In this paper, we propose an analytical method to compute the collateral liquidation probability in decentralized finance (DeFi) stablecoin single-collateral lending. Our approach models the collateral exchange rate as a zero-drift geometric Brownian motion, and derives the probability of it crossing the liquidation threshold. Unlike most existing methods that rely on computationally intensive simulations such as Monte Carlo, our formula provides a lightweight, exact solution. This advancement offers a more efficient alternative for risk assessment in DeFi platforms.
Although modern blockchains almost universally produce blocks at fixed intervals, existing models still lack an analytical formula for the loss-versus-rebalancing (LVR) incurred by Automated Market Makers (AMMs) liquidity providers in this setting. Leveraging tools from random walk theory, we derive the following closed-form approximation for the per block per unit of liquidity expected LVR under constant block time: \[ \overline{\mathrm{ARB}}= \frac{\,σ_b^{2}} {\,2+\sqrt{2π}\,γ/(|ζ(1/2)|\,σ_b)\,}+O\!\bigl(e^{-\mathrm{const}\tfracγ{σ_b}}\bigr)\;\approx\; \frac{σ_b^{2}}{\,2 + 1.7164\,γ/σ_b}, \] where $σ_b$ is the intra-block asset volatility, $γ$ the AMM spread and $ζ$ the Riemann Zeta function. Our large Monte Carlo simulations show that this formula is in fact quasi-exact across practical parameter ranges. Extending our analysis to arbitrary block-time distributions as well, we demonstrate both that--under every admissible inter-block law--the probability that a block carries an arbitrage trade converges to a universal limit, and that only constant block spacing attains the asymptotically minimal LVR. This shows that constant block intervals provide the best possible protection against arbitrage for liquidity providers.
Agustin Muñoz Gonzalez, Juan Ignacio Sequeira, Ariel Dembling
This work analytically characterizes impermanent loss for automated market makers (AMMs) in decentralized markets such as Uniswap or Balancer (CPMM). We derive a static replication formula for the pool's value using a combination of European calls and puts. Furthermore, we establish a result guaranteeing hedging coverage for all final prices within a predefined interval. These theoretical results motivate a numerical example where we illustrate the strangle strategy using real cryptocurrency options data from Deribit, one of the most liquid markets available.
Robert Ghrist, Julian Gould, Miguel Lopez, Hans Riess
Modern financial networks involve complex obligations that transcend simple monetary debts: multiple currencies, prioritized claims, supply chain dependencies, and more. We present a mathematical framework that unifies and extends these scenarios by recasting the classical Eisenberg-Noe model of financial clearing in terms of lattice liability networks. Each node in the network carries a complete lattice of possible states, while edges encode nominal liabilities. Our framework generalizes the scalar-valued clearing vectors of the classical model to lattice-valued clearing sections, preserving the elegant fixed-point structure while dramatically expanding its descriptive power. Our main theorem establishes that such networks possess clearing sections that themselves form a complete lattice under the product order. This structure theorem enables tractable analysis of equilibria in diverse domains, including multi-currency financial systems, decentralized finance with automated market makers, supply chains with resource transformation, and permission networks with complex authorization structures. We further extend our framework to chain-complete lattices for term structure models and multivalued mappings for complex negotiation systems. Our results demonstrate how lattice theory provides a natural language for understanding complex network dynamics across multiple domains, creating a unified mathematical foundation for analyzing systemic risk, resource allocation, and network stability.
Decentralized lending protocols within the decentralized finance ecosystem enable the lending and borrowing of crypto-assets without relying on traditional intermediaries. Interest rates in these protocols are set algorithmically and fluctuate according to the supply and demand for liquidity. In this study, we propose an agent-based model tailored to a decentralized lending protocol and determine the optimal interest rate model. When the responses of the agents are linear with respect to the interest rate, the optimal solution is derived from a system of Riccati-type ODEs. For nonlinear behaviors, we propose a Monte-Carlo estimator, coupled with deep learning techniques, to approximate the optimal solution. Finally, after calibrating the model using block-by-block data, we conduct a risk-adjusted profit and loss analysis of the liquidity pool under industry-standard interest rate models and benchmark them against the optimal interest rate model.
This paper examines a semi-analytical approach for pricing American options in time-inhomogeneous models characterized by negative interest rates (for equity/FX) or negative convenience yields (for commodities/cryptocurrencies). Under such conditions, exercise boundaries may exhibit a "floating" structure - dynamically appearing and disappearing. For example, a second exercise boundary could emerge within the computational domain and subsequently both could collapse, demanding specialized pricing methodologies.
This paper develops a rigorous mathematical framework for analyzing Concentrated Liquidity Market Makers (CLMMs) in Decentralized Finance (DeFi) within a continuous-time setting. We model the evolution of liquidity profiles as measure-valued processes and characterize their dynamics under continuous trading. Our analysis encompasses two critical aspects of CLMMs: the mechanics of concentrated liquidity provision and the strategic behavior of arbitrageurs. We examine three distinct arbitrage models -- myopic, finite-horizon, and infinite-horizon with discounted and ergodic controls -- and derive closed-form solutions for optimal arbitrage strategies under each scenario. Importantly, we demonstrate that the presence of trading fees fundamentally constrains the admissible price processes, as the inclusion of fees precludes the existence of diffusion terms in the price process to avoid infinite fee generation. This finding has significant implications for CLMM design and market efficiency.
Decentralized finance (DeFi) has revolutionized the financial landscape, with protocols like Uniswap offering innovative automated market-making mechanisms. This article explores the development of a backtesting framework specifically tailored for concentrated liquidity market makers (CLMM). The focus is on leveraging the liquidity distribution approximated using a parametric model, to estimate the rewards within liquidity pools. The article details the design, implementation, and insights derived from this novel approach to backtesting within the context of Uniswap V3. The developed backtester was successfully utilized to assess reward levels across several pools using historical data from 2023 (pools Uniswap v3 for pairs of altcoins, stablecoins and USDC/ETH with different fee levels). Moreover, the error in modeling the level of rewards for the period under review for each pool was less than 1%. This demonstrated the effectiveness of the backtester in quantifying liquidity pool rewards and its potential in estimating LP's revenues as part of the pool rewards, as focus of our next research. The backtester serves as a tool to simulate trading strategies and liquidity provision scenarios, providing a quantitative assessment of potential returns for liquidity providers (LP). By incorporating statistical tools to mirror CLMM pool liquidity dynamics, this framework can be further leveraged for strategy enhancement and risk evaluation for LPs operating within decentralized exchanges. • Develop a methodology for backtesting liquidity provision in a CFMM. • Enhance CFMM backtesting by leveraging GPU acceleration for faster computation. • Showcase the practicality of CFMM backtesting using actual Uniswap pool data.
In decentralized finance, any individual can pool their assets into an automated market maker (AMM) -- herein we focus on the constant product market maker (CPMM) -- in exchange for a claim on a fraction of future pool assets and fees earned from the market making operations. This position is represented by a liquidity token, whose prevailing on-chain price is effectively the initial deposited assets. Though this price is well-defined, we treat the liquidity token as a derivative position in the prices of the underlying assets for the CPMM in order to deduce risk-neutral pricing and hedging formulas, not dissimilar to the Black-Scholes result. Adopting this perspective, in a frictionless environment, hedging the CPMM liquidity token under fair valuation should produce a riskless process, which therefore grows at the risk-free rate, something that is not seen in empirical case studies under the prevailing price. With our novel pricing formula, we construct a method to calibrate a volatility to data which provides an updated (non-market) valuation which is consistent with the (near-continuous) replication strategy out-of-sample. We conclude with a discussion of novel AMM design considerations motivated by this derivative-pricing perspective.
Rob McLaughlin, Nir Chemaya, Dingyue Liu, Dahlia Malkhi
This paper introduces a trade ordering rule that aims to reduce intra-block price volatility in Automated Market Maker (AMM) powered decentralized exchanges. The ordering rule introduced here, Clever Look-ahead Volatility Reduction (CLVR), operates under the (common) framework in decentralized finance that allows some entities to observe trade requests before they are settled, assemble them into "blocks", and order them as they like. On AMM exchanges, asset prices are continuously and transparently updated as a result of each trade and therefore, transaction order has high financial value. CLVR aims to order transactions for traders' benefit. Our primary focus is intra-block price stability (minimizing volatility), which has two main benefits for traders: it reduces transaction failure rate and allows traders to receive closer prices to the reference price at which they submit their transactions accordingly. We show that CLVR constructs an ordering which approximately minimizes price volatility with a small computation cost and can be trivially verified externally.
An Pham Ngoc Nguyen, Martin Crane, Thomas Conlon, Marija Bezbradica
Herding behavior has become a familiar phenomenon to investors, with potential dangers of both undervaluing and overvaluing assets, while also threatening market stability. This study contributes to the literature on herding behavior by using a recent dataset, covering the most impactful events of recent years. To our knowledge, this is the first study examining herding behavior across three different types of investment vehicle and also the first study observing herding at a community (subset) level. Specifically, we first explore this phenomenon in each separate type of investment vehicle, namely stocks, US ETFs and cryptocurrencies, using the Cross-Sectional Absolute Deviation model. We find mostly similar herding patterns for stocks and US ETFs. Subsequently, the same experiment is implemented on a combination of all three investment vehicles. For a deeper investigation, we adopt graph-based techniques including the Minimum Spanning Tree and Louvain community detection to partition the combination into smaller subsets to detect herding behavior for each subset. We find that herding behavior exists at all times across all types of investment vehicle at a subset level, although perhaps not at the superset level, and that this herding behavior tends to stem from specific events that solely impact that subset of assets. Lastly, we explore herding by examining the financial contagion effects between these types of investment vehicle. Results show that US ETFs not only have a tendency to propagate similar trading behaviors in stocks and especially cryptocurrencies but also show self-reinforcing herding behavior, acting as drivers of their own trends.
This research proposes a novel arbitrage approach in multivariate pair trading, termed the Optimal Trading Technique (OTT). We present a method for selectively forming a "bucket" of fiat currencies anchored to cryptocurrency for monitoring and exploiting trading opportunities simultaneously. To address quantitative conflicts from multiple trading signals, a novel bi-objective convex optimization formulation is designed to balance investor preferences between profitability and risk tolerance. We understand that cryptocurrencies carry significant financial risks. Therefore this process includes tunable parameters such as volatility penalties and action thresholds. In experiments conducted in the cryptocurrency market from 2020 to 2022, which encompassed a vigorous bull run followed by a bear run, the OTT achieved an annualized profit of 15.49%. Additionally, supplementary experiments detailed in the appendix extend the applicability of OTT to other major cryptocurrencies in the post-COVID period, validating the model's robustness and effectiveness in various market conditions. The arbitrage operation offers a new perspective on trading, without requiring external shorting or holding the intermediate during the arbitrage period. As a note of caution, this study acknowledges the high-risk nature of cryptocurrency investments, which can be subject to significant volatility and potential loss.
The study examines whether fama-french equity factors can effectively explain the idiosyncratic risk and return characteristics of Bitcoin. By incorporating Fama-french factors, the explanatory power of these factors on Bitcoin's excess returns over various moving average periods is tested through applications of several statistical methods. The analysis aims to determine if equity market factors are significant in explaining and modeling systemic risk in Bitcoin.
We study how trading fees and continuous-time arbitrage affect the profitability of liquidity providers (LPs) in Geometric Mean Market Makers (G3Ms). We use stochastic reflected diffusion processes to analyze the dynamics of a G3M model under the arbitrage-driven market [Milionis et al. 2022a. “Automated Market Making and Loss-Versus-Rebalancing.” arXiv e-prints]. Our research focuses on calculating LP wealth and extends the findings of Tassy and White [Tassy and White. 2020. “Growth Rate of a Liquidity Provider's Wealth in xy = c Automated Market Makers.”] for the constant product market maker (Uniswap v2) to a broader range of G3Ms, including Balancer. This allows us to calculate the long-term expected logarithmic growth of LP wealth, offering new insights into the complex dynamics of AMMs and their implications for LPs in decentralized finance.
Recent empirical evidence has highlighted the crucial role of jumps in both price and volatility within the cryptocurrency market. In this paper, we integrate price--volatility co-jumps and volatility short-term dependency into a coherent model framework, featuring fractional stochastic volatility. We particularly focus on inverse options, including the emerging Quanto inverse options and their power-type generalizations, aiming at mitigating cryptocurrency exchange rate risk and adjusting inherent risk exposure. Characteristic function-based pricing--hedging formulas are derived for these inverse options. The model framework is applied to asymmetric Laplace jump-diffusions and Gaussian-mixed tempered stable-type processes, employing three types of fractional kernels, for an extensive empirical analysis involving model calibration on two independent Bitcoin options data sets, during and after the COVID-19 pandemic. Key insights from our theoretical analysis and empirical findings include: (1) the superior performance of fractional stochastic-volatility models compared to various benchmark models, including those incorporating jumps and stochastic volatility, along with high computational efficiency when utilizing a piecewise kernel, (2) the practical necessity of considering jumps in both price and volatility, along with rough volatility, in pricing and hedging cryptocurrency options, (3) stability of calibrated parameter values in line with stylized facts.
Mikkel Bennedsen, Kim Christensen, Peter Christensen
We develop a framework for composite likelihood estimation of parametric continuous-time stationary Gaussian processes. We derive the asymptotic theory of the associated maximum composite likelihood estimator. We implement our approach on a pair of models that have been proposed to describe the random log-spot variance of financial asset returns. A simulation study shows that it delivers good performance in these settings and improves upon a method-of-moments estimation. In an empirical investigation, we inspect the dynamic of an intraday measure of the spot log-realized variance computed with high-frequency data from the cryptocurrency market. The evidence supports a mechanism, where the short- and long-term correlation structure of stochastic volatility are decoupled in order to capture its properties at different time scales. This is further backed by an analysis of the associated spot log-trading volume.
An intriguing link between a wide range of problems occurring in physics and financial engineering is presented. These problems include the evolution of small perturbations of linear flows in hydrodynamics, the movements of particles in random fields described by the Kolmogorov and Klein-Kramers equations, the Ornstein-Uhlenbeck and Feller processes, and their generalizations. They are reduced to affine differential and pseudo-differential equations and solved in a unified way by using Kelvin waves and developing a comprehensive math framework for calculating transition probabilities and expectations. Kelvin waves are instrumental for studying the well-known Black-Scholes, Heston, and Stein-Stein models and more complex path-dependent volatility models, as well as the pricing of Asian options, volatility and variance swaps, bonds, and bond options. Kelvin waves help to solve several cutting-edge problems, including hedging the impermanent loss of Automated Market Makers for cryptocurrency trading. This title is also available as Open Access on Cambridge Core.
Austin Adams, Ciamac C. Moallemi, Sara Reynolds, Dan Robinson
Automated market makers (AMMs) have emerged as the dominant market mechanism for trading on decentralized exchanges implemented on blockchains. This paper presents a single mechanism that targets two important unsolved problems for AMMs: reducing losses to informed orderflow, and maximizing revenue from uninformed orderflow. The ``auction-managed AMM'' works by running a censorship-resistant onchain auction for the right to temporarily act as ``pool manager'' for a constant-product AMM. The pool manager sets the swap fee rate on the pool, and also receives the accrued fees from swaps. The pool manager can exclusively capture some arbitrage by trading against the pool in response to small price movements, and also can set swap fees incorporating price sensitivity of retail orderflow and adapting to changing market conditions, with the benefits from both ultimately accruing to liquidity providers. Liquidity providers can enter and exit the pool freely in response to changing rent, though they must pay a small fee on withdrawal. We prove that under certain assumptions, this AMM should have higher liquidity in equilibrium than any standard, fixed-fee AMM.
