# Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of
Pricing Functions

Philippe Bergault111Université Paris Dauphine-PSL, Ceremade, 75116, Paris,
France<EMAIL_ADDRESS>Louis Bertucci222Institut Louis
Bachelier, 75002, Paris, France<EMAIL_ADDRESS>David Bouba333Swaap Labs<EMAIL_ADDRESS>Olivier Guéant444Université Paris 1
Panthéon-Sorbonne, Centre d’Economie de la Sorbonne, 106 Boulevard de
l’Hôpital, 75642 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract

With the emergence of decentralized finance, new trading mechanisms called
Automated Market Makers have appeared. The most popular Automated Market
Makers are Constant Function Market Makers. They have been studied both
theoretically and empirically. In particular, the concept of impermanent loss
has emerged and explains part of the profit and loss of liquidity providers in
Constant Function Market Makers. In this paper, we propose another mechanism
in which price discovery does not solely rely on liquidity takers but also on
an external exchange rate or price oracle. We also propose to compare the
different mechanisms from the point of view of liquidity providers by using a
mean / variance analysis of their profit and loss compared to that of agents
holding assets outside of Automated Market Makers. In particular, inspired by
Markowitz’ modern portfolio theory, we manage to obtain an efficient frontier
for the performance of liquidity providers in the idealized case of a perfect
oracle. Beyond that idealized case, we show that even when the oracle is
lagged and in the presence of adverse selection by liquidity takers and
systematic arbitrageurs, optimized oracle-based mechanisms perform better than
popular Constant Function Market Makers.

Key words: Automated market makers, cryptocurrencies, DeFi, oracles,
stochastic optimal control.

##  1 Introduction

Since the early days of the Decentralized Finance (DeFi) era, Automated Market
Makers (AMMs) have been some of the largest DeFi protocols on public
blockchains. Although it is nothing more than a smart contract on a
blockchain, an AMM should be regarded, from a financial point of view, as a
liquidity pool of two assets555There exist AMMs functioning with pools
involving more than two assets, but we focus on the two-asset case throughout
this paper. involving two types of agents: liquidity providers (referred to as
LPs) and liquidity takers (referred to as LTs or, more simply, traders). LPs
supply reserves to the pool, usually in both assets of the pool. In exchange,
they become entitled to a share of the pool that is in line with their supply.
LTs, in turn, use AMMs to trade, that is to swap a given quantity of one asset
against the other. The exchange rate proposed by an AMM is typically based on
a pre-defined and public formula that depends on the reserves, the transaction
size and the direction of the swap. This function is often called the pricing
function, and, although it must be defined in the contract, it can use
external data through what is called oracles.

AMMs constitute a new paradigm beyond (i) that of dealer markets (like the
global FX market, see [62, 63]) where specific agents – usually banks –
provide liquidity to clients and hold risk on their balance sheet, (ii) that
of classical order-driven markets organized around limit order books (like
most stock markets), and (iii) that of dark pools introduced in the last
decades (see [51]). Unlike in dealer markets, any agent can provide liquidity
through an AMM. Unlike in limit order books, prices are automatically set by
the protocol. Unlike in dark pools, the available liquidity is visible and the
price is not defined solely by importing that of another venue. Most
importantly, LPs do not provide liquidity to LTs in the case of AMMs: LPs
provide liquidity to the pool and LTs take liquidity from the pool.666In
particular (like for most DeFi applications) AMMs are fully collateralized and
neither LPs nor LTs are exposed to any sort of counterparty risk. To be more
precise, LPs are exposed to a technological – or cyber – risk, but as long as
the smart contract works as expected, they will be able to withdraw their
share of the reserves. The novelty of AMMs raises a lot of theoretical
questions from an economic point of view, in relation to the classical market
microstructure literature, but also optimization and quantitative questions
related to the quantitative finance and financial engineering literature.

This paper is a contribution to the quantitative finance literature on AMMs
focusing on the relation between the design of AMMs and the performance of
LPs. Using standard convex analysis, we recover the now well-known result
that, in the absence of transaction fees, posting liquidity into a Constant
Function Market Maker777We refer to [59] for a pedagogical introduction to
CFMMs and the classical pricing functions. Examples of popular CFMMs include
Uniswap (see [2, 3, 4]), Balancer (see [56]), Curve (see [35, 36]), etc. An
interesting empirical analysis of Uniswap v3 is [54]. (CFMM) exposes to a
concave payoff that is inferior to that of holding coins outside of the
pool.888This led to the now classical concept of impermanent loss that is
widely used in the DeFi world. The “impermanent” trait of such loss lies in
the fact that losses vanish when the exchange rate reverts back to its
original value (at the time the liquidity was provided). This nonpositive and
concave payoff comes from the fact that price discovery in CFMMs is left to
LTs who act as arbitrageurs and make money out of the pool. Transaction fees
should therefore compensate a risk that is fundamentally related to the design
of CFMMs and competition between CFMMs cannot reduce transaction fees below a
threshold that depends on market conditions. We therefore plead in this paper
for the design of oracle-based AMMs that are more complicated than CFMMs in
that they do not solely rely on LTs for price discovery.

In our paper, we explore indeed AMMs in which the pricing function uses
external information about current market prices. Even if a blockchain only
has knowledge about on-chain activity, it is possible to feed a smart contract
with external data through oracles (see [27] for a discussion). This situation
is quite similar to that of a traditional market maker in dealer markets who
provides quotes based on an estimation of the price or exchange rate
(typically a mid-price imported from an electronic platform or a composite
price) that is available at the same time. Instead of being entirely passive,
the AMM can update its bid and offer not only after a trade or a provision or
redemption of liquidity, but also after the oracle has fired a price update.
One difference with the traditional finance case is however that a dealer in
FX or corporate bond markets can have and frequently has short positions
whereas liquidity must always be present in the pool in the case of an AMM.

In order to compare different AMMs (from the point of view of LPs999Due to the
challenges posed by impermanent loss, much of the existing literature has
sought to optimize AMM designs with a focus on LPs. Our work aligns with this
perspective, drawing additional inspiration from research works on optimal
market making in OTC markets, which predominantly consider the dealers’
standpoint. It is worth noting, however, that an economic perspective on AMM
design should ideally encompass both LTs and LPs. In the below literature
review, there are a few papers tackling equilibrium issues encompassing both
LPs and LTs but not with designs like ours.), we build in this paper a simple
mean-variance framework inspired by the so-called modern portfolio theory of
the 1950s. However, unlike in the Markowitz case, we always consider the
Hodl101010Hodl is slang in the cryptocurrency community for holding a
cryptocurrency. In traditional finance, this would be called “buy and hold”.
strategy as a benchmark. An agent faces indeed an alternative: providing
liquidity by posting coins into an AMM or holding them. The Hodl benchmark is
also intimately linked to the notion of impermanent loss that is ubiquitous in
the field.

We also borrow from the modern portfolio theory the concept of efficient
portfolios which becomes in our case efficient market making strategies, i.e.
efficient pricing functions. Indeed, in addition to comparing the risk /
return profile of different existing AMMs, we are interested in computing the
maximum extra return that a LP could expect for a given level of tracking
error with respect to Hodl.

Although the efficient frontier cannot be computed in closed form, it can
easily be approximated using a model inspired by the modern literature on
market making. Interestingly, we show that popular CFMMs exhibit poor
performances relatively to our approximation of the theoretical efficient
frontier in a complete information framework. However, even in the presence of
a lagged oracle and adverse selection, an optimized oracle-based AMM is able
to get close to that theoretical efficient frontier.

