i.e. when one is really sure that
the flow is imbalanced.

(a) Bond 1.1

(b) Bond 2.1

(c) Bond 3.1

(d) Bond 4.1

Figure 9: Micro-price for different bonds as a function of $\pi^{0,1}$ when
$\pi^{1,0}=0$.

(a) Bond 1.1

(b) Bond 2.1

(c) Bond 3.1

(d) Bond 4.1

Figure 10: Micro-price for different bonds as a function of $\pi^{0,1}$ when
$\pi^{1,0}=0.3$.
Figure 11: Micro-price for Bond 1.1 as a function of $\pi^{0,1}$ and
$\pi^{1,0}$.
Figure 12: Micro-price for Bond 2.1 as a function of $\pi^{0,1}$ and
$\pi^{1,0}$.
Figure 13: Micro-price for Bond 3.1 as a function of $\pi^{0,1}$ and
$\pi^{1,0}$.
Figure 14: Micro-price for Bond 4.1 as a function of $\pi^{0,1}$ and
$\pi^{1,0}$.

###  4.4 Fair Transfer Price

Let us now come to the case of Fair Transfer Prices. For that purpose, we need
to fit S-curves, choose a risk aversion parameter and solve a HJB equation.

####  4.4.1 Estimation of S-curves

For the functions $f^{b}$ and $f^{a}$ defined in Section 3.2.1, we assume a
logistic form. We noticed no systematic difference between the bid and ask
sides. Consequently, we considered

$f^{b}(\delta)=f^{a}(\delta)=\frac{1}{1+e^{\alpha+\frac{\beta}{\delta^{0}}\delta}}=:f(\delta),$

where $\delta^{0}$ is the current composite bid-ask spread of the bond, and
the parameters $\alpha$ and $\beta$ are estimated with a logistic regression.

Figure 15: Intensity function $f$ for the different sectors.

With this parametrization, the functions appeared to be almost uniform accross
sectors (as shown in Figure 15), and we therefore estimated, for the sake of
simplicity, a single S-curve using the entire dataset, independently of the
sector. We obtained the following (rounded) values: $\alpha=-0.7$ and
$\beta=3.1$.

####  4.4.2 Solving HJB equations

Our concept of FTP relies on the bid and ask quotes of a theoretical market
maker that knows the current state of the market. To solve the stochastic
optimal control problem of that market maker and obtain the associated quotes,
one needs to compute numerically the value functions.

Let us recall that, in the model of Section 3, the value functions
$(\theta^{j_{b},j_{a}})_{1\leq j_{b}\leq m_{b},1\leq j_{a}\leq m_{a}}$ of the
market maker satisfy the following system of Hamilton-Jacobi-Bellman (HJB)
equations:212121We state the equations in the general case, i.e. not in the
case of the extension of Section 2.3.1, although our illustrations rely on the
exchangeability assumption.

$\partial_{t}\theta^{j_{b},j_{a}}(t,q)+\kappa(\lambda^{j_{a},a}-\lambda^{j_{b},b})q-\frac{1}{2}\gamma\sigma^{2}q^{2}+\sum_{1\leq
k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\theta^{k_{b},k_{a}}(t,q)$
$+z\lambda^{j_{b},b}H^{b}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q+z)}{z}\right)+z\lambda^{j_{a},a}H^{a}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q-z)}{z}\right)=0$

with terminal condition $\theta^{j_{b},j_{a}}(T,q)=0$.

We need to compute or approximate numerically the solution of this system of
equations in order to compute FTPs. A natural approach is to use a Euler
scheme, preferably implicit. In that case, relevant boundary conditions can be
chosen by adding risk limits to the inventory of the theoretical market maker,
and the equations become

$\partial_{t}\theta^{j_{b},j_{a}}(t,q)+\kappa(\lambda^{j_{a},a}-\lambda^{j_{b},b})q-\frac{1}{2}\gamma\sigma^{2}q^{2}+\sum_{1\leq
k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\theta^{k_{b},k_{a}}(t,q)$
$+z\lambda^{j_{b},b}\mathds{1}_{\\{q+z\leq\bar{q}\\}}H^{b}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q+z)}{z}\right)+z\lambda^{j_{a},a}\mathds{1}_{\\{q-z\geq-\bar{q}\\}}H^{a}\left(\frac{\theta^{j_{b},j_{a}}(t,q)-\theta^{j_{b},j_{a}}(t,q-z)}{z}\right)=0$

with terminal condition $\theta^{j_{b},j_{a}}(T,q)=0$, where $\bar{q}>0$
corresponds to the risk limit, i.e. the market maker refuses any trade that
would bring the inventory out of the interval $[-\bar{q},\bar{q}]$. If
$\bar{q}$ is large enough, this has almost no impact on the bid and ask quotes
of the market maker at $q=0$ that are used to compute FTPs.