We propose a novel approach to marked Hawkes kernel inference which we name the moment-based neural Hawkes estimation method. Hawkes processes are fully characterized by their first- and second-order statistics through a Fredholm integral equation of the second kind. Using recent advances in solving partial differential equations with physics-informed neural networks, we provide a numerical procedure to solve this integral equation in high dimension. Together with an adapted training pipeline, we give a generic set of hyperparameters that produces robust results across a wide range of kernel shapes. We conduct an extensive numerical validation on simulated data. We finally propose two applications of the method to the analysis of the microstructure of cryptocurrency markets. In a first application, we extract the influence of volume on the arrival rate of BTC-USD trades and in a second application we analyze the causality relationships and their directions amongst a universe of 15 cryptocurrency pairs in a centralized exchange.
In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating It^o's formula we first compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We also show that our results extend easily to Quanto-Inverse options. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew. Finally, we provide an empirical application using Bitcoin options traded on Debirit to show how our theoretical formulas can be used to model real market data of such options.
We extend the application and test the performance of a recently introduced volatility prediction framework encompassing LSTM and rough volatility. Our asset class of interest is cryptocurrencies, at the beginning of the "crypto-winter" in 2022. We first show that to forecast volatility, a universal LSTM approach trained on a pool of assets outperforms traditional models. We then consider a parsimonious parametric model based on rough volatility and Zumbach effect. We obtain similar prediction performances with only five parameters whose values are non-asset-dependent. Our findings provide further evidence on the universality of the mechanisms underlying the volatility formation process.
Consider a finite set of trade orders and automated market makers (AMMs) at some state. We propose a solution to the problem of finding an equilibrium price vector to execute all the orders jointly with corresponding optimal AMMs swaps. The solution is based on Brouwer's fixed-point theorem. We discuss computational aspects relevant for realistic situations in public blockchain activity.
Abstract Cryptocurrencies and Bitcoin, in particular, are prone to wild swings resulting in frequent jumps in prices, making them historically popular for traders to speculate. It is claimed in recent literature that Bitcoin price is influenced by sentiment about the Bitcoin system. Transaction, as well as the popularity, have shown positive evidence as potential drivers of Bitcoin price. This study introduces a bivariate jump-diffusion model to capture the dynamics of Bitcoin prices and the Bitcoin sentiment indicator, integrating trading volumes or Google search trends with Bitcoin price movements. We derive a closed-form solution for the Bitcoin price and the associated Black–Scholes equation for Bitcoin option valuation. The resulting partial differential equation for Bitcoin options is solved using an artificial neural network, and the model is validated with data from highly volatile stocks. We further test the model’s robustness across a broad spectrum of parameters, comparing the results to those obtained through Monte Carlo simulations. Our findings demonstrate the model’s practical significance in accurately predicting Bitcoin price movements and option values, providing a reliable tool for traders, analysts, and risk managers in the cryptocurrency market.
It is a well-documented fact that the correlation function of the returns on two "related" assets is generally increasing as a function of the horizon $h$ of these returns. This phenomenon, termed the Epps Effect, holds true in a wide variety of markets, and there is a large body of literature devoted to its theoretical justification. Our focus here is to describe and understand a deviation to the Epps effect, observed in the context of the foreign exchange and cryptocurrency markets. Specifically, we document a sharp local maximum of the cross-correlation function of returns on the Euro EUR/USD and Bitcoin BTC/USD pairs as a function of $h$. Our claim is that this anomaly reveals the activity of short-term momentum traders.