In recent years, many research works have been carried out on AMMs, especially
CFMMs, with a special focus on Constant Product Market Makers (CPMMs) and
their generalizations. Motivated by the emergence of Uniswap and the volume
exchanged through it, the authors of [11] studied the main properties of
CPMMs. In particular, they showed that, in the presence of arbitrageurs, the
exchange rate proposed by a CPMM for small transactions should be in a range
around that of competing venues – the width of the range being determined by
transaction fees. Then, they studied the main properties of CPMMs: the no-
splitting property, the no-depletion property, etc. They also studied the
payoff of LPs in a CPMM as a function of an external asset price when there is
no transaction fees (see also [33] and an extension to the case of liquidity
concentration as in Uniswap v3 in [34]). The returns of LPs are also studied
in [24] and [37] for pricing functions that are geometric means. In
particular, impermanent loss is studied for several pricing functions.

Beyond CPMMs, CFMMs have been studied by the authors of [7] in a general
multi-coin setting. They introduced a natural notion of trading set and showed
that its convexity is key to obtain desirable properties. In the no-
transaction-fee case, they also obtained a formula for the value of the pool
that involves Legendre-Fenchel transforms (see also [9] and Section 2 for a
discussion). In particular, they showed that the returns of LPs between two
liquidity provision or redemption events suffered from impermanent loss for
all CFMMs. The same group of authors with additional coauthors then shed a new
light on their previous works in [6] by embedding several problems involving
CFMMs in a convex setting.

A recent and important work to better understand the payoff of LPs in CFMMs is
[57]. Because the value of a CFMM is a concave function of the exchange rate
between the two assets, Itô’s formula leads to a decomposition111111When the
price process is a martingale this is the Doob-Meyer decomposition of the
payoff. of the payoff of LPs into two terms: a stochastic integral
corresponding to the payoff of a self-financing strategy and a nonincreasing
and nonpositive term. They use this decomposition to claim that, at least
theoretically, part of the risk can be hedged away. In particular, if
continuous-time hedging with no friction was possible and if an AMM or an LP
implemented the hedging strategy, only part of the impermanent loss would
remain that they call Loss-Versus-Rebalancing (LVR).121212In the very recent
paper [58] they added transaction costs and discrete-time trading and obtained
asymptotic results with respect to the frequency of blocks.

Beyond the properties of CFMMs and the returns of LPs, several questions have
also been addressed in the literature. The question of the optimal fees is
discussed in [38]. That of the take rate (the proportion of the fees kept by
the protocol) is addressed in [41]. Strategic liquidity provision is the topic
of several papers. With a viewpoint rooted into the classical Kyle/Glosten-
Milgrom literature on market microstructure, [12] studies liquidity provision
in a CPMM with a focus on equilibrium issues. Equilibrium questions are also
discussed in [48]. The authors of [30] and [60] discuss the impact of
liquidity concentration (as in Uniswap v3) on liquidity provision strategies.
A microstructural and game-theoretical approach is proposed in [26] in which
the authors discuss, among many interesting topics, the influence of
volatility on the adoption of AMMs. The coexistence of limit order books and
AMMs and the competition between them are discussed in [13], [16] and [52].
The execution of orders in CFMMs has also been studied in several papers: [10]
tackles the static problem of optimal routing across a set of several CFMMs
while [29] studies a dynamic problem of optimal execution à la Almgren-Chriss
on a CFMM.

In this paper, we go beyond CFMMs and propose an optimized oracle-based AMM.
Our ideas and the model we use are inspired by the recent literature on market
making. The quantitative literature on market making initially started in the
1980s with the seminal works [49, 50] (see also the papers [5] and [61] from
the 1980s). It was revived in 2008 in [14] where the authors use stochastic
optimal control tools. Following the latter paper, market making has become an
important research strand in quantitative finance over the last decade. The
authors of [44] presented a thorough analysis of the problem introduced in
[14], and proposed closed-form approximations of the optimal quotes in the
case of exponential intensities. Instead of the expected utility framework of
[14], our model uses the objective function introduced in [32]. Since then, a
lot of new features have been progressively added to the initial models: the
impact of parameters ambiguity is studied in [28], in [21] the authors
considered a framework with various trade sizes, etc. Also, specific models
for different asset classes have been proposed: for stock options in [15], for
foreign exchange currencies in [17], [18] and [19], and for bonds in
[45].131313See the books [31] and [42] for a detailed bibliography on market
making. Our model extends the recent market making models tackling the problem
faced by FX dealers, in which exchange rate dynamics are geometric rather than
arithmetic, by considering instead of the total profit and loss (PnL) of the
liquidity provider, only the extra money (positive or negative) beyond Hodl.

The remaining of the paper is organized as follows. In Section 2, we introduce
notation and recall classical results about the returns of LPs in CFMMs in a
concise manner. Section 3 goes beyond the case of CFMMs and derives
approximations of efficient pricing functions for an AMM with complete
information (in particular a perfect exchange rate oracle). Section 4 relaxes
the assumptions of Section 3 to be closer to reality by considering
misspecification of the parameters, a lagged oracle and adverse selection by
arbitrageurs.

##  2 The payoff of LPs in CFMMs: a primer

###  2.1 Notation and definition of the protocol

CFMMs constitute one of the simplest forms of AMMs. In CFMM protocols, the
exchange rate for a given pair of coins or tokens is determined by the
function of (i) the reserves in the liquidity pool and (ii) the size (and
side) of the prospective transaction. In what follows, we recall the
functioning of two-currency CFMMs and the now classical analysis of the PnL of
LPs in these protocols. It is noteworthy that, because ownership over the CFMM
reserves is defined proportionally to the amounts deposited by LPs, one can
consider – without loss of generality – the special case of a 1-LP-only
system.141414As in most papers of the literature, our analysis is valid
between any two liquidity provision/redemption events. In other words, we
assume that liquidity evolves according to the swaps placed by LTs only.

In what follows, we shall denote by $(q_{t}^{0})_{t\geq 0}$ and
$(q_{t}^{1})_{t\geq 0}$ the two processes for the reserves corresponding
respectively to the number of coins of currency 0 and currency 1 in the pool
(time $0$ corresponds to the initial posting of reserves). In the case of a
CFMM with no transaction fees, the proposed exchange rate is typically such
that the quantity $f(q^{0}_{t},q^{1}_{t})$ remains constant151515Of course, in
the case of provision/redemption of liquidity, the quantity changes, hence the
above footnote. before and after a trade where the function
$f:\mathbb{R}_{+}^{*}\times\mathbb{R}_{+}^{*}\rightarrow\mathbb{R}$ is
typically increasing with respect to both of its variables, reflecting the
intent to swap one currency for the other. For what follows, it is more
practical to use an alternative formulation based on level sets. We assume
that reserves always satisfy the equation $q_{t}^{0}=\psi(q_{t}^{1})$ where
the function $\psi:\mathbb{R}_{+}^{*}\to\mathbb{R}_{+}^{*}$ satisfies the
following properties:

  * •

$\psi$ is decreasing – this reflects that one currency is swapped for the
other

  * •

$\lim_{q^{1}\to 0^{+}}\psi(q^{1})=+\infty$ and
$\lim_{q^{1}\to+\infty}\psi(q^{1})=0$ – to prevent depletion of one of the two
currencies

  * •

$\psi$ is strictly convex – convexity161616The strictness is important only to
differentiate the Legendre transform of $\psi$ below. guarantees the absence
of arbitrage opportunities

  * •

$\psi$ is continuously differentiable – to simplify the analysis (it is the
case in practice).