Euler schemes can be time-consuming when the number of states $m_{b}\times
m_{a}$ is large, or even unfeasible if the one-asset market making model is
replaced by a multi-asset one. Using the same approach as in [6], we propose
in the following paragraphs a quadratic approximation of the value functions.

Let us replace the Hamiltonian functions $H^{b}$ and $H^{a}$ by the quadratic
functions

$\check{H}^{b}:p\mapsto\alpha^{b}_{0}+\alpha^{b}_{1}p+\frac{1}{2}\alpha^{b}_{2}p^{2}\quad\textrm{and}\quad\check{H}^{a}:p\mapsto\alpha^{a}_{0}+\alpha^{a}_{1}p+\frac{1}{2}\alpha^{a}_{2}p^{2}.$

A natural choice for the functions $\check{H}^{b}$ and $\check{H}^{a}$ derives
from Taylor expansions around $p=0$. In that case, we have

$\forall
i\in\\{0,1,2\\},\quad\alpha^{b}_{i}={H^{b}}^{(i)}(0)\quad\textrm{and}\quad\alpha^{a}_{i}={H^{a}}^{(i)}(0).$

For $(j_{b},j_{a})\in\\{1,\ldots,m_{b}\\}\times\\{1,\ldots,m_{a}\\}$, we
denote by $\check{\theta}^{j_{b},j_{a}}$ the approximation of
$\theta^{j_{b},j_{a}}$ associated with the functions $\check{H}^{b}$ and
$\check{H}^{a}$. The functions $(\check{\theta}^{j_{b},j_{a}})_{1\leq
j_{b}m_{b},1\leq j_{a}\leq m_{a}}$ verify

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{t}\check{\theta}^{j_{b},j_{a}}(t,q)+\kappa(\lambda^{j_{a},a}-\lambda^{j_{b},b})q-\frac{1}{2}\gamma\sigma^{2}q^{2}$
$\displaystyle+\sum_{1\leq k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}\check{\theta}^{k_{b},k_{a}}(t,q)+z\left(\lambda^{j_{b},b}\alpha^{b}_{0}+\lambda^{j_{a},a}\alpha^{a}_{0}\right)$
$\displaystyle+\left(\lambda^{j_{b},b}\alpha^{b}_{1}\left(\check{\theta}^{j_{b},j_{a}}(t,q)-\check{\theta}^{j_{b},j_{a}}(t,q+z)\right)+\lambda^{j_{a},a}\alpha^{a}_{1}\left(\check{\theta}^{j_{b},j_{a}}(t,q)-\check{\theta}^{j_{b},j_{a}}(t,q-z)\right)\right)$
$\displaystyle+\frac{1}{2z}\left(\lambda^{j_{b},b}\alpha^{b}_{2}\left(\check{\theta}^{j_{b},j_{a}}(t,q)-\check{\theta}^{j_{b},j_{a}}(t,q+z)\right)^{2}+\lambda^{j_{a},a}\alpha^{a}_{2}\left(\check{\theta}^{j_{b},j_{a}}(t,q)-\check{\theta}^{j_{b},j_{a}}(t,q-z)\right)^{2}\right),$

and of course we consider the terminal condition
$\check{\theta}^{j_{b},j_{a}}(T,q)=0$.

To write the approximations of the value functions in a simple way, let us
introduce for $i\in\\{0,1,2\\}$ and $k\in\mathbb{N}$

$\Delta_{i,k}^{b}=\alpha^{b}_{i}z^{k}\quad\text{and}\quad\Delta_{i,k}^{a}=\alpha^{a}_{i}z^{k}.\vspace{-0.1cm}$

For $(j_{b},j_{a})\in\\{1,\ldots,m_{b}\\}\times\\{1,\ldots,m_{a}\\}$, let us
consider three differentiable functions $A_{j_{b},j_{a}}:[0,T]\to\mathbb{R}$,
$B_{j_{b},j_{a}}:[0,T]\to\mathbb{R}$, and $C_{j_{b},j_{a}}:[0,T]\to\mathbb{R}$
solutions of the system of ordinary differential equations

