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Liquidity Providers on Automated Market Makers generate millions of USD in transaction fees daily. However, the net value of a Liquidity Position is vulnerable to price changes in the underlying assets in the pool. The dominant measure of loss in a Liquidity Position is Impermanent Loss. Impermanent Loss for Constant Function Market Makers has been widely studied. We propose a new metric to measure Liquidity Position…
In this paper, we focus on finding the optimal hedging strategy of a credit index option using reinforcement learning. We take a practical approach, where the focus is on realism i.e. discrete time, transaction costs; even testing our policy on real market data. We apply a state of the art algorithm, the Trust Region Volatility Optimization (TRVO) algorithm and show that the derived hedging strategy outperforms the p…
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…
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 …
We derive the stochastic price process for tokens whose sole price discovery mechanism is a constant-product automated market maker (AMM). When the net flow into the pool follows a diffusion, the token price follows a constant elasticity of variance (CEV) process, nesting Black-Scholes as the limiting case of infinite liquidity. We obtain closed-form European option prices and introduce liquidity-adjusted Greeks. The…
Modelling stock prices via jump processes is common in financial markets. In practice, to hedge a contingent claim one typically uses the so-called delta-hedging strategy. This strategy stems from the Black--Merton--Scholes model where it perfectly replicates contingent claims. From the theoretical viewpoint, there is no reason for this to hold in models with jumps. However in practice the delta-hedging strategy is w…
In the theory of riskfree hedges in continuous time finance, one can start with the delta-hedge and derive the option pricing equation, or one can start with the replicating, self-financing hedging strategy and derive both the delta-hedge and the option pricing partial differential equation. Approximately reversible trading is implicitly assumed in both cases. The option pricing equation is not restricted to the stan…
This study deals with the problem of pricing European currency options in discrete time setting, whose prices follow the fractional Black Scholes model with transaction costs. Both the pricing formula and the fractional partial differential equation for European call currency options are obtained by applying the delta-hedging strategy. Some Greeks and the estimator of volatility are also provided. The empirical studi…
This paper studies empirical deep hedging for S&P 500 index options under a local downside-shortfall reward. It moves beyond performance comparison by asking what the learned hedge does, when it fails, and whether it can be made auditable. TD3 agents are compared with a daily-updated Black-Scholes delta hedge on the same option episodes. In walk-forward tests from 2015 to 2023, the agents usually learn a systematic d…
This paper investigates the hedging performance of pegged foreign exchange market in a regime switching (RS) model introduced in a recent paper by Drapeau, Wang and Wang (2019). We compare two prices, an exact solution and first order approximation and provide the bounds for the error. We provide exact RS delta, approximated RS delta as well as mean variance hedging strategies for this specific model and compare thei…
The domain of hedge fund investments is undergoing significant transformation, influenced by the rapid expansion of data availability and the advancement of analytical technologies. This study explores the enhancement of hedge fund investment performance through the integration of machine learning techniques, the application of PolyModel feature selection, and the analysis of fund size. We address three critical ques…
We develop an optimal currency hedging strategy for fund managers who own foreign assets to choose the hedge tenors that maximize their FX carry returns within a liquidity risk constraint. The strategy assumes that the offshore assets are fully hedged with FX forwards. The chosen liquidity risk metric is Cash Flow at Risk (CFaR). The strategy involves time-dispersing the total nominal hedge value into future time buc…
In volatile financial markets, balancing risk and return remains a significant challenge. Traditional approaches often focus solely on equity allocation, overlooking the strategic advantages of options trading for dynamic risk hedging. This work presents DeltaHedge, a multi-agent framework that integrates options trading with AI-driven portfolio management. By combining advanced reinforcement learning techniques with…
Automated Market Makers based on concentrated liquidity, such as Uniswap v3, significantly improve capital efficiency but expose Liquidity Providers (LPs) to adverse selection costs, formalized as Loss-Versus-Rebalancing (LVR). While theoretical literature quantifies these costs, the interplay between realistic blockchain microstructure and endogenous pricing mechanisms remains under-explored. This paper develops a g…
We study cash-flow forecasting for derivatives used in liquidity management and clarify its relation to risk-neutral valuation and replication. While it is well known that expectations under different measures (e.g., $\mathbb{P}$ vs. $\mathbb{Q}$) can yield different undiscounted cash-flows, further inconsistencies arise when payment times are stochastic. We show that using discounting sensitivities (funding-curve he…
Understanding how market participants react to shocks like scheduled macroeconomic news is crucial for both traders and policymakers. We develop a calibrated data generation process DGP that embeds four stylized trader archetypes retail, pension, institutional, and hedge funds into an extended CAPM augmented by CPI surprises. Each agents order size choice is driven by a softmax discrete choice rule over small, medium…
We consider an equity market subject to risk from both unhedgeable shocks and default. The novelty of our work is that to partially offset default risk, investors may dynamically trade in a credit default swap (CDS) market. Assuming investment opportunities are driven by functions of an underlying diffusive factor process, we identify the certainty equivalent for a constant absolute risk aversion investor with a semi…
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…