In this setting, if a client wants to sell $\Delta q^{1}>0$ coins of currency
1 to the pool at date $t$, he/she will receive $\Delta q^{0}$ coins of
currency 0 where

$q_{t}^{0}-\Delta q^{0}=\psi(q_{t}^{1}+\Delta
q^{1}),\quad\text{i.e.}\quad\frac{\Delta q^{0}}{\Delta
q^{1}}=-\frac{\psi(q_{t}^{1}+\Delta q^{1})-\psi(q_{t}^{1})}{\Delta q^{1}}.$

Symmetrically, if a client wants to buy $\Delta q^{1}>0$ coins of currency 1
from the pool at date $t$, he/she will pay $\Delta q^{0}$ coins of currency 0
where

$q_{t}^{0}+\Delta q^{0}=\psi(q_{t}^{1}-\Delta
q^{1}),\quad\text{i.e.}\quad\frac{\Delta q^{0}}{\Delta
q^{1}}=\frac{\psi(q_{t}^{1}-\Delta q^{1})-\psi(q_{t}^{1})}{\Delta q^{1}}.$

Of course, the convexity of $\psi$ ensures that

$-\frac{\psi(q_{t}^{1}+\Delta q^{1})-\psi(q_{t}^{1})}{\Delta
q^{1}}\leq\frac{\psi(q_{t}^{1}-\Delta q^{1})-\psi(q_{t}^{1})}{\Delta q^{1}},$

thereby precluding the existence of arbitrage opportunities within the pool.

###  2.2 No-arbitrage assumption and PnLs

Assuming there exists at time $t$ an external market exchange rate $S_{t}$ for
currency 1 (in terms of currency $0$) at which infinitesimal quantities could
be traded, we clearly see from the above equations that there would be an
arbitrage opportunity at time $t$ if $S_{t}$ was not equal to
$-\psi^{\prime}(q_{t}^{1})$. Neglecting the existence of bid-ask spreads even
for tiny transactions, we write $S_{t}=-\psi^{\prime}(q_{t}^{1})$ and decide
to evaluate (in currency 0 terms) any position in currency 1 with exchange
rate $S_{t}$.

In such an idealized setting, the PnL at time $t$ (in currency 0
terms)171717The PnL needs to be accounted in a given currency. We arbitrarily
choose currency 0 in what follows. of the representative LP is therefore

$\text{PnL}_{t}=\left(q_{t}^{0}+S_{t}q_{t}^{1}\right)-\left(q_{0}^{0}+S_{0}q_{0}^{1}\right)$

while that of the same agent who would not have posted reserves in the pool
would be

$\text{PnL}^{\text{Hodl}}_{t}=\left(q_{0}^{0}+S_{t}q_{0}^{1}\right)-\left(q_{0}^{0}+S_{0}q_{0}^{1}\right)=(S_{t}-S_{0})q_{0}^{1}.$

To compare those two PnLs, let us define $\psi^{*}$ the Legendre-Fenchel
transform181818We restrict the function to the interior of its domain. of
$\psi$, i.e.

$\psi^{*}:p\in\mathbb{R}^{*}_{-}\mapsto\sup_{q\in\mathbb{R}^{*}_{+}}pq-\psi(q).$

The maximizer $q$ in the definition of $\psi^{*}(p)$ is uniquely defined by
$\psi^{\prime}(q)=p$ and it is hence $q_{t}^{1}$ when $p=-S_{t}$. We therefore
obtain

$\psi^{*}(-S_{t})=-q_{t}^{1}S_{t}-\psi(q_{t}^{1})=-q_{t}^{1}S_{t}-q_{t}^{0}\quad\text{and}\quad\psi^{*^{\prime}}(-S_{t})=q_{t}^{1}.$

In particular, we have,

$\text{PnL}_{t}=\psi^{*}(-S_{0})-\psi^{*}(-S_{t})\quad\text{and}\quad\text{PnL}^{\text{Hodl}}_{t}=(S_{t}-S_{0})\psi^{*^{\prime}}(-S_{0}).$

Because $\psi^{*}$ is strictly convex, its graph lies above the tangent line
at all points and we have therefore

$\text{PnL}_{t}-\text{PnL}^{\text{Hodl}}_{t}=\psi^{*}(-S_{0})-\psi^{*}(-S_{t})-(S_{t}-S_{0})\psi^{*^{\prime}}(-S_{0})\leq
0,$

with equality if and only if $S_{t}=S_{0}$. This result corresponds to the
notion of impermanent loss since the loss vanishes if the exchange rate goes
back to its value when reserves were added to the pool.

###  2.3 Analysis and discussion

The above inequality is fundamental as it claims that, under mild assumptions,
there is no benefit in providing liquidity through a CFMM with no fees. In
other words, a minimal amount of fees is necessary to encourage liquidity
provision. In particular, competition between CFMMs cannot decrease fees to
zero.

The above computations prove that the payoff of a LP in a CFMM with no fees is
not only nonpositive but also concave, i.e. similar to that of an option
seller. Assuming that $\psi^{*}$ is twice continuously differentiable and
applying Itô’s formula as in [57], we see that

$\text{PnL}_{t}-\text{PnL}^{\text{Hodl}}_{t}=\int_{0}^{t}(\psi^{*^{\prime}}(-S_{s})-\psi^{*^{\prime}}(-S_{0}))dS_{s}-\frac{1}{2}\int_{0}^{t}\psi^{*^{\prime\prime}}(-S_{s})d\langle
S\rangle_{s}$

where $\langle\cdot\rangle$ denoted quadratic variation. The first term can be
(theoretically) hedged away through a self-financing strategy with
$-(\psi^{*^{\prime}}(-S_{s})-\psi^{*^{\prime}}(-S_{0}))$ coins of asset $1$ at
time $s$, while the second term, called LVR in [57], is always nonpositive
because $\psi^{*}$ is convex.

Regarding the first term, continuous-time hedging with no friction is a
classical theoretical assumption. However, in practice, hedging raises a lot
of questions given the high frequency at which trades happen in AMMs: when
should we hedge? on which venue? should we cross the spread in limit order
books or post limit orders? what about our market impact? etc. Furthermore,
hedging requires to trade on another venue and may be complicated to implement
inside an AMM: LPs should carry out the hedging process by themselves, all the
more since LPs might not want to be hedged at the level of each AMM but rather
at the level of their portfolio to benefit from netting effects. In any case,
one can argue that part of the risk could be hedged away, leading therefore to
a reduction of the fees charged by CFMMs to LTs in order to compensate LPs.
This is an important route for future research in the field.

Regarding the second term, it is intrinsically related to the main problem of
CFMM protocols. In a CFMM, the price discovery process indeed occurs solely
thanks to trades with LTs, and LTs therefore extract value from LPs. To avoid
this value extraction, an interesting idea consists in using a price oracle.
This is the purpose of this paper.

##  3 Efficient pricing functions in the complete information case

###  3.1 Modeling framework

In this section, we consider a filtered probability space
$\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{F}=(\mathcal{F}_{t})_{t\geq
0}\right)$ satisfying the usual conditions.

In the previous section, the automated market making protocol did not rely on
any external information to propose exchange rates. It indeed proposed
exchange rates for various transaction sizes based only on the reserves lying
in the pool at a given time. We now consider the theoretical and idealized
case where an exogenous market exchange rate is known, for instance the mid-
price on a centralized exchange based on a limit order book, like those of
Binance, Kraken or Coinbase. This exchange rate is indicative rather than
tradable (equivalently, we assume that the AMM is not able to trade on other
venues) even for infinitesimal sizes but we still denote by
$(S_{t})_{t\in\mathbb{R}_{+}}$ the market exchange rate process, stating the
value of currency 1 in terms of currency 0. We assume for it the following
dynamics:

$dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t},$

where $\mu\in\mathbb{R}$ is a known deterministic drift, $\sigma>0$ a known
deterministic volatility and $\left(W_{t}\right)_{t\in\mathbb{R}_{+}}$ a
standard Brownian motion.

To study the PnL of LPs, we need assumptions on the demand of LTs. To build a
parsimonious model, we consider as in [19] that transaction sizes are labelled
in the accounting currency (currency 0) and that transactions are decomposed
into two parts: one part corresponding to the market exchange rate and another
part corresponding to a markup (that might very rarely be a discount) that is
accounted in currency 0, whatever the side of the transaction, for the sake of
simplicity. More precisely, if a client wants to buy $z$ coins of currency 0
at time $t$, then $z/S_{t}$ coins of currency 1 will be asked and
$z\delta^{1,0}(t,z)$ out of the total of $z$ coins of currency 0 will not be
transferred to him/her. Symmetrically, if a client wants to sell $z$ coins of
currency 0 at time $t$, then $z/S_{t}$ coins of currency 1 will be offered to
him/her and $z\delta^{0,1}(t,z)$ extra coins of currency 0 will be asked as a
markup.191919$\delta^{0,1}(t,z)$ and $\delta^{1,0}(t,z)$ converted in basis
points, could be regarded as “mid”-to-bid and ask-to-“mid” in basis points.
Everything works indeed almost as if the prices proposed for swapping were
respectively $S_{t}(1-\delta^{1,0}(t,z))$ (bid) and
$S_{t}(1+\delta^{0,1}(t,z))$ (ask). While $S_{t}$ serves as an indicative
price independent of size $z$, the ultimate exchange rate depends on the size
(and direction) of the transaction.

We assume that the markups $\left(\delta^{0,1},\delta^{1,0}\right)$ belong to

$\begin{split}\mathcal{A}:=\Bigg{\\{}\delta=\left(\delta^{0,1},\delta^{1,0}\right):\Omega\times[0,T]&\times\mathbb{R}_{+}^{*}\mapsto\mathbb{R}^{2}\bigg{|}\delta\text{
is }\mathcal{P}\otimes\mathcal{B}(\mathbb{R}_{+}^{*})\text{-measurable }\\\
\qquad\qquad\qquad\qquad&\text{and
}\delta^{0,1}(t,z)\wedge\delta^{1,0}(t,z)\geq-C\ \mathbb{P}\otimes dt\otimes
dz\ a.e.\Bigg{\\}},\end{split}$

for a given (large) constant $C>0$. Here, $\mathcal{P}$ denotes the
$\sigma$-algebra of $\mathbb{F}$-predictable subsets of $\Omega\times[0,T]$
and $\mathcal{B}(\mathbb{R}_{+}^{*})$ denotes the Borelian sets of
$\mathbb{R}_{+}^{*}$.

In the model, to simplify the analysis, we accumulate these markups in a
process $(X_{t})_{t\in[0,T]}$, separated from the reserves. Its dynamics is

$dX_{t}=\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{0,1}(t,z)J^{0,1}(dt,dz)+\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{1,0}(t,z)J^{1,0}(dt,dz),$

with $X_{0}=0$, where $J^{0,1}(dt,dz)$ and $J^{1,0}(dt,dz)$ are two
$\mathbb{R}_{+}^{*}$-marked point processes modelling transactions through
which the AMM sells currency 1 and receives currency 0 (for $J^{0,1}(dt,dz)$)
and transactions through which the AMM sells currency 0 and receives currency
1 (for $J^{1,0}(dt,dz)$).

These marked point processes also allow to write the dynamics of the reserves:

$dq^{0}_{t}=\int_{z\in\mathbb{R}_{+}^{*}}z\left(J^{0,1}(dt,dz)-J^{1,0}(dt,dz)\right)\quad\text{and}\quad
dq^{1}_{t}=\int_{z\in\mathbb{R}_{+}^{*}}\frac{z}{S_{t}}\left(J^{1,0}(dt,dz)-J^{0,1}(dt,dz)\right).$

Because we are interested in the PnL of a LP compared to that of an agent who
would have held the coins outside of the AMM, we introduce the following two
processes:

$\left(Y^{0}_{t}\right)_{t\in\mathbb{R}_{+}}=\left((q^{0}_{t}-q^{0}_{0})\right)_{t\in\mathbb{R}_{+}}\quad\text{and}\quad\left(Y^{1}_{t}\right)_{t\in\mathbb{R}_{+}}=\left((q^{1}_{t}-q^{1}_{0})S_{t}\right)_{t\in\mathbb{R}_{+}}.$

Their dynamics are given by:

$dY^{0}_{t}=\int_{z\in\mathbb{R}_{+}^{*}}\\!\\!z\left(J^{0,1}(dt,dz)-J^{1,0}(dt,dz)\right)\text{
and }dY^{1}_{t}=\mu Y^{1}_{t}dt+\sigma
Y^{1}_{t}dW_{t}+\int_{z\in\mathbb{R}_{+}^{*}}\\!\\!z\left(J^{1,0}(dt,dz)-J^{0,1}(dt,dz)\right).$

We assume that the processes $J^{0,1}(dt,dz)$ and $J^{1,0}(dt,dz)$ have known
intensity kernels given respectively by
$(\nu^{0,1}_{t}(dz))_{t\in\mathbb{R}_{+}}$ and
$(\nu^{1,0}_{t}(dz))_{t\in\mathbb{R}_{+}}$, verifying

$\nu^{0,1}_{t}(dz)=\Lambda^{0,1}\left(z,\delta^{0,1}(t,z)\right)\mathds{1}_{\\{q^{1}_{t-}\geq\frac{z}{S_{t}}\\}}m(dz)\quad\text{and}\quad\nu^{1,0}_{t}(dz)=\Lambda^{1,0}\left(z,\delta^{1,0}(t,z)\right)\mathds{1}_{\\{q^{0}_{t-}\geq
z\\}}m(dz),$

where $m$ is a measure (typically Lebesgue or discrete) and $\Lambda^{0,1}$
and $\Lambda^{1,0}$ are called the intensity functions of the processes
$J^{0,1}(dt,dz)$ and $J^{1,0}(dt,dz)$ respectively. In the standard literature
on market making (see [18] or [21], for instance), these intensity functions
(which correspond to the demand curve of LTs) are assumed of the logistic
type, i.e.

$\Lambda^{0,1}(z,\delta)=\lambda^{0,1}(z)\frac{1}{1+e^{\alpha^{0,1}(z)+\beta^{0,1}(z)\delta}}\quad\text{and}\quad\Lambda^{1,0}(z,\delta)=\lambda^{1,0}(z)\frac{1}{1+e^{\alpha^{1,0}(z)+\beta^{1,0}(z)\delta}},\vspace{-0.1cm}$

where $\lambda^{0,1}(z)m(dz)$ and $\lambda^{1,0}(z)m(dz)$ describe the maximum
number of transactions of size in $[z,z+dz]$ per unit of time (the height of
the demand curve) and $\alpha^{0,1}(z)$, $\beta^{0,1}(z)$, $\alpha^{1,0}(z)$,
and $\beta^{1,0}(z)$ the shape of the demand curve, in particular the
sensitivity to the markups. It is noteworthy that indicator functions
represent the impossibility for the AMM to propose exchange rates for
transactions that cannot occur because reserves are too low in the demanded
currency.