$\begin{cases}\displaystyle{A_{j_{b},j_{a}}^{\prime}}(t)=&2\left(\lambda^{j_{b},b}\Delta^{b}_{2,1}+\lambda^{j_{a},a}\Delta^{a}_{2,1}\right)A_{j_{b},j_{a}}(t)^{2}-\frac{1}{2}\gamma\sigma^{2}\\\
&-\sum_{1\leq k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}A_{k_{b},k_{a}}(t)\\\
{B_{j_{b},j_{a}}^{\prime}}(t)=&2\left(\lambda^{j_{b},b}\Delta^{b}_{1,1}-\lambda^{j_{a},a}\Delta^{a}_{1,1}\right)A_{j_{b},j_{a}}(t)+2\left(\lambda^{j_{b},b}\Delta^{b}_{2,2}-\lambda^{j_{a},a}\Delta^{a}_{2,2}\right)A_{j_{b},j_{a}}(t)^{2}\\\
&+\kappa(\lambda^{j_{a},a}-\lambda^{j_{b},b})+2\left(\lambda^{j_{b},b}\Delta^{b}_{2,1}+\lambda^{j_{a},a}\Delta^{a}_{2,1}\right)A_{j_{b},j_{a}}(t)B_{j_{b},j_{a}}(t)\\\
&-\sum_{1\leq k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}B_{k_{b},k_{a}}(t)\\\
{C_{j_{b},j_{a}}^{\prime}}(t)=&\left(\lambda^{j_{b},b}\Delta^{b}_{0,1}+\lambda^{j_{a},a}\Delta^{a}_{0,1}\right)+\left(\lambda^{j_{b},b}\Delta^{b}_{1,2}+\lambda^{j_{a},a}\Delta^{a}_{1,2}\right)A_{j_{b},j_{a}}(t)\\\
&+\left(\lambda^{j_{b},b}\Delta^{b}_{1,1}-\lambda^{j_{a},a}\Delta^{a}_{1,1}\right)B_{j_{b},j_{a}}(t)+\frac{1}{2}\left(\lambda^{j_{b},b}\Delta^{b}_{2,3}+\lambda^{j_{a},a}\Delta^{a}_{2,3}\right)A_{j_{b},j_{a}}(t)^{2}\\\
&+\frac{1}{2}\left(\lambda^{j_{b},b}\Delta^{b}_{2,1}+\lambda^{j_{a},a}\Delta^{a}_{2,1}\right)B_{j_{b},j_{a}}(t)^{2}+\left(\lambda^{j_{b},b}\Delta^{b}_{2,2}-\lambda^{j_{a},a}\Delta^{a}_{2,2}\right)A_{j_{b},j_{a}}(t)B_{j_{b},j_{a}}(t)\\\
&-\sum_{1\leq k_{b}\leq m_{b},1\leq k_{a}\leq
m_{a}}Q_{(j_{b}-1)m_{a}+j_{a},(k_{b}-1)m_{a}+k_{a}}C_{k_{b},k_{a}}(t),\end{cases}$

with terminal conditions $A_{j_{b},j_{a}}(T)=0$, $B_{j_{b},j_{a}}(T)=0$ and
$C_{j_{b},j_{a}}(T)=0$.

Then, for all
$(j_{b},j_{a})\in\\{1,\ldots,m_{b}\\}\times\\{1,\ldots,m_{a}\\}$, we have:

$\check{\theta}^{j_{b},j_{a}}(t,q)=-q^{2}A_{j_{b},j_{a}}(t)-qB_{j_{b},j_{a}}(t)-C_{j_{b},j_{a}}(t).\vspace{-0.1cm}$

Moreover, asymptotic results on value functions continue to hold on their
approximations.

This kind of approximations has been used in [4, 5] with great success in
terms of risk management. We investigate the quality of the approximation in
terms of FTPs below.

####  4.4.3 FTP in practice

To compute FTPs as proposed in Section 3.2.1, we still have to choose the risk
aversion of the theoretical market maker. A natural way to choose $\gamma$ is
to calibrate it to composite bid and ask prices, i.e. we assume that the
quotes of the theoretical market maker correspond to that of the market when
inventory is equal to $0$.

