arXiv · arXiv · 2021
This study derives the expected liquidity cost when performing the delta hedging process of a European option. This cost is represented by an integration formula that includes European option prices and a certain function depending on the delta process. We first define a unit liquidity cost and then show that the liquidity cost is a multiplication of the unit liquidity cost, stock price, supply curve parameter, and t…
Kyungsub Lee, Byoung Ki Seo
arXiv · arXiv · 2019
We propose an option approach for pricing bond illiquidity that is reminiscent of the celebrated work of Longstaff (1995) on the non-marketability of some non-dividend-paying shares in IPOs. This approach describes a quite common situation in the fixed income market: it is rather usual to find issuers that, besides liquid benchmark bonds, issue some other bonds that either are placed to a small number of investors in…
Roberto Baviera, Aldo Nassigh, Emanuele Nastasi
arXiv · arXiv · 2018
A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel…
Janine Balter, Alexander J. McNeil
arXiv · arXiv · 2024
This paper introduces a novel stochastic model for credit spreads. The stochastic approach leverages the diffusion of default intensities via a CIR++ model and is formulated within a risk-neutral probability space. Our research primarily addresses two gaps in the literature. The first is the lack of credit spread models founded on a stochastic basis that enables continuous modeling, as many existing models rely on fa…
Mohamed Ben Alaya, Ahmed Kebaier, Djibril Sarr
arXiv · arXiv · 2023
In this paper, we consider a generic interest rate market in the presence of roll-over risk, which generates spreads in spot/forward term rates. We do not require classical absence of arbitrage and rely instead on a minimal market viability assumption, which enables us to work in the context of the benchmark approach. In a Markovian setting, we extend the control theoretic approach of Gombani & Runggaldier (2013) and…
Claudio Fontana, Simone Pavarana, Wolfgang J. Runggaldier
arXiv · arXiv · 2026
Automated market makers (AMMs) are a cornerstone of decentralised finance (DeFi). Constant product markets with concentrated liquidity, such as UniswapV3, are now a well-established design. In these markets, liquidity providers (LPs) face a sequential decision problem: they must decide when to rebalance their positions and which price ranges to allocate capital to as market conditions evolve. We formulate dynamic liq…
Georgios Chionas, Charalampos Kleitsikas, Stefanos Leonardos, Leandro Sánchez-Betancourt, Carmine Ventre
arXiv · arXiv · 2026
We develop a variational formulation of Kyle's model of informed trading that accommodates stochastic liquidity and multiple traded assets. The main equilibrium result is stated first: under a martingale dual condition, a matrix-valued martingale depth process generates a linear-Gaussian equilibrium with stochastic matrix-valued price impact. We derive this martingale from a primal-dual problem, inspired by causal op…
Ibrahim Ekren, Evangelos A. Nikitopoulos, Lu Vy
arXiv · arXiv · 2026
This paper develops a robust mathematical framework for Constant Function Market Makers (CFMMs) by transitioning from traditional token reserve analyses to a coordinate system defined by price and intrinsic liquidity. We establish a canonical parametrization of the bonding curve that ensures dimensional consistency across diverse trading functions, such as those employed by Uniswap and Balancer, and demonstrate that …
Jimmy Risk, Shen-Ning Tung, Tai-Ho Wang
arXiv · arXiv · 2025
We study the economic viability of liquidity provision in decentralised exchanges (DEXs) within a structural framework in which market outcomes are endogenous. We formulate strategic interactions as a sequential game: a risk-averse liquidity provider (LP) sets the supply of liquidity in the DEX and a costly dynamic replication strategy in a centralised exchange (CEX), price-sensitive traders determine trading volumes…
Fayçal Drissi, Xuchen Wu, Sebastian Jaimungal
arXiv · arXiv · 2025
We find the equilibrium contract that an automated market maker (AMM) offers to their strategic liquidity providers (LPs) in order to maximize the order flow that gets processed by the venue. Our model is formulated as a leader-follower stochastic game, where the venue is the leader and a representative LP is the follower. We derive approximate closed-form equilibrium solutions to the stochastic game and analyze the …
Alif Aqsha, Philippe Bergault, Leandro Sánchez-Betancourt
arXiv · arXiv · 2025
Although the valuation of life contingent assets has been thoroughly investigated under the framework of mathematical statistics, little financial economics research pays attention to the pricing of these assets in a non-arbitrage, complete market. In this paper, we first revisit the Fundamental Theorem of Asset Pricing (FTAP) and the short proof of it. Then we point out that discounted asset price is a martingale on…
Patrick Ling
arXiv · arXiv · 2025
We study a multi-agent setting in which brokers transact with an informed trader. Through a sequential Stackelberg-type game, brokers manage trading costs and adverse selection with an informed trader. In particular, supplying liquidity to the informed traders allows the brokers to speculate based on the flow information. They simultaneously attempt to minimize inventory risk and trading costs with the lit market bas…
Ryan Donnelly, Zi Li
arXiv · arXiv · 2024
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 pri…
Maxim Bichuch, Zachary Feinstein
arXiv · arXiv · 2024
We formulate quantum computing solutions to a large class of dynamic nonlinear asset pricing models using algorithms, in theory exponentially more efficient than classical ones, which leverage the quantum properties of superposition and entanglement. The equilibrium asset pricing solution is a quantum state. We introduce quantum decision-theoretic foundations of ambiguity and model/parameter uncertainty to deal with …
Eric Ghysels, Jack Morgan
arXiv · arXiv · 2023
Liquidity providers are currently incentivised to provide liquidity through the LP Incentives Programme on dYdX. Based on the various parameters - makerVolume, depths and spreads, they are rewarded accordingly based on their activities. Given the maturity of the BTC and ETH markets, alongside other altcoins which enjoy a consistent amount of liquidity, this paper aims to update the formula to encourage more active an…
Colin Chan
arXiv · arXiv · 2023
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 behav…
Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden
arXiv · arXiv · 2023
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 impo…
Niccolò Bardoscia, Alessandro Nodari
arXiv · arXiv · 2022
We study the formation of an optimal interbank network in a model where banks control both their supply of liquidity, through cash reserves, and their exposures to other banks' risky projects. The value of each bank's project may suddenly decline depending on their cash reserves and both the occurence and magnitude of liquidity shocks. In two distinct settings, we solve the system-wide optimal control problem and obt…
Daniel E. Rigobon, Ronnie Sircar