The PnL minus the PnL of Hodl at time $T$, hereafter the excess PnL, of a LP
is therefore given by

$\displaystyle X_{T}+Y^{0}_{T}+Y^{1}_{T}$
$\displaystyle=\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{0,1}(t,z)J^{0,1}(dt,dz)+\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{1,0}(t,z)J^{1,0}(dt,dz)$
$\displaystyle\qquad+\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}\\!\\!z\left(J^{0,1}(dt,dz)-J^{1,0}(dt,dz)\right)+\int_{0}^{T}\mu
Y^{1}_{t}dt+\int_{0}^{T}\sigma Y^{1}_{t}dW_{t}$
$\displaystyle\qquad+\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}\\!\\!z\left(J^{1,0}(dt,dz)-J^{0,1}(dt,dz)\right)$
$\displaystyle=\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{0,1}(t,z)J^{0,1}(dt,dz)+\int_{0}^{T}\int_{z\in\mathbb{R}_{+}^{*}}z\delta^{1,0}(t,z)J^{1,0}(dt,dz)$
$\displaystyle\qquad+\int_{0}^{T}\mu Y^{1}_{t}dt+\int_{0}^{T}\sigma
Y^{1}_{t}dW_{t}.$ (1)

We see that the excess PnL can be decomposed into two parts: one corresponding
to the accumulated markups and the other one representing the deviation from
the Hodl strategy reserves. In particular, it contains jump terms and a
Brownian term.202020As in the recent article [57], one can argue that the term
$\int_{0}^{T}\mu Y^{1}_{t}dt+\int_{0}^{T}\sigma
Y^{1}_{t}dW_{t}=\int_{0}^{T}(q^{1}_{t}-q^{1}_{0})dS_{t}$ could be hedged, at
least theoretically.

###  3.2 Towards an efficient frontier

We now derive the optimal strategy in this framework with complete information
for an objective function that is not exactly a mean-variance one but a more
practical one for us in order to use the tools of stochastic optimal control
theory. In fact, we only consider the variance of $\int_{0}^{T}\sigma
Y^{1}_{t}dW_{t}$ which is a very reasonable proxy for the variance of the
excess PnL, as most of the risk comes from the Brownian term and not from the
drift or the jump terms.

More precisely, for each $\gamma>0$,212121The parameter $\gamma$ can be
interpreted in two ways. One can see it as a risk aversion parameter or as a
Lagrange multiplier, like in the Markowitz framework. we introduce the
following stochastic optimal control problem:

$\sup_{(\delta^{0,1},\delta^{1,0})\in\mathcal{A}}\mathbb{E}\Bigg{[}\int\limits_{0}^{T}\Bigg{\\{}\int_{z\in\mathbb{R}_{+}^{*}}\Big{(}z\delta^{0,1}(t,z)\Lambda^{0,1}(z,\delta^{0,1}(t,z))\mathds{1}_{\\{q^{1}_{t-}\geq\frac{z}{S_{t}}\\}}$
$\qquad\qquad\qquad\qquad+z\delta^{1,0}(t,z)\Lambda^{1,0}(z,\delta^{1,0}(t,z))\mathds{1}_{\\{q^{0}_{t-}\geq
z\\}}\Big{)}m(dz)+\mu
Y^{1}_{t}-\frac{\gamma}{2}\sigma^{2}(Y^{1}_{t})^{2}\Bigg{\\}}dt\Bigg{]}.$

This stochastic optimal control problem has $4$ state variables and is
therefore hardly tractable, even numerically. Nevertheless, it is noteworthy
that for moderate values of $\mu$, the quadratic penalty provides an incentive
to keep the composition of the pool close to the initial one.222222In pratice,
the choice of $\gamma$ should be contingent on the pool size and the level of
liquidity. For example, a higher $\gamma$ value might be suitable when the
pool size is smaller and/or liquidity demand is high. Conversely, a reduced
$\gamma$ could be adopted when the pool is more substantial, and liquidity
demand is lesser. Notably, the value of $\gamma$ could be adjusted at each
time of liquidity provision/redemption. In this paper, as in most of the
literature, we analyse the PnL of LPs between two times of liquidity
provision/redemption and $\gamma$ is fixed. Therefore, the no-depletion
constraints (which translated into indicators in the above formula) can be
regarded as superfluous, and we subsequently approximate the problem by
removing the latter terms.232323In fact, in our numerical examples, we observe
that, with the markups obtained using this approach, the reserves remain
positive at all times, i.e. the constraints are not binding. In other words,
we consider the following modified objective function:

$\displaystyle\mathbb{E}\Bigg{[}\int\limits_{0}^{T}\Bigg{\\{}\int_{z\in\mathbb{R}_{+}^{*}}\Big{(}z\delta^{0,1}(t,z)\Lambda^{0,1}(z,\delta^{0,1}(t,z))+z\delta^{1,0}(t,z)\Lambda^{1,0}(z,\delta^{1,0}(t,z))\Big{)}m(dz)+\mu
Y^{1}_{t}-\frac{\gamma}{2}\sigma^{2}(Y^{1}_{t})^{2}\Bigg{\\}}dt\Bigg{]}.$

It is noteworthy that this new problem only depends on one state variable,
$Y^{1}$, whose dynamic is Markovian. It can be addressed with classical tools
of stochastic optimal control. For that purpose, we introduce the value
function $\theta:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ associated with
this stochastic optimal control problem. It is well known that $\theta$ solves
the following Hamilton-Jacobi-Bellman equation:242424If continuous-time
hedging with no friction was possible as in [57], the HJB equation would be
Eq. (2) with $\mu=0$ and $\sigma=0$. This would lead to $\theta(t,y)=C(T-t)$
for some constant $C$ independent of $y$ and an optimal strategy independent
of $Y^{1}_{t}$. A softer way to consider the possibility to hedge could be, as
in [17], to add a term $\sup_{v}v\partial_{y}\theta-L(v)$ with
$L:v\mapsto\psi|v|+\eta|v|^{1+\phi}$ an execution cost function as in the
optimal execution literature. In particular, the cost of liquidity and the
need to cross the spread would be modeled.

$\begin{cases}\\!&0=\partial_{t}\theta(t,y)+\mu
y\left(1+\partial_{y}\theta(t,y)\right)-\frac{\gamma}{2}\sigma^{2}y^{2}+\frac{1}{2}\sigma^{2}y^{2}\partial^{2}_{yy}\theta(t,y)\\\
\\!&\qquad+\text{\scalebox{0.6}[1.0]{$\bigint$}}_{\\!\\!\mathbb{R}_{+}^{*}}\left(zH^{0,1}\left(z,\frac{\theta(t,y)-\theta(t,y-z)}{z}\right)+zH^{1,0}\left(z,\frac{\theta(t,y)-\theta(t,y+z)}{z}\right)\right)m(dz),\\\
\\!&\theta(T,y)=0,\end{cases}$ (2)

where

$H^{0,1}:(z,p)\in\mathbb{R}_{+}^{*}\times\mathbb{R}\mapsto\underset{\delta\geq-C}{\sup}\
\Lambda^{0,1}(z,\delta)(\delta-p)\text{ and
}H^{1,0}:(z,p)\in\mathbb{R}_{+}^{*}\times\mathbb{R}\mapsto\underset{\delta\geq-C}{\sup}\
\Lambda^{1,0}(z,\delta)(\delta-p).$

Using the same ideas as in [43], we see that for $i\neq j\in\\{0,1\\}$, the
supremum in the definition of $H^{i,j}(z,p)$ is reached at a unique
$\bar{\delta}^{i,j}(z,p)$ given by

$\bar{\delta}^{i,j}(z,p)=(\Lambda^{i,j})^{-1}\left(z,-\partial_{p}{H^{i,j}}(z,p)\right)$

where for all $z$, $(\Lambda^{i,j})^{-1}(z,.)$ denotes the inverse of the
function $\Lambda^{i,j}(z,.)$. Moreover, the markups that maximize our
modified objective function are obtained in the following form

$\displaystyle\delta^{0,1*}(t,z)=\bar{\delta}^{0,1}\left(z,\frac{\theta(t,Y^{1}_{t-})-\theta(t,Y^{1}_{t-}-z)}{z}\right)$
(3)

and

$\displaystyle\delta^{1,0*}(t,z)=\bar{\delta}^{1,0}\left(z,\frac{\theta(t,Y^{1}_{t-})-\theta(t,Y^{1}_{t-}+z)}{z}\right).$
(4)

In practice, we proceed in two steps in order to solve the above stochastic
optimal control problem. First, we approximate numerically the solution to the
HJB equation (2). For that purpose, we employ operator splitting in order to
deal separately with the differential terms and the non-local terms. For the
differential terms, we used an implicit scheme with Neumann conditions at the
boundaries ($\pm\bar{y}$ for $\bar{y}$ chosen large). As for the non-local
terms, we used a discrete measure $m$ and applied a Newton-Raphson algorithm
to resolve the implicit scheme at each time step (we assume that the AMM does
not accept trades that go beyond the boundaries $\pm\bar{y}$). This approach
proved to be computationally efficient. Once the value function $\theta$ is
approximated, the second step consists in plugging the approximation of
$\theta$ into Equations (3) and (4) to obtain the associated markups.

###  3.3 Numerical example

In what follows, we are going to illustrate the excess PnL associated with
various strategies. In order to illustrate our findings, we use throughout the
paper a market simulator based on a discrete-time version of the above
framework, with the following realistic parameters:

  * •

currency 0: USD, currency 1: ETH

  * •

Initial exchange rate: 1600 USD per ETH

  * •

Drift: $\mu=0\text{ day}^{-1}$

  * •

Volatility: $\sigma=1\text{ year}^{-\frac{1}{2}}$

  * •

Single transaction size: 4000 USD (i.e. $m$ is a Dirac mass)

  * •

Intensity functions: $\lambda^{0,1}=\lambda^{1,0}=100\text{ day}^{-1}$,
$\alpha^{0,1}=\alpha^{1,0}=-1.8$, $\beta^{0,1}=\beta^{1,0}=1300\text{
bps}^{-1}$ (this corresponds to an average of 86 trades per day when the
proposed exchange rate always equals the market exchange rate, 96 trades per
day when the proposed exchange rate is the market exchange rate improved by 10
bps, and 62 trades per day when the proposed exchange rate is the market
exchange rate worsen by 10 bps)

  * •

Initial inventory: $2000000$ USD and $1250$ ETH

  * •

Time horizon: $T=0.5\text{ day}.$

We plot in Figure 1 the performance of AMMs following different strategies:

  * •

Naive oracle-based strategies consisting in choosing $\delta^{0,1}$ and
$\delta^{1,0}$ constant

  * •

CPMM strategies with various transaction fees

  * •

CFMM strategies as decribed in [35, 36]

  * •

An oracle-based strategy documented in [23]

  * •

Oracle-based strategies associated with the markups we derived from the above
stochastic optimal control problem.

Figure 1: Performance of strategies in terms of mean / standard deviation of
excess PnL. In blue: naive strategies with constant
$\delta^{0,1},\delta^{1,0}$ (the number next to the point corresponds to the
value of $\delta^{0,1}$ and $\delta^{1,0}$ in bps). In grey: CPMM with fees
(the numbers next to the points correspond to the transaction fees in bps). In
pink: CFMM without market exchange rate oracle as described in [35, 36] for
different sets of realistic parameters. In purple $(\star)$: AMM with market
exchange rate oracle as described in [23]. In green: approximation of the
efficient frontier, obtained using the optimal markups for different levels of
risk aversion (the numbers next to the points correspond to the value of
$\gamma$).

We used Monte-Carlo simulations with 1000 trajectories. The (approximation of
the) efficient frontier is parameterized by $\gamma$. Unsurprisingly, when
$\gamma$ is large (i.e. large risk aversion), the optimal strategy consists in
providing liquidity at a very high cost, and the resulting risk / return
profile gets close to the origin $(0,0)$. As $\gamma$ decreases, the efficient
frontier describes an increasing curve that stops (unlike in the Markowitz
case) at $\gamma=0$, at a point corresponding to (optimized) constant markups
and to the maximum expected excess PnL.

Although this first graph corresponds to an idealized world where the market
exchange rate is perfectly known at all times (and where there is therefore no
arbitrage), it sheds light on the intrinsic limitations of (oracle-free) CFMMs
compared to oracle-based AMMs. In our setting, CFMMs indeed underperform
optimal strategies, as expected, but also naive strategies, both in terms of
expected excess PnL and in terms of standard deviation. It is noteworthy that
implementing a hedging strategy for CFMMs would reduce the standard deviation
but leave negative expected excess PnLs in our setting.

These results, however need to be qualified. Naive strategies with constant
transaction fees lack of robustness: in the presence of an arbitrage
opportunity due to information asymmetry or mispricing, the pool would indeed
be depleted from one of the two assets! In the next section, we show that our
optimized oracle-based strategies are, however, robust to misspecifications,
lags in price oracles and the introduction of adverse selection.

##  4 Beyond the idealized case: misspecification, incomplete information and
introduction of adverse selection

In practice, an AMM can be designed with some values of the drift, volatility
and liquidity parameters, but different values may realize. Furthermore, the
drift is highly unpredictable, volatility is not constant but clustered, while
liquidity varies depending on market conditions.252525For models taking
account of the stochastic nature of volatility and liquidity, see [22].
Consequently, it is of the utmost importance to study the impact of parameters
misspecification on the risk / return profile of allegedly efficient
strategies, i.e. what happens if the realized drift, volatility and liquidity
parameters do not match those used in the smart contract.

In Figure 2, we consider the case where the actual liquidity parameters
$\lambda^{0,1}$ and $\lambda^{1,0}$ one inputs in the simulator are the same
as in the above section, but the liquidity parameters of the strategy are set
to $\lambda^{0,1}=\lambda^{1,0}=50\text{ day}^{-1}$. We compare the results to
the efficient frontier and observe that the misspecified strategies remain
almost exactly on the efficient frontier, with a shift toward higher risk
aversions. Similarly, we consider in Figure 3 the case where the actual
volatility one inputs in the simulator is equal to 100% ($\sigma=1\text{
year}^{-\frac{1}{2}}$) as above, but the volatility used to compute the
strategy is 120% ($\sigma=1.2\text{ year}^{-\frac{1}{2}}$). We compare the
results to the efficient frontier and observe the same phenomenon as for
liquidity parameters. These results can be explained theoretically. Indeed, in
the absence of drift and ignoring the Laplacian term
$\frac{1}{2}\sigma^{2}y^{2}\partial^{2}_{yy}\theta(t,y)$, it can be shown that
when $\lambda^{0,1}=\lambda^{1,0}=:\lambda$, misspecifying $\lambda$ or
$\sigma$ has the same effect as choosing another $\gamma$, because the
solution to Equation (2) depends only on the ratio
$\frac{\lambda}{\gamma\sigma^{2}}$. This is in line with the observation made
in [18].

Figure 2: Performance of strategies in terms of mean / standard deviation of
excess PnL when $\lambda^{0,1}$ and $\lambda^{1,0}$ are misspecified. In
green: efficient frontier, obtained with the efficient strategy for different
levels of risk aversion with complete information. In pink: performance of the
misspecified strategy obtained with $\lambda^{0,1}=\lambda^{1,0}=50\text{
day}^{-1}$ for different levels of risk aversion. The numbers next to the
points correspond to the value of $\gamma$.
Figure 3: Performance of strategies in terms of mean / standard deviation of
excess PnL when $\sigma$ is misspecified. In green: efficient frontier,
obtained with the efficient strategy for different levels of risk aversion
with complete information. In pink: performance of the misspecified strategy
obtained with $\sigma=1.2\text{ year}^{-\frac{1}{2}}$ for different levels of
risk aversion. The numbers next to the points correspond to the value of
$\gamma$.