The optimal strategy of the theoretical market maker is obtained by solving
numerically the HJB equation, using an implicit Euler scheme, and through a
quadratic approximation of the value function. Depending on the numerical
method we use, $\gamma$ calibrated to composite bid and ask prices takes
different values.222222The values of $\gamma$ vary across bonds. This comes
from our choice of a simple market making model to illustrate our concepts.
However, in terms of FTP, the results obtained with the two numerical methods
are almost identical, as shown in Table 4 (FTP (a) corresponds to the Euler
scheme and FTP (b) to the quadratic approximation). As for micro-prices, in
practice, one can never be certain to be in any given state, and the FTP has
to be computed as an expectation over the different possible states, depending
on the current estimate. Therefore the FTPs exhibited in Table 4 correspond to
bounds for the FTPs that would be used in practice. Notice that the
adjustments given by FTPs are of lower magnitude than those suggested by
micro-prices. As for micro-prices, we study how FTPs evolve depending on the
probabilities. Figure 16 documents FTPs as a function of $\pi^{0,1}$, when
$\pi^{1,0}=0$ while Figure 17 documents FTPs as a function of $\pi^{0,1}$,
when $\pi^{1,0}=0.3$. We see that adjustments are always small. This is linked
to the fact that, even when the market is imbalanced, market makers can
slightly skew their quotes to deter risk-increasing trades and transform
requests into trades when trades would result in a less risky position (less
inventory in absolute value in our case). This strongly relies on our implicit
assumption that S-curves are the same independently of the liquidity regime.
However, we found no empirical evidence of the influence of intensities on
fill rates.

Bond |  $\gamma$ (a) |  $\gamma$ (b) | Bid price | Ask price |  $\pi^{\\!1,0}\\!\\!=\\!\\!1\\!\\!\\!$ : FTP (a) | FTP (b) |  $\pi^{\\!0,1}\\!\\!=\\!\\!1\\!\\!\\!$ : FTP (a) | FTP (b)
---|---|---|---|---|---|---|---|---
1.1 | $4.5\cdot 10^{-9}$ | $5.1\cdot 10^{-9}$ | 103.098 | 104.088 | 103.458 | 103.458 | 103.728 | 103.729
1.2 | $8.9\cdot 10^{-9}$ | $9.1\cdot 10^{-9}$ | 96.514 | 97.600 | 97.092 | 97.092 | 97.122 | 97.122
1.3 | $4.4\cdot 10^{-8}$ | $5.2\cdot 10^{-8}$ | 98.631 | 99.661 | 99.038 | 99.037 | 99.254 | 99.255
1.4 | $8.5\cdot 10^{-7}$ | $1.6\cdot 10^{-6}$ | 93.049 | 95.325 | 94.167 | 94.172 | 94.207 | 94.202
2.1 | $6.1\cdot 10^{-8}$ | $6.9\cdot 10^{-8}$ | 99.291 | 100.355 | 99.682 | 99.681 | 99.964 | 99.965
2.2 | $7.0\cdot 10^{-8}$ | $8.3\cdot 10^{-8}$ | 98.603 | 99.936 | 99.106 | 99.104 | 99.433 | 99.435
2.3 | $1.1\cdot 10^{-7}$ | $1.2\cdot 10^{-7}$ | 98.815 | 100.483 | 99.554 | 99.553 | 99.743 | 99.744
2.4 | $1.3\cdot 10^{-7}$ | $1.6\cdot 10^{-7}$ | 97.570 | 100.235 | 98.824 | 98.824 | 98.981 | 98.981
3.1 | $4.9\cdot 10^{-7}$ | $5.6\cdot 10^{-7}$ | 94.674 | 96.001 | 95.195 | 95.193 | 95.480 | 95.482
3.2 | $6.1\cdot 10^{-7}$ | $7.6\cdot 10^{-7}$ | 91.860 | 92.927 | 92.364 | 92.365 | 92.423 | 94.422
3.3 | $7.0\cdot 10^{-7}$ | $9.6\cdot 10^{-7}$ | 96.484 | 97.790 | 97.104 | 97.107 | 97.169 | 97.166
3.4 | $4.3\cdot 10^{-7}$ | $7.7\cdot 10^{-7}$ | 94.220 | 95.458 | 94.815 | 94.824 | 94.860 | 94.851
4.1 | $1.2\cdot 10^{-7}$ | $1.3\cdot 10^{-7}$ | 102.151 | 103.112 | 102.523 | 102.525 | 102.740 | 102.738
4.2 | $1.3\cdot 10^{-7}$ | $1.7\cdot 10^{-7}$ | 104.327 | 105.242 | 104.691 | 104.701 | 104.878 | 104.868
4.3 | $1.8\cdot 10^{-7}$ | $2.2\cdot 10^{-7}$ | 104.293 | 105.355 | 104.697 | 104.706 | 104.951 | 104.942
4.4 | $1.5\cdot 10^{-8}$ | $1.6\cdot 10^{-8}$ | 107.991 | 108.884 | 108.377 | 108.377 | 108.498 | 108.498
Table 4: FTP for the different bonds in imbalanced states.

