Option Hedging

Model-Free Deep Hedging with Transaction Costs and Light Data Requirements ArXiv ID: 2505.22836 “View on arXiv” Authors: Pierre Brugière, Gabriel Turinici Abstract Option pricing theory, such as the Black and Scholes (1973) model, provides an explicit solution to construct a strategy that perfectly hedges an option in a continuous-time setting. In practice, however, trading occurs in discrete time and often involves transaction costs, making the direct application of continuous-time solutions potentially suboptimal. Previous studies, such as those by Buehler et al. (2018), Buehler et al. (2019) and Cao et al. (2019), have shown that deep learning or reinforcement learning can be used to derive better hedging strategies than those based on continuous-time models. However, these approaches typically rely on a large number of trajectories (of the order of $10^5$ or $10^6$) to train the model. In this work, we show that using as few as 256 trajectories is sufficient to train a neural network that significantly outperforms, in the Geometric Brownian Motion framework, both the classical Black & Scholes formula and the Leland model, which is arguably one of the most effective explicit alternatives for incorporating transaction costs. The ability to train neural networks with such a small number of trajectories suggests the potential for more practical and simple implementation on real-time financial series. ...

Unified Approach for Hedging Impermanent Loss of Liquidity Provision ArXiv ID: 2407.05146 “View on arXiv” Authors: Unknown Abstract We develop static and dynamic approaches for hedging of the impermanent loss (IL) of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols. We provide detailed definitions and formulas for computing the IL to unify different definitions occurring in the existing literature. We show that the IL can be seen a contingent claim with a non-linear payoff for a fixed maturity date. Thus, we introduce the contingent claim termed as IL protection claim which delivers the negative of IL payoff at the maturity date. We apply arbitrage-based methods for valuation and risk management of this claim. First, we develop the static model-independent replication method for the valuation of IL protection claim using traded European vanilla call and put options. We extend and generalize an existing method to show that the IL protection claim can be hedged perfectly with options if there is a liquid options market. Second, we develop the dynamic model-based approach for the valuation and hedging of IL protection claims under a risk-neutral measure. We derive analytic valuation formulas using a wide class of price dynamics for which the characteristic function is available under the risk-neutral measure. As base cases, we derive analytic valuation formulas for IL protection claim under the Black-Scholes-Merton model and the log-normal stochastic volatility model. We finally discuss estimation of risk-reward of LP staking using our results. ...