Finally, we consider in Figure 4 the case where the parameters of the strategy
correspond to those used in the previous section with no drift, but the drift
one inputs to compute the strategy is equal to 40% ($\mu=0.4\text{
year}^{-1}$). We compare the results to the efficient frontier and observe
that the misspecified strategies remain close to the efficient frontier. It is
noteworthy that we did not compute the point corresponding to no risk aversion
($\gamma=0$) because the optimal strategy of any risk-neutral agent with a
view on the drift is to get an infinite directional position: here, this means
that the AMM should sell one of the two assets to buy the other.

Figure 4: Performance of strategies in terms of mean / standard deviation of
excess PnL when $\mu$ is misspecified. In green: efficient frontier, obtained
with the efficient strategy for different levels of risk aversion with
complete information. In pink: performance of the misspecified strategy
obtained with $\mu=0.4\text{ year}^{-1}$ for different levels of risk
aversion. The numbers next to the points correspond to the value of $\gamma$.

Misspecification can be a problem, but the main problem faced by AMMs is
adverse selection. It is important to recall that the main problem of CFMMs is
that value is extracted by LTs. Of course, in the above model with complete
information, the market exchange rate is known at all times and adverse
selection does not exist. To introduce adverse selection, we assume that the
AMM cannot observe the market exchange rate perfectly but rather with a lag,
while the demand curves of LTs are centered on the right (current) price. We
show the results in Figure 5 and clearly see that performances move away from
the efficient frontier as the delay increases. However, the performance
remains good for reasonable values of the lag.

Partial information regarding the market exchange rate can sometimes result in
arbitrage opportunities for LTs. These arbitrage opportunities are already
taken into account in the demand curves modeled by the intensity functions,
though not in a systematic way. In other words, if a price appears to be very
good for LTs, the probability that a transaction occurs is very high in our
model. In practice however, there exists a category of agents who
systematically exploit arbitrage opportunities: they trade with the AMM until
arbitrage opportunities disappear.262626Of course, these trades come with a
cost for arbitrageurs, who still have to pay the fees associated with a swap
on an other (centralized) exchange. In our case, we assume a proportional cost
of $7.5\text{ bps}$. Moreover, note that in our simulations, arbitrageurs
trade a size that is optimal for them – and thus does not necessarily
correspond to the size of the other trades. The above analysis regarding
market exchange rate oracles can then only be complete if we take into account
these arbitrageurs. In practice, arbitrageurs can represent a significant
volume as soon as the market exchange rate moves outside of the spread offered
by the AMM, and we compare in Figure 6 the performance of the efficient
strategies in the presence of a 10-second delayed oracle and with
arbitrageurs. We also provide in Figure 7 a complete risk / return analysis of
the different strategies studied in this paper, in the presence of
arbitrageurs and incomplete information (for those relying on oracles).

Figure 5: Performance of the efficient strategies in terms of mean / standard
deviation of excess PnL, obtained by playing the efficient strategies for
different levels of risk aversion with different oracle delays: complete
information (in green), 10-second delay (in yellow), 30-second delay (in
blue), 1-minute delay (in purple), 5-minute delay (in pink), 30-minute delay
(in red). The number next to the point corresponds to the value of $\gamma$.
Figure 6: Performance of the previous optimal strategy in terms of mean /
standard deviation of excess PnL. In green: efficient frontier, obtained by
playing the optimal strategy for different levels of risk aversion with
complete information. In yellow: performance of the same optimal strategy for
different levels of risk aversion with a lagged oracle. In orange: performance
of the same optimal strategy for different levels of risk aversion with a
lagged oracle and with arbitrage flow. The number next to the point
corresponds to the value of $\gamma$.
Figure 7: Performance of strategies in terms of mean / standard deviation of
excess PnL with a lagged oracle and with arbitrage flow. In grey: CPMM with
fees. In pink: CFMM without market exchange rate oracle as described in [35,
36] for different sets of realistic parameters. In purple ($\star$): AMM with
market exchange rate oracle as described in [23]. In orange: performance of
the efficient strategies for different levels of risk aversion.

What we observe in Figure 6 is that the presence of arbitrageurs who
systematically and continuously keep the offered exchange rate in a narrow
range around the market exchange rate tends to increase the expected excess
PnL and reduce risk. This may sound counter-intuitive, but it comes from the
fact that those arbitrageurs actually “protect” the AMM against trades with
traditional LTs at prices even further away from the market exchange rate
(this could be related to the findings of [58]). Of course, the efficient
frontier documented in Figure 6 does not take into account this additional
flow. Building a model that would internalize both delayed oracles and
(systematic) arbitrageurs and allow to derive optimal strategies in this
context remains an open problem.

Figure 7 confirms that the use of a market exchange rate oracle, even delayed,
really makes a difference in terms of risk / return profile. Nevertheless, it
is important to note that introducing an oracle creates a fundamentally
different protocol which relies on external data. One of the issues with
oracles (as noted in [65] and then in [55]), is that they could be
manipulated. Using an oracle in an AMM protocol should therefore be performed
carefully in order for LPs to really achieve the promised improved risk-
adjusted performance (for a detailed explanation of the next generation of
decentralized oracles see [25]).

##  5 Conclusion

In this paper, we provide an analysis of AMMs in a mean / standard deviation
framework inspired by both Markowitz’ modern portfolio theory and the recent
literature on optimal market making. We show that traditional CFMMs (including
CPMMs) with different levels of transaction fees perform poorly relative to
the theoretical efficient frontier and very often exhibit negative excess PnL.
We also show that allowing an AMM to get information about the current market
exchange rate (through an oracle) can significantly improve performance. Such
an oracle-based AMM would quote a bid / ask spread around the current market
exchange rate based on its reserves. This design significantly reduces the
volatility of the excess PnL while delivering a positive excess PnL on
average. Our results are robust to the presence of a lagged oracle and to the
introduction of adverse selection by LTs and arbitrageurs. Nevertheless, while
introducing an oracle in the AMM design can significantly improve the risk-
adjusted performance of LPs, it comes at the cost that the oracle itself
should be carefully designed to avoid introducing additional attack vectors.

## Statement and acknowledgment

The research carried out for this paper benefited from the support of the
Research Program “Decentralized Finance and Automated Market Makers”, a
Research Program under the aegis of Institut Europlace de Finance, in
partnership with Apesu / Swaap Labs.

The content of this article has been presented at several conferences and
seminars: BlockSem seminar (Paris, France), FFEA 2nd Spring Workshop on
FinTech (Ghent, Belgium), The 3rd workshop on Decentralized Finance (Bol,
Croatia), DeFOx seminar (Oxford, UK), Euro Working Group on Commodities and
Financial Modelling meeting (Rome, Italy), SIAM Conference on Financial
Mathematics and Engineering (Philadelphia, USA), AMaMeF conference (Bielefeld,
Germany), Florence-Paris Workshop on Mathematical Finance (Florence, Italy),
Frontiers in Quantitative Finance Seminar (London, UK), Séminaire Bachelier
(Paris, France), 16th Financial Risks International Forum (Paris, France),
London-Paris Bachelier Workshop on Mathematical Finance (London, UK),
Blockchain@X-OMI Workshop on Blockchain and Decentralized Finance (Palaiseau,
France), Apéro DeFi (Paris, France). The discussions that took place during
these events have substantially contributed to improving the presentation of
our results.

We extend our sincere gratitude to the two anonymous referees for their
insights and constructive feedback, which greatly enhanced the quality of our
article.

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††thanks: Work performed in the framework of R&D for future accelerator
operation modes at the European XFEL and financed by the European XFEL GmbH.

