Hedging

Constrained deep learning for pricing and hedging european options in incomplete markets ArXiv ID: 2511.20837 “View on arXiv” Authors: Nicolas Baradel Abstract In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets. ...

Tail-Safe Stochastic-Control SPX-VIX Hedging: A White-Box Bridge Between AI Sensitivities and Arbitrage-Free Market Dynamics ArXiv ID: 2510.15937 “View on arXiv” Authors: Jian’an Zhang Abstract We present a white-box, risk-sensitive framework for jointly hedging SPX and VIX exposures under transaction costs and regime shifts. The approach couples an arbitrage-free market teacher with a control layer that enforces safety as constraints. On the market side, we integrate an SSVI-based implied-volatility surface and a Cboe-compliant VIX computation (including wing pruning and 30-day interpolation), and connect prices to dynamics via a clipped, convexity-preserving Dupire local-volatility extractor. On the control side, we pose hedging as a small quadratic program with control-barrier-function (CBF) boxes for inventory, rate, and tail risk; a sufficient-descent execution gate that trades only when risk drop justifies cost; and three targeted tail-safety upgrades: a correlation/expiry-aware VIX weight, guarded no-trade bands, and expiry-aware micro-trade thresholds with cooldown. We prove existence/uniqueness and KKT regularity of the per-step QP, forward invariance of safety sets, one-step risk descent when the gate opens, and no chattering with bounded trade rates. For the dynamics layer, we establish positivity and second-order consistency of the discrete Dupire estimator and give an index-coherence bound linking the teacher VIX to a CIR-style proxy with explicit quadrature and projection errors. In a reproducible synthetic environment mirroring exchange rules and execution frictions, the controller reduces expected shortfall while suppressing nuisance turnover, and the teacher-surface construction keeps index-level residuals small and stable. ...

Volatility Calibration via Automatic Local Regression ArXiv ID: 2509.16334 “View on arXiv” Authors: Ruozhong Yang, Hao Qin, Charlie Che, Liming Feng Abstract Managing exotic derivatives requires accurate mark-to-market pricing and stable Greeks for reliable hedging. The Local Volatility (LV) model distinguishes itself from other pricing models by its ability to match observable market prices across all strikes and maturities with high accuracy. However, LV calibration is fundamentally ill-posed: finite market observables must determine a continuously-defined surface with infinite local volatility parameters. This ill-posed nature often causes spiky LV surfaces that are particularly problematic for finite-difference-based valuation, and induces high-frequency oscillations in solutions, thus leading to unstable Greeks. To address this challenge, we propose a pre-calibration smoothing method that can be integrated seamlessly into any LV calibration workflow. Our method pre-processes market observables using local regression that automatically minimizes asymptotic conditional mean squared error to generate denoised inputs for subsequent LV calibration. Numerical experiments demonstrate that the proposed pre-calibration smoothing yields significantly smoother LV surfaces and greatly improves Greek stability for exotic options with negligible additional computational cost, while preserving the LV model’s ability to fit market observables with high fidelity. ...

DeltaHedge: A Multi-Agent Framework for Portfolio Options Optimization ArXiv ID: 2509.12753 “View on arXiv” Authors: Feliks Bańka, Jarosław A. Chudziak Abstract 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 an ensembled options-based hedging strategy, DeltaHedge enhances risk-adjusted returns and stabilizes portfolio performance across varying market conditions. Experimental results demonstrate that DeltaHedge outperforms traditional strategies and standalone models, underscoring its potential to transform practical portfolio management in complex financial environments. Building on these findings, this paper contributes to the fields of quantitative finance and AI-driven portfolio optimization by introducing a novel multi-agent system for integrating options trading strategies, addressing a gap in the existing literature. ...

Nested Optimal Transport Distances ArXiv ID: 2509.06702 “View on arXiv” Authors: Ruben Bontorno, Songyan Hou Abstract Simