We present a formal framework for the aggregation of financial markets mediated by arbitrage. Our main tool is to characterize markets via utility functions and to employ a one-to-one correspondence to limit order book states. Inspired by the theory of thermodynamics, we argue that the arbitrage-mediated aggregation mechanism gives rise to a market-dynamical entropy, which quantifies the loss of liquidity caused by aggregation. As a concrete guiding example, we illustrate our general approach with the Uniswap v2 automated market maker protocol used in decentralized cryptocurrency exchanges, which we characterize as a so-called ideal market. We derive its equivalent limit order book representation and explicitly compute the arbitrage-mediated aggregation of two liquidity pools of the same asset pair with different marginal prices. We also discuss future directions of research in this emerging theory of market dynamics.
Guillermo Angeris, Tarun Chitra, Theo Diamandis, Alex Evans · 5 authors
Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or 'axioms') which encompass and generalize many of the known results for CFMMs in the literature, without requiring strong conditions such as differentiability or homogeneity. One particular consequence of this framework is that every CFMM has a (unique) canonical trading function that is nondecreasing, concave, and homogeneous, showing that many results known only for homogeneous trading functions are actually fully general. We also show that CFMMs satisfy a number of intuitive and geometric composition rules, and give a new proof, via conic duality, of the equivalence of the portfolio value function and the trading function. Many results are extended to the general setting where the CFMM is not assumed to be path-independent, but only one trade is allowed. Finally, we show that all 'path-independent' CFMMs have a simple geometric description that does not depend on any notion of a 'trading history'.
Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden
We consider the impact of trading fees on the profits of arbitrageurs trading against an automated market maker (AMM) or, equivalently, on the adverse selection incurred by liquidity providers (LPs) due to arbitrage. We extend the model of Milionis et al. [2022] for a general class of two asset AMMs to introduce both fees and discrete Poisson block generation times. In our setting, we are able to compute the expected instantaneous rate of arbitrage profit in closed form. When the fees are low, in the fast block asymptotic regime, the impact of fees takes a particularly simple form: fees simply scale down arbitrage profits by the fraction of blocks which present profitable trading opportunities to arbitrageurs. This fraction decreases with an increasing block rate, hence our model yields an important practical insight: faster blockchains will result in reduced LP losses. Further introducing gas fees (fixed costs) in our model, we show that, in the fast block asymptotic regime, lower gas fees lead to smaller losses for LPs.
We develop a formalism for reasoning about trading on decentralized exchanges on blockchains and a formulation of a particular form of maximal extractable value (MEV) that represents the total arbitrage opportunity extractable from on-chain liquidity. We use this formalism to prove that for blockchains with deterministic block times whose liquidity pools satisfy some natural properties that are satisfied by pools in practice, this form of MEV is invariant under changes to the ordering mechanism of the blockchain and distribution of block times. We do this by characterizing the MEV as the profit of a particularly simple arbitrage strategy when left uncontested. These results can inform design of blockchain protocols by ruling out designs aiming to increase trading opportunity by changing the ordering mechanism or shortening block times.
Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden
In decentralized finance ("DeFi"), automated market makers (AMMs) enable traders to programmatically exchange one asset for another. Such trades are enabled by the assets deposited by liquidity providers (LPs). The goal of this paper is to characterize and interpret the optimal (i.e., profit-maximizing) strategy of a monopolist liquidity provider, as a function of that LP's beliefs about asset prices and trader behavior. We introduce a general framework for reasoning about AMMs based on a Bayesian-like belief inference framework, where LPs maintain an asset price estimate. In this model, the market maker (i.e., LP) chooses a demand curve that specifies the quantity of a risky asset to be held at each dollar price. Traders arrive sequentially and submit a price bid that can be interpreted as their estimate of the risky asset price; the AMM responds to this submitted bid with an allocation of the risky asset to the trader, a payment that the trader must pay, and a revised internal estimate for the true asset price. We define an incentive-compatible (IC) AMM as one in which a trader's optimal strategy is to submit its true estimate of the asset price, and characterize the IC AMMs as those with downward-sloping demand curves and payments defined by a formula familiar from Myerson's optimal auction theory. We generalize Myerson's virtual values, and characterize the profit-maximizing IC AMM. The optimal demand curve generally has a jump that can be interpreted as a "bid-ask spread," which we show is caused by a combination of adverse selection risk (dominant when the degree of information asymmetry is large) and monopoly pricing (dominant when asymmetry is small). This work opens up new research directions into the study of automated exchange mechanisms from the lens of optimal auction theory and iterative belief inference, using tools of theoretical computer science in a novel way.
In traditional finance, the Black & Scholes model has guided almost 50 years of derivatives pricing, defining a standard to model any volatility-based product. With the rise of Decentralized Finance (DeFi) and constant product Automated Market Makers (AMMs), Liquidity Providers (LPs) are playing an increasingly important role in markets functioning, but, as the recent bear market highlighted, they are exposed to important risks such as Impermanent Loss (IL). In this paper, we tailor the formulas introduced by Black & Scholes to DeFi, proposing a method to calculate the greeks of an LP. We also introduce Impermanent Gain, a product that LPs can use to hedge their position and traders can use to bet on a rise in volatility and benefit from large market moves.
With the increasing adoption of the Proof of Stake (PoS) blockchain, it is timely to study the economy created by such blockchain. In this chapter, we will survey recent progress on the trading and wealth evolution in a cryptocurrency where the new coins are issued according to the PoS protocol. We first consider the wealth evolution in the PoS protocol assuming no trading, and focus on the problem of decentralisation. Next we consider each miner's trading incentive and strategy through the lens of optimal control, where the miner needs to trade off PoS mining and trading. Finally, we study the collective behavior of the miners in a PoS trading environment by a mean field model. We use both stochastic and analytic tools in our study. A list of open problems are also presented.
Automated Market Makers (AMMs) are a class of smart contracts on Ethereum and other blockchains that "make markets" autonomously. In other words, AMMs stand ready to trade with other market participants that interact with them, at the conditions determined by the AMM. In this this paper, which relies on the existing and growing corpus of literature available, we review and present the key mathematical and quantitative finance aspects that underpin their operations, including the interesting relationship between AMMs and derivatives pricing and hedging.
In recent years, with the development of easy to use learning environments, implementing and reproducible benchmarking of reinforcement learning algorithms has been largely accelerated by utilizing these frameworks. In this article, we introduce the Dynamic Fee learning Environment (DyFEn), an open-source real-world financial network model. It can provide a testbed for evaluating different reinforcement learning techniques. To illustrate the promise of DyFEn, we present a challenging problem which is a simultaneous multi-channel dynamic fee setting for off-chain payment channels. This problem is well-known in the Bitcoin Lightning Network and has no effective solutions. Specifically, we report the empirical results of several commonly used deep reinforcement learning methods on this dynamic fee setting task as a baseline for further experiments. To the best of our knowledge, this work proposes the first virtual learning environment based on a simulation of blockchain and distributed ledger technologies, unlike many others which are based on physics simulations or game platforms.
Within this work we consider an axiomatic framework for Automated Market Makers (AMMs). AMMs are smart contracts that set prices for swaps on a pool of assets. By imposing reasonable axioms on the underlying utility function, we are able to characterize the properties of the swap size of the assets and of the resulting pricing oracle. In providing these general axioms, we define a novel measure of price impacts that can be used to quantify those costs between different AMM constructions. We have analyzed many existing AMMs and shown that the vast majority of them satisfy our axioms. We have also considered the question of fees and divergence loss. In doing so, we have proposed a new fee structure so as to make the AMM indifferent to transaction splitting. Finally, we have proposed a novel AMM that has nice analytical properties and provides a large range over which there is no divergence loss.
A trading system is said to be {robust} if it generates a robust return regardless of market direction. To this end, a consistently positive expected trading gain is often used as a robustness metric for a trading system. In this paper, we propose a new class of trading policies called the {double linear policy} in an asset trading scenario when the transaction costs are involved. Unlike many existing papers, we first show that the desired robust positive expected gain may disappear when transaction costs are involved. Then we quantify under what conditions the desired positivity can still be preserved. In addition, we conduct heavy Monte-Carlo simulations for an underlying asset whose prices are governed by a geometric Brownian motion with jumps to validate our theory. A more realistic backtesting example involving historical data for cryptocurrency Bitcoin-USD is also studied.
Recent financial bubbles such as the emergence of cryptocurrencies and "meme stocks" have gained increasing attention from both retail and institutional investors. In this paper, we propose a game-theoretic model on optimal liquidation in the presence of an asset bubble. Our setup allows the influx of players to fuel the price of the asset. Moreover, traders will enter the market at possibly different times and take advantage of the uptrend at the risk of an inevitable crash. In particular, we consider two types of crashes: an endogenous burst which results from excessive selling, and an exogenous burst which cannot be anticipated and is independent from the actions of the traders. The popularity of asset bubbles suggests a large-population setting, which naturally leads to a mean field game (MFG) formulation. We introduce a class of MFGs with varying entry times. In particular, an equilibrium will depend on the entry-weighted average of conditional optimal strategies. To incorporate the exogenous burst time, we adopt the method of progressive enlargement of filtrations. We prove existence of MFG equilibria using the weak formulation in a generalized setup, and we show that the equilibrium strategy can be decomposed into before-and-after-burst segments, each part containing only the market information. We also perform numerical simulations of the solution, which allow us to provide some intriguing results on the relationship between the bubble burst and equilibrium strategies.
Stochastic clocks represent a class of time change methods for incorporating trading activity into continuous-time financial models, with the ability to deal with typical asymmetrical and tail risks in financial returns. In this paper we propose a significant improvement of stochastic clocks for the same objective but without decreasing the number of trades or changing the trading intensity. Our methodology targets any Lévy subordinator, or more generally any process of nonnegative independent increments, and is based on various choices of regulating kernels motivated from repeated averaging. By way of a hyperparameter linked to the degree of regulation, arbitrarily large skewness and excess kurtosis of returns can be easily achieved. Generic-time Laplace transforms, characterizing triplets, and cumulants of the regulated clocks and subsequent mixed models are analyzed, serving purposes ranging from statistical estimation and option price calibration to simulation techniques. Under specified jump--diffusion processes and tempered stable processes, a robust moment-based estimation procedure with profile likelihood is developed and a comprehensive empirical study involving S\&P500 and Bitcoin daily returns is conducted to demonstrate a series of desirable effects of the proposed methods.
Erdinc Akyildirim, Matteo Gambara, Josef Teichmann, Syang Zhou
Anomaly detection is the process of identifying abnormal instances or events in data sets which deviate from the norm significantly. In this study, we propose a signatures based machine learning algorithm to detect rare or unexpected items in a given data set of time series type. We present applications of signature or randomized signature as feature extractors for anomaly detection algorithms; additionally we provide an easy, representation theoretic justification for the construction of randomized signatures. Our first application is based on synthetic data and aims at distinguishing between real and fake trajectories of stock prices, which are indistinguishable by visual inspection. We also show a real life application by using transaction data from the cryptocurrency market. In this case, we are able to identify pump and dump attempts organized on social networks with F1 scores up to 88% by means of our unsupervised learning algorithm, thus achieving results that are close to the state-of-the-art in the field based on supervised learning.
Over 90% of exchange trading on crypto options has always been on the Deribit platform. This centralised crypto exchange only lists inverse products because they do not accept fiat currency. Currently, fiat-based traders can only make deposits in bitcoin, although they can withdraw both bitcoin and ether to their on-chain wallets. Likewise, other major crypto options platforms only list crypto--stablecoin trading pairs in so-called direct options, which are similar to the standard crypto options listed by the CME except the U.S. dollar is replaced by a stablecoin version. Until now a clear mathematical exposition of these products has been lacking. We discuss the sources of market incompleteness in direct and inverse options and compare their pricing and hedging characteristics. Then we discuss the useful applications of currency protected "quanto" direct and inverse options for fiat-based traders and describe their pricing and hedging characteristics, all in the Black-Scholes setting.
Purpose This paper aims to demonstrate a dynamic cointegration-based pairs trading strategy, including an optimal look-back window framework in the cryptocurrency market and evaluate its return and risk by applying three different scenarios. Design/methodology/approach This study uses the Engle-Granger methodology, the Kapetanios-Snell-Shin test and the Johansen test as cointegration tests in different scenarios. This study calibrates the mean-reversion speed of the Ornstein-Uhlenbeck process to obtain the half-life used for the asset selection phase and look-back window estimation. Findings By considering the main limitations in the market microstructure, the strategy of this paper exceeds the naive buy-and-hold approach in the Bitmex exchange. Another significant finding is that this study implements a numerous collection of cryptocurrency coins to formulate the model’s spread, which improves the risk-adjusted profitability of the pairs trading strategy. Besides, the strategy’s maximum drawdown level is reasonably low, which makes it useful to be deployed. The results also indicate that a class of coins has better potential arbitrage opportunities than others. Originality/value This research has some noticeable advantages, making it stand out from similar studies in the cryptocurrency market. First is the accuracy of data in which minute-binned data create the signals in the formation period. Besides, to backtest the strategy during the trading period, this study simulates the trading signals using best bid/ask quotes and market trades. This study exclusively takes the order execution into account when the asset size is already available at its quoted price (with one or more period gaps after signal generation). This action makes the backtesting much more realistic.
A model is proposed for Bitcoin prices that takes into account market attention. Market attention, modeled by a mean-reverting Cox-Ingersoll-Ross processes, affects the volatility of Bitcoin returns, with some delay. The model is affine and tractable, with closed formulae for the conditional characteristic functions with respect to both the conventional and a delayed filtration. This leads to semi-closed formulae for European call and put prices. A maximum likelihood estimation procedure is provided, as well as a method for changing to a risk-neutral measure. The model compares very well against classical and attention-based models when tested on real data.
Constant Function Market Makers (CFMMs) are a family of automated market makers that enable censorship-resistant decentralized exchange on public blockchains. Arbitrage trades have been shown to align the prices reported by CFMMs with those of external markets. These trades impose costs on Liquidity Providers (LPs) who supply reserves to CFMMs. Trading fees have been proposed as a mechanism for compensating LPs for arbitrage losses. However, large fees reduce the accuracy of the prices reported by CFMMs and can cause reserves to deviate from desirable asset compositions. CFMM designers are therefore faced with the problem of how to optimally select fees to attract liquidity. We develop a framework for determining the value to LPs of supplying liquidity to a CFMM with fees when the underlying process follows a general diffusion. Focusing on a popular class of CFMMs which we call Geometric Mean Market Makers (G3Ms), our approach also allows one to select optimal fees for maximizing LP value. We illustrate our methodology by showing that an LP with mean-variance utility will prefer a G3M over all alternative trading strategies as fees approach zero.
We consider the hedging problem where a futures position can be automatically liquidated by the exchange without notice. We derive a semi-closed form for an optimal hedging strategy with dual objectives - to minimise both the variance of the hedged portfolio and the probability of liquidations due to insufficient collateral. The optimal solution depends on the statistical characteristics of the spot and futures extreme returns and parameters that characterise the hedger by loss aversion, choice of leverage and collateral management. An empirical analysis of bitcoin shows that the optimal strategy combines superior hedge effectiveness with a reduction in the probability of liquidation. We compare the performance of seven major direct and inverse hedging instruments traded on five different exchanges, based on minute-level data. We also link this performance to novel speculative trading metrics, which differ markedly between venues.
We investigate the behaviour of cryptocurrencies using data for bitcoin, ethereum and ripple which account for over 70% of the cryptocurrency market. We demonstrate that $α$-stable distribution is an appropriately sufficient model for highly speculative cryptocurrencies which outperforms other heavy tailed distributions that are used in financial econometrics. We find that the maximum likelihood method proposed by DuMouchel (1971) produces estimates that fit the cryptocurrency return data much better than the quantile based approach of McCulloch (1986) and sample characteristic method by Koutrouvelis (1980). The empirical results show that the leptokurtic feature presented in cryptocurrency return data can be captured by an $α$-stable distribution. The findings highlight that $α$-stable distribution is not only parsimonious with its four free parameters but also a creative model that is close to reality. This paper covers early reports and literature on cryptocurrencies and stable distributions.
This paper introduces new methods for analysing the extreme and erratic behaviour of time series to evaluate the impact of COVID-19 on cryptocurrency market dynamics. Across 51 cryptocurrencies, we examine extreme behaviour through a study of distribution extremities, and erratic behaviour through structural breaks. First, we analyse the structure of the market as a whole and observe a reduction in self-similarity as a result of COVID-19, particularly with respect to structural breaks in variance. Second, we compare and contrast these two behaviours, and identify individual anomalous cryptocurrencies. Tether (USDT) and TrueUSD (TUSD) are consistent outliers with respect to their returns, while Holo (HOT), NEXO (NEXO), Maker (MKR) and NEM (XEM) are frequently observed as anomalous with respect to both behaviours and time. Even among a market known as consistently volatile, this identifies individual cryptocurrencies that behave most irregularly in their extreme and erratic behaviour and shows these were more affected during the COVID-19 market crisis.
This book, which is in Spanish, provides detailed descriptions, including over 550 mathematical formulas, for over 150 trading strategies across a host of asset classes (and trading styles). This includes stocks, options, fixed income, futures, ETFs, indexes, commodities, foreign exchange, convertibles, structured assets, volatility (as an asset class), real estate, distressed assets, cash, cryptocurrencies, miscellany (such as weather, energy, inflation), global macro, infrastructure, and tax arbitrage. Some strategies are based on machine learning algorithms (such as artificial neural networks, Bayes, k-nearest neighbors). We also give: source code for illustrating out-of-sample backtesting with explanatory notes; around 2,000 bibliographic references; and over 900 glossary, acronym and math definitions. The presentation is intended to be descriptive and pedagogical. ----- Este libro proporciona descripciones detalladas, que incluyen más de 550 fórmulas matemáticas, para más de 150 estrategias de trading para una gran cantidad de clases de activos y estilos de trading. Esto incluye acciones, opciones, bonos (renta fija), futuros, ETFs, índices, commodities, divisas, bonos convertibles, activos estructurados, volatilidad (como clase de activos), bienes inmuebles, activos en distress, efectivo, criptomonedas, misceláneos (como clima, energía, inflación), macro global, infraestructura y arbitraje impositivo. Algunas estrategias se basan en algoritmos de aprendizaje automático (como redes neuronales artificiales, Bayes, k vecinos más cercanos). El libro también incluye código para backtesting fuera de la muestra con notas explicativas; cerca de 2,000 referencias bibliográficas; más de 900 términos que comprenden el glosario, acrónimos y definiciones matemáticas. La presentación pretende ser descriptiva y pedagógica.
We discuss the idea of a purely algorithmic universal world iCurrency set forth in [Kakushadze and Liew, 2014] (https://ssrn.com/abstract=2542541) and expanded in [Kakushadze and Liew, 2017] (https://ssrn.com/abstract=3059330) in light of recent developments, including Libra. Is Libra a contender to become iCurrency? Among other things, we analyze the Libra proposal, including the stability and volatility aspects, and discuss various issues that must be addressed. For instance, one cannot expect a cryptocurrency such as Libra to trade in a narrow band without a robust monetary policy. The presentation in the main text of the paper is intentionally nontechnical. It is followed by an extensive appendix with a mathematical description of the dynamics of (crypto)currency exchange rates in target zones, mechanisms for keeping the exchange rate from breaching the band, the role of volatility, etc.
Dorje C. Brody, Lane P. Hughston, Bernhard K. Meister
A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our analysis suggests that strict local martingales can be used for modelling the pricing kernels associated with virtual currencies based on distributed ledger technologies.