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arXiv · arXiv · 2023

Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision

Constant product markets with concentrated liquidity (CL) are the most popular type of automated market makers. In this paper, we characterise the continuous-time wealth dynamics of strategic LPs who dynamically adjust their range of liquidity provision in CL pools. Their wealth results from fee income, the value of their holdings in the pool, and rebalancing costs. Next, we derive a self-financing and closed-form op…

Álvaro Cartea, Fayçal Drissi, Marcello Monga

arXiv · arXiv · 2022

Static Replication of Impermanent Loss for Concentrated Liquidity Provision in Decentralised Markets

This article analytically characterizes the impermanent loss of concentrated liquidity provision for automatic market makers in decentralised markets such as Uniswap. We propose two static replication formulas for the impermanent loss by a combination of European calls or puts with strike prices supported on the liquidity provision price interval. It facilitates liquidity providers to hedge permanent loss by trading …

Jun Deng, Hua Zong, Yun Wang

arXiv · arXiv · 2021

UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider

Uniswap is a decentralized exchange (DEX) and was first launched on November 2, 2018 on the Ethereum mainnet [1] and is part of an Ecosystem of products in Decentralized Finance (DeFi). It replaces a traditional order book type of trading common on centralized exchanges (CEX) with a deterministic model that swaps currencies (or tokens/assets) along a fixed price function determined by the amount of currencies supplie…

Andreas A. Aigner, Gurvinder Dhaliwal

arXiv · arXiv · 2026

Bank Run Exposure in a Paycheck-to-Paycheck Economy with Loss-Averse Depositors

We develop a behavioural model of bank run exposure in a paycheck-to-paycheck economy with loss averse depositors. Income is received through demand deposits, and consumption ratcheting embeds reference dependence in a parsimonious asset-pricing framework. We show that sufficiently high subjective bad-state probabilities endogenously increase liquidity demand and generate equilibrium stress states supporting bank run…

G. Charles-Cadogan

arXiv · arXiv · 2026

Axient: On-Chain Credit and Loss Allocation for Leveraged Event Markets: A Venue-Agnostic Protocol for Traders, Credit Providers, Market Makers, and Liquidation Backstops

A physically backed leveraged event position requires real credit: if collateral C receives leverage L, the protocol supplies (L-1)C and uses the combined amount to acquire recognized event exposure. This paper develops a venue-agnostic on-chain credit architecture for that capital layer and an endogenous model of its capital market. It separates traders, Senior Credit LPs, market makers, liquidators, and Liquidation…

Maksym Nechepurenko

arXiv · arXiv · 2026

Uniform-Loss Automated Market Making for Prediction Markets

Automated market makers (AMMs) for prediction markets descend from market scoring rules, where a mechanism operator subsidizes a market to aggregate beliefs about uncertain events. The existing literature has focused on bounding the total worst-case loss to the subsidizer, but has not addressed how that loss is distributed across price states or over time. We use the framework of loss-versus-rebalancing (LVR) to stud…

Ciamac C. Moallemi, Dan Robinson, Brian Zhu

arXiv · arXiv · 2026

Optimal Dynamic Fees for Automated Market Makers: A Stochastic Control Approach to Loss-Versus-Rebalancing

We study the fee policy of a liquidity provider (LP) in a constant-product automated market maker (AMM) whose fee can be adjusted continuously, as enabled by programmable hooks. Building on the loss-versus-rebalancing (LVR) framework of Milionis et al. (2022) and its extension to nonzero fees by Milionis et al. (2024), we model the LP's wealth relative to the continuously rebalanced benchmark as a controlled process …

Farbod Ghasemlu

arXiv · arXiv · 2025

Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options

This paper mathematically models a constant-function automated market maker (CFAMM) position as a portfolio of exotic options, known as perpetual American continuous-installment (CI) options. This model replicates an AMM position's delta at each point in time over an infinite time horizon, thus taking into account the perpetual nature and optionality to withdraw of liquidity provision. This framework yields two key t…

Srisht Fateh Singh, Reina Ke Xin Li, Samuel Gaskin, Yuntao Wu, Jeffrey Klinck

arXiv · arXiv · 2024

Rebalancing-versus-Rebalancing: Improving the fidelity of Loss-versus-Rebalancing

Automated Market Makers (AMMs) hold assets and are constantly being rebalanced by external arbitrageurs to match external market prices. Loss-versus-rebalancing (LVR) is a pivotal metric for measuring how an AMM pool performs for its liquidity providers (LPs) relative to an idealised benchmark where rebalancing is done not via the action of arbitrageurs but instead by trading with a perfect centralised exchange with …

Matthew Willetts, Christian Harrington

arXiv · arXiv · 2024

Impermanent loss and loss-vs-rebalancing I: some statistical properties

There are two predominant metrics to assess the performance of automated market makers and their profitability for liquidity providers: 'impermanent loss' (IL) and 'loss-versus-rebalance' (LVR). In this short paper we shed light on the statistical aspects of both concepts and show that they are more similar than conventionally appreciated. Our analysis uses the properties of a random walk and some analytical properti…

Abe Alexander, Lars Fritz

arXiv · arXiv · 2021

Impermanent Loss in Uniswap v3

AMMs are autonomous smart contracts deployed on a blockchain that make markets between different assets that live on that chain. In this paper we are examining a specific class of AMMs called Constant Function Market Makers whose trading profile, ignoring fees, is determined by their bonding curve. This class of AMM suffers from what is commonly referred to as Impermanent Loss, which we have previously identified as …