(a) Bond 1.1

(b) Bond 2.1

(c) Bond 3.1

(d) Bond 4.1

Figure 16: FTP for different bonds as a function of $\pi^{0,1}$ when
$\pi^{1,0}=0$.

(a) Bond 1.1

(b) Bond 2.1

(c) Bond 3.1

(d) Bond 4.1

Figure 17: FTP for different bonds as a function of $\pi^{0,1}$ when
$\pi^{1,0}=0.3$.

## Conclusion

In this paper, we developed a new approach to model liquidity in corporate
bond markets. This approach, which can be used in any OTC market where clients
request dealers for prices, relies on Markov-modulated Poisson processes. The
statistical estimation procedure we proposed is based on an EM algorithm and
can be used either at the bond level or at a more macroscopic level (at the
sector level for instance). Although asymmetric states are hard to identify
with great confidence, we showed that flow imbalances contain information
about the evolution of the price and use flow asymmetries to generalize the
notion of micro-price proposed by Stoikov in the context of markets organized
around limit order books. We also coined a new concept inspired by the recent
market making literature: Fair Transfer Price. It is related to the quotes
proposed by a market maker who takes flow imbalances into account and
therefore projects liquidity asymmetries onto the price space. We noticed that
the price adjustments associated with FTP are often small, smaller than those
associated with micro-prices.

## Acknowledgment

This research has been conducted with the support of J.P. Morgan and under the
aegis of the Institut Louis Bachelier. The ideas presented in this paper do
not necessarily reflect the views or practices at J.P. Morgan. The authors
would like to thank Morten Andersen (J.P. Morgan), Gabriele Butti (J.P.
Morgan), and Nabil Nouaman (J.P. Morgan) for the numerous and insightful
discussions they had on the subject.

## Data availability statement

Due to confidentiality reasons, the data used in this article cannot be made
publicly available.