# High gradients at SRF photoinjector cavities with low RRR copper cathode
plug screwed to the cavity back wall

E. Vogel<EMAIL_ADDRESS>J. Sekutowicz  D. Bazyl  T. Büttner  B. van der
Horst  J. Iversen  D. Klinke  A. Muhs
D. Reschke  S. Sägebarth  P. Schilling  M. Schmökel  L. Steder  J.-H. Thie  H.
Weise  M. Wiencek  Deutsches Elektronen-Synchrotron DESY, Notkestraße 85,
D-22607 Hamburg, Germany

###### Abstract

In recent years we increased the typical maximum peak field on axis gradients
obtained in L-band superconducting RF (SRF) photoinjector cavities at vertical
tests to around 55 MV/m. This was achieved with niobium cathode plugs directly
screwed to the cavity back wall omitting an RF choke filter and a load lock
system for cathodes. Copper demonstrated being a suitable cathode material in
normal conducting injector cavities used at X-Ray Free Electron Lasers (XFELs)
operating with pulsed RF. In this article we present the first experimental
confirmation that peak field on axis gradients around 55 MV/m and beyond can
be achieved in L-band SRF photoinjector cavities with copper cathode plugs
screwed to the cavity back wall. We view this as a major milestone for the
development of a high gradient photoinjector operating continuous wave (CW).

††preprint: APS/123-QED

##  I Introduction

High gradient photoinjector cavities enable “pancake” emission of beams and
direct matching into subsequent L-band SRF linacs [1]. An additional buncher
section as required in other setups [2] can be omitted. For facilities
operating CW and High-Duty-Cycle (HDC), L-band SRF photoinjector cavities
provide the required gradients. They are the first choice for the planned HDC
operation of the European XFEL [3, 4].

Our cavity design (see Fig. 1) consists of a cathode plug screwed to the
cavity back wall [5, 6]. Indium is used for sealing and for good thermal
contact. Well established surface treatments like high pressure rinsing (HPR)
can be applied after cathode insertion in a clean room. A main prerequisite
are photocathodes which are robust against the exposure to air and the usual
SRF cavity cleaning procedures. Consequently our cathode material choice is
mainly limited to metals.

Figure 1: Photoinjector cavity with a cathode plug (red) screwed into a hole
on the cavity backside. Figure 2: Copper cathode plug in the SRF cavity back
wall.

In recent years we developed the surface treatment procedures for our cavities
showing now typical maximum peak field on axis gradients around 55 MV/m in
vertical tests with niobium cathode plugs [7].

However, the quantum efficiency (QE) of bulk niobium is around $10^{-7}$ and
too low to use it as the emitter [8, 9]. For the European XFEL, we need a QE
of $10^{-4}$ or higher to generate 100 pC electron bunches with a repetition
rate of at least 100 kHz. Copper is our preferred cathode choice. It has a
moderate work function of 4.6 eV, excellent RF and thermal characteristics and
the tendency causing dark current is minimal. Surface laser cleaning and
additional surface treatments on copper cathodes offers the possibility of
achieving a QE of at least $10^{-4}$ [10, 11].

##  II The RF field in the cavity

For beam generation and acceleration our L-band 1.6-cell SRF photoinjector
cavities will be operated at 1.3 GHz with the TM010 $\pi$-mode. The cathode
plane is retracted from the back wall of the cavity by 450 $\mu$m to optimize
the beam dynamics near the cathode surface. This reduces the peak electric
field at the cathode to 85 $\%$ of the maximal peak electric field along the
axis. To study the performance of the cavity half-cell with the cathode plug
in more detail, we also measure the TM010 0-mode in our vertical tests. At the
$\pi$-mode the peak electric field is almost identical in both cells. In
contrast at the 0-mode, the peak field is twice as high in the half-cell than
in the full cell. Both modes differ by about 18 MHz in frequency. The axial
electric field distribution of the TM010 0- and $\pi$-mode is illustrated in
Figure 3 and the electric and magnetic field maps of the TM010 modes can be
found in Figure 4.

Figure 3: Distribution of the peak electric field on the cavity’s axis for the
TM010 0- and $\pi$-mode. Figure 4: Illustration of the electric and the
magnetic field distribution of the TM010 0- and $\pi$-mode.

At the TM010 modes the surface magnetic field is zero at the radial center of
the cathode, likewise the surface current. It increases with the radial
offset, Fig. 5. Due to the small cathode radius of 2 mm, the surface magnetic
field at the cathode is a factor of 42 lower as compared to the peak surface
magnetic field in the equator region of the cavity. This combined with the
good thermal conductivity of copper justified the expectation that the low
dissipated RF power at the cathode should not affect the superconductivity in
the remaining cavity surface.

Figure 5: Magnetic field distribution at the surface of the cathode for the
TM010 $\pi$-mode. The field amplitude is normalized for 1 J stored energy in
the photoinjector cavity.

##  III Vertical tests with copper plugs

In May 2023 we received the most recent manufactured prototype cavities 16G09
and 16G10 after final buffered chemical polishing (BCP) treatment. Initial
tests with niobium cathode plugs showed the expected maximum peak field on
axis gradients around 55 MV/m at the $\pi$-mode for 16G10. At the first cool
down 16G09 suffered from a cold leak. Re-tightening all sealing connections
and no additional cleaning was performed. Afterwards, 16G09 was leak tight but
showed a significant lower gradient in the first test limited by strong
radiation.

Figure 6: Photograph of the inner surface of the back wall of a photoinjector
cavity with inserted copper cathode plug.

In August, we fitted both cavities with cathode plugs made of industrial-grade
copper (Fig. 2) with comparatively low heat conductivity (residual resistivity
ratio (RRR) around 30) and performed vertical tests. The cathode insertion was
followed by the usual HPR and 90 °C baking [7]. Figure 6 shows the inner
surface of the back wall of a SRF photoinjector cavity, photographed during a
clean room optical inspection after the successful vertical tests.

Figure 7 contains the successful vertical test results obtained with the
cavities 16G09 and 16G10. Both cavities reached maximum peak field on axis
gradients around 55 MV/m in the $\pi$-mode and about 65 MV/m in the 0-mode.

We assume the difference of the maximum peak field on axis between the
$\pi$-mode and the 0-mode being caused by losses in the stainless steel vacuum
flange connected to the cavity beam tube for the vertical tests. For a given
maximum peak field on axis the remaining field is higher near the vacuum
flange in the case of the $\pi$-mode, Fig. 3. Dedicated flanges coated with
copper or niobium to verify this assumption and minimize the effect for future
tests are in preparation.

Figure 7: Q vs E curves describing the successful test results of the SRF
photoinjector cavities 16G09 and 16G10 with copper cathode plugs and the first
test of 16G10 with niobium cathode plug for comparison. The design goal
indicated reflects the specification for the TM010 $\pi$-mode in terms of the
Q-value and the peak electric field values.

However, there is almost no performance degradation with the copper cathode
plugs as compared to the niobium cathode plug. The high gradients are still
achieved. The Q-values appear to be slightly lower with copper cathode plugs
compared to those with niobium cathode plugs, Fig. 7.

We intend to try other heat treatment parameters and apply electro polishing
at the cavities and using copper with high heat conductivity for the cathode
plugs to reduce the Q-slope in general.

Field emission and dark current are of particular interest at photoinjector
cavities. In our tests, some field emission was measurable in the first test
runs. Conditioning took place and in the later tests shown in Figure 7 the
field emission detected decreased to irrelevant low levels. Until now, we did
not implement measures to optimize the cavity cathode hole rim. We plan this
for the future.

##  IV Summary

We proofed for the first time, that high gradients can be obtained with normal
conducting cathode plugs with comparatively low heat conductivity screwed to
the back wall of L-band SRF photoinjector cavities. This enables many new
options for cathodes used in our cavities. Using higher-purity copper for
cathode plugs enhances heat transfer from the cathode to the liquid helium
bath.

##  V Acknowledgments

The authors acknowledge the significant contributions from numerous colleagues
at all institutes joining the effort for an SRF photoinjector cavity with a
cathode plug directly screwed to the back wall. Many people from industry
contribute to this effort as well.

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