Delta Hedging

Application of Deep Reinforcement Learning to At-the-Money S&P 500 Options Hedging ArXiv ID: 2510.09247 “View on arXiv” Authors: Zofia Bracha, Paweł Sakowski, Jakub Michańków Abstract This paper explores the application of deep Q-learning to hedging at-the-money options on the S&P500 index. We develop an agent based on the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm, trained to simulate hedging decisions without making explicit model assumptions on price dynamics. The agent was trained on historical intraday prices of S&P500 call options across years 2004–2024, using a single time series of six predictor variables: option price, underlying asset price, moneyness, time to maturity, realized volatility, and current hedge position. A walk-forward procedure was applied for training, which led to nearly 17~years of out-of-sample evaluation. The performance of the deep reinforcement learning (DRL) agent is benchmarked against the Black–Scholes delta-hedging strategy over the same period. We assess both approaches using metrics such as annualized return, volatility, information ratio, and Sharpe ratio. To test the models’ adaptability, we performed simulations across varying market conditions and added constraints such as transaction costs and risk-awareness penalties. Our results show that the DRL agent can outperform traditional hedging methods, particularly in volatile or high-cost environments, highlighting its robustness and flexibility in practical trading contexts. While the agent consistently outperforms delta-hedging, its performance deteriorates when the risk-awareness parameter is higher. We also observed that the longer the time interval used for volatility estimation, the more stable the results. ...

Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options ArXiv ID: 2508.02971 “View on arXiv” Authors: Srisht Fateh Singh, Reina Ke Xin Li, Samuel Gaskin, Yuntao Wu, Jeffrey Klinck, Panagiotis Michalopoulos, Zissis Poulos, Andreas Veneris Abstract 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 theoretical results: (a) It proves that the AMM’s adverse-selection cost, loss-versus-rebalancing (LVR), is analytically identical to the continuous funding fees (the time value decay or theta) earned by the at-the-money CI option embedded in the replicating portfolio. (b) A special case of this model derives an AMM liquidity position’s delta profile and boundaries that suffer approximately constant LVR, up to a bounded residual error, over an arbitrarily long forward window. Finally, the paper describes how the constant volatility parameter required by the perpetual option can be calibrated from the term structure of implied volatilities and estimates the errors for both implied volatility calibration and LVR residual error. Thus, this work provides a practical framework enabling liquidity providers to choose an AMM liquidity profile and price boundaries for an arbitrarily long, forward-looking time window where they can expect an approximately constant, price-independent LVR. The results establish a rigorous option-theoretic interpretation of AMMs and their LVR, and provide actionable guidance for liquidity providers in estimating future adverse-selection costs and optimizing position parameters. ...

Perpetual Demand Lending Pools ArXiv ID: 2502.06028 “View on arXiv” Authors: Unknown Abstract Decentralized perpetuals protocols have collectively reached billions of dollars of daily trading volume, yet are still not serious competitors on the basis of trading volume with centralized venues such as Binance. One of the main reasons for this is the high cost of capital for market makers and sophisticated traders in decentralized settings. Recently, numerous decentralized finance protocols have been used to improve borrowing costs for perpetual futures traders. We formalize this class of mechanisms utilized by protocols such as Jupiter, Hyperliquid, and GMX, which we term~\emph{“Perpetual Demand Lending Pools”} (PDLPs). We then formalize a general target weight mechanism that generalizes what GMX and Jupiter are using in practice. We explicitly describe pool arbitrage and expected payoffs for arbitrageurs and liquidity providers within these mechanisms. Using this framework, we show that under general conditions, PDLPs are easy to delta hedge, partially explaining the proliferation of live hedged PDLP strategies. Our results suggest directions to improve capital efficiency in PDLPs via dynamic parametrization. ...

Enhancing Deep Hedging of Options with Implied Volatility Surface Feedback Information ArXiv ID: 2407.21138 “View on arXiv” Authors: Unknown Abstract We present a dynamic hedging scheme for S&P 500 options, where rebalancing decisions are enhanced by integrating information about the implied volatility surface dynamics. The optimal hedging strategy is obtained through a deep policy gradient-type reinforcement learning algorithm. The favorable inclusion of forward-looking information embedded in the volatility surface allows our procedure to outperform several conventional benchmarks such as practitioner and smiled-implied delta hedging procedures, both in simulation and backtesting experiments. The outperformance is more pronounced in the presence of transaction costs. ...

Enhancing Black-Scholes Delta Hedging via Deep Learning ArXiv ID: 2407.19367 “View on arXiv” Authors: Unknown Abstract This paper proposes a deep delta hedging framework for options, utilizing neural networks to learn the residuals between the hedging function and the implied Black-Scholes delta. This approach leverages the smoother properties of these residuals, enhancing deep learning performance. Utilizing ten years of daily S&P 500 index option data, our empirical analysis demonstrates that learning the residuals, using the mean squared one-step hedging error as the loss function, significantly improves hedging performance over directly learning the hedging function, often by more than 100%. Adding input features when learning the residuals enhances hedging performance more for puts than calls, with market sentiment being less crucial. Furthermore, learning the residuals with three years of data matches the hedging performance of directly learning with ten years of data, proving that our method demands less data. ...

Is the difference between deep hedging and delta hedging a statistical arbitrage? ArXiv ID: 2407.14736 “View on arXiv” Authors: Unknown Abstract The recent work of Horikawa and Nakagawa (2024) claims that under a complete market admitting statistical arbitrage, the difference between the hedging position provided by deep hedging and that of the replicating portfolio is a statistical arbitrage. This raises concerns as it entails that deep hedging can include a speculative component aimed simply at exploiting the structure of the risk measure guiding the hedging optimisation problem. We test whether such finding remains true in a GARCH-based market model, which is an illustrative case departing from complete market dynamics. We observe that the difference between deep hedging and delta hedging is a speculative overlay if the risk measure considered does not put sufficient relative weight on adverse outcomes. Nevertheless, a suitable choice of risk measure can prevent the deep hedging agent from engaging in speculation. ...

Construction and Hedging of Equity Index Options Portfolios ArXiv ID: 2407.13908 “View on arXiv” Authors: Unknown Abstract This research presents a comprehensive evaluation of systematic index option-writing strategies, focusing on S&P500 index options. We compare the performance of hedging strategies using the Black-Scholes-Merton (BSM) model and the Variance-Gamma (VG) model, emphasizing varying moneyness levels and different sizing methods based on delta and the VIX Index. The study employs 1-minute data of S&P500 index options and index quotes spanning from 2018 to 2023. The analysis benchmarks hedged strategies against buy-and-hold and naked option-writing strategies, with a focus on risk-adjusted performance metrics including transaction costs. Portfolio delta approximations are derived using implied volatility for the BSM model and market-calibrated parameters for the VG model. Key findings reveal that systematic option-writing strategies can potentially yield superior returns compared to buy-and-hold benchmarks. The BSM model generally provided better hedging outcomes than the VG model, although the VG model showed profitability in certain naked strategies as a tool for position sizing. In terms of rehedging frequency, we found that intraday hedging in 130-minute intervals provided both reliable protection against adverse market movements and a satisfactory returns profile. ...