Stefan Loesch, Nate Hindman, Mark B Richardson, Nicholas Welch

arXiv · arXiv · 2020

Loss-Given-Default Modeling by Post-Last Passage Time Process

This study proposes a stochastic model for loss-given-default (LGD) which provides the LGD distribution based on credit market and company-specific financial conditions. The model utilizes last passage time of a linear diffusion (representing firm value) to a certain threshold point, after which default occurs as a surprising event. By treating the post-last passage time process in a continuum of the original process…

Masahiko Egami, Rusudan Kevkhishvili

arXiv · arXiv · 2008

Default correlation, cluster dynamics and single names: The GPCL dynamical loss model

We extend the common Poisson shock framework reviewed for example in Lindskog and McNeil (2003) to a formulation avoiding repeated defaults, thus obtaining a model that can account consistently for single name default dynamics, cluster default dynamics and default counting process. This approach allows one to introduce significant dynamics, improving on the standard "bottom-up" approaches, and to achieve true consist…

Damiano Brigo, Andrea Pallavicini, Roberto Torresetti

arXiv · arXiv · 2026

A Censored Transformed Model for Proportional Outcomes with Boundary Mass and an Application to Loss Given Default Modeling

We introduce the zero-one censored transformed normal (ZOC-TN) model for proportional responses with potential probability mass at the boundaries 0 and 1. The model combines a censored Gaussian variable with a two-parameter affine-logit transformation on the interior (0,1). We characterize the transformation parameters, establish large-sample properties, and relate the affine-logit specification to broader classes of…

Yuan Christopher Qiang, Fabio Sigrist

arXiv · arXiv · 2026

From Control Boundary to Insurance Claim: Reconstructing AI-Mediated Losses Through the CER Framework

AI losses that arise through an insured organization's generative or agentic AI system require state reconstruction, not merely event reconstruction, because the relevant state changes as the system reasons, retrieves, calls tools, and acts. The relevant question is not only what loss occurred, but what the system was allowed to do, what it actually did, and whether that reconstructed loss can support insurance claim…

Alex Leung, Rex Zhang, Kentaroh Toyoda, SiewMei Loh

arXiv · arXiv · 2025

Assessment of loan losses after default

The paper shows how to determine the loss on an LGD borrower's loan after default, with or without preparation of a separate model. LGD after default is estimated taking into account the average repayment period of the defaulted loan, knowledge of volumes, moments of default and repayments, the rate or other parameters in the vector of determinants. The calculation of the average repayment period for overdue loans is…

Pomazanov Mikhail

arXiv · arXiv · 2025

Pool Value Replication (CPM) and Impermanent Loss Hedging

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…

Agustin Muñoz Gonzalez, Juan Ignacio Sequeira, Ariel Dembling

Wiki Entities · 21

Backpropagation

Backpropagation computes gradients of a scalar loss with respect to every weight by applying the chain rule backwards through the computational graph.

Cross-Entropy Loss

Cross-entropy measures how well a predicted distribution q matches a target distribution p. For one-hot labels it reduces to −log q(y), the usual classification and language-model loss.

Gradient Descent

Gradient descent updates parameters against the gradient of a loss: θ ← θ − η ∇_θ L. Stochastic and mini-batch variants make the method tractable on large datasets.

Overfitting

Overfitting is when a model fits training idiosyncrasies instead of the transferable regularity, so held-out or live error rises even as train loss falls.

Scaling Laws

Scaling laws are empirical power laws relating language-model loss to parameter count, data, and compute, used to plan pretraining rather than guess.

Banking

Bank Capital Ratio

Bank Capital Ratio — Loss-absorbing equity buffer determining lending capacity and dividend policy.

Credit Default Swap

A CDS is a bilateral contract that pays the loss on a reference credit after a credit event — default insurance quoted as a spread.

Investment Grade

Investment grade is a credit rating of BBB− / Baa3 or better — a regulatory and mandate bucket, not a promise of no loss.

Loss Given Default

LGD is 1 minus recovery — the fraction of exposure lost when default happens.

Recovery Rate

Recovery is what a claim is worth after default, as a fraction of par — the complement of loss given default.

Financial Crises

Barings 1995

Barings Bank was wiped out in 1995 by Nick Leeson’s hidden Nikkei futures losses in Singapore — a rogue-trader plus failed control story, not a macro crisis.

Liquidity

Clearing Member Default Waterfall

Clearing Member Default Waterfall — Loss-allocation sequence after a clearing member fails.

Mathematics

Kullback–Leibler Divergence

KL divergence KL(P‖Q) = E_P[log dP/dQ] is the expected extra log-loss from using Q when the truth is P — the loss behind cross-entropy training and many variational methods.

Microstructure

Adverse Selection

Adverse selection is the expected loss a liquidity provider takes when the other side is informed — the Glosten–Milgrom reason spreads exist even with no inventory.

Microstructure

Stop-Loss Order

A stop-loss becomes a market (or stop-limit) order once a trigger trades — a planned exit that can become a gap-out.

Disposition Effect

The disposition effect is the habit of selling winners and keeping losers — realizing gains, papering losses, versus a mark-to-market rule.

Expected Shortfall

Expected shortfall is the average loss beyond VaR — a coherent tail measure that asks how bad the bad days are.

Loss Aversion

Loss aversion is the empirical fact that losses hurt more than equal gains please — a kink at the reference point, not a risk-aversion parameter.

Maximum Drawdown Control

Maximum Drawdown Control — Rules that de-risk after losses to preserve capital and investor mandates.