## References

  * [1] Marco Avellaneda and Sasha Stoikov.  High-frequency trading in a limit order book.  Quantitative Finance, 8(3):217–224, 2008.
  * [2] Bastien Baldacci, Philippe Bergault, Joffrey Derchu, and Mathieu Rosenbaum.  On bid and ask side-specific tick sizes.  to appear in SIAM Journal on Financial Mathematics, 2024.
  * [3] Alexander Barzykin, Philippe Bergault, and Olivier Guéant.  Algorithmic market making in dealer markets with hedging and market impact.  Mathematical Finance, 33(1):41–79, 2023.
  * [4] Alexander Barzykin, Philippe Bergault, and Olivier Guéant.  Market making by an foreign exchange dealer.  Risk Magazine, August 2022.
  * [5] Alexander Barzykin, Philippe Bergault, and Olivier Guéant.  Dealing with multi-currency inventory risk in foreign exchange cash markets.  Risk Magazine, March 2023.
  * [6] Philippe Bergault, David Evangelista, Olivier Guéant, and Douglas Vieira.  Closed-form approximations in multi-asset market making.  Applied Mathematical Finance, 28(2):101–142, 2021.
  * [7] Philippe Bergault and Olivier Guéant.  Size matters for otc market makers: general results and dimensionality reduction techniques.  Mathematical Finance, 31(1):279–322, 2021.
  * [8] Julius Bonart and Fabrizio Lillo.  A continuous and efficient fundamental price on the discrete order book grid.  Physica A: Statistical Mechanics and its Applications, 503:698–713, 2018.
  * [9] Alvaro Cartea, Ryan Donnelly, and Sebastian Jaimungal.  Enhancing trading strategies with order book signals.  Applied Mathematical Finance, 25(1):1–35, 2018.
  * [10] Álvaro Cartea, Sebastian Jaimungal, and José Penalva.  Algorithmic and high-frequency trading.  Cambridge University Press, 2015.
  * [11] Álvaro Cartea, Sebastian Jaimungal, and Jason Ricci.  Buy low, sell high: A high frequency trading perspective.  SIAM Journal on Financial Mathematics, 5(1):415–444, 2014.
  * [12] Rama Cont and Adrien De Larrard.  Price dynamics in a Markovian limit order market.  SIAM Journal on Financial Mathematics, 4(1):1–25, 2013.
  * [13] Khalil Dayri and Mathieu Rosenbaum.  Large tick assets: implicit spread and optimal tick size.  Market Microstructure and Liquidity, 1(01):1550003, 2015.
  * [14] Sylvain Delattre, Christian Y. Robert, and Mathieu Rosenbaum.  Estimating the efficient price from the order flow: a Brownian Cox process approach.  Stochastic Processes and their Applications, 123(7):2603–2619, 2013\.
  * [15] Jens Dick-Nielsen.  Liquidity biases in TRACE.  The Journal of Fixed Income, 19(2):43–55, 2009.
  * [16] Jens Dick-Nielsen.  How to clean enhanced TRACE data.  Available at SSRN 2337908, 2014.
  * [17] Wolfgang Fischer and Kathleen Meier-Hellstern.  The markov-modulated poisson process (mmpp) cookbook.  Performance evaluation, 18(2):149–171, 1993.
  * [18] Lawrence R. Glosten and Paul R. Milgrom.  Bid, ask and transaction prices in a specialist market with heterogeneously informed traders.  Journal of financial economics, 14(1):71–100, 1985.
  * [19] Martin D. Gould and Julius Bonart.  Queue imbalance as a one-tick-ahead price predictor in a limit order book.  Market Microstructure and Liquidity, 2(02):1650006, 2016.
  * [20] Olivier Guéant, Charles-Albert Lehalle, and Joaquin Fernandez-Tapia.  Dealing with the inventory risk: a solution to the market making problem.  Mathematics and financial economics, 7(4):477–507, 2013.
  * [21] Olivier Guéant and Jiang Pu.  Mid-price estimation for European corporate bonds: a particle filtering approach.  Market microstructure and liquidity, 4(01n02):1950005, 2018.
  * [22] Olivier Guéant.  The Financial Mathematics of Market Liquidity: From optimal execution to market making, volume 33.  CRC Press, 2016.
  * [23] Olivier Guéant.  Optimal market making.  Applied Mathematical Finance, 24(2):112–154, 2017.
  * [24] Olivier Guéant and Iuliia Manziuk.  Optimal control on graphs: existence, uniqueness, and long-term behavior.  ESAIM: Control, Optimisation and Calculus of Variations, 26:22, 2020\.
  * [25] Thibault Jaisson.  Liquidity and impact in fair markets.  Market Microstructure and Liquidity, 1(02):1550010, 2015.
  * [26] Charles-Albert Lehalle and Othmane Mounjid.  Limit order strategic placement with adverse selection risk and the role of latency.  Market Microstructure and Liquidity, 3(01):1750009, 2017.
  * [27] Ananth Madhavan, Matthew Richardson, and Mark Roomans.  Why do security prices change? A transaction-level analysis of NYSE stocks.  The Review of Financial Studies, 10(4):1035–1064, 1997.
  * [28] Kathleen S Meier-Hellstern.  A fitting algorithm for markov-modulated poisson processes having two arrival rates.  European Journal of Operational Research, 29(3):370–377, 1987.
  * [29] Roel Oomen.  Last look.  Quantitative Finance, 17(7):1057–1070, 2017.
  * [30] Christian Y. Robert and Mathieu Rosenbaum.  A new approach for the dynamics of ultra-high-frequency data: The model with uncertainty zones.  Journal of Financial Econometrics, 9(2):344–366, 2011.
  * [31] Tobias Rydén.  An EM algorithm for estimation in Markov-modulated Poisson processes.  Computational Statistics & Data Analysis, 21(4):431–447, 1996\.
  * [32] Sasha Stoikov.  The micro-price: a high-frequency estimator of future prices.  Quantitative Finance, 18(12):1959–1966, 2018.
  * [33] Sasha Stoikov, Peter Decrem, Yikai Hua, and Anne Shen.  The Microstructure of Cointegrated Assets.  Available at SSRN 3824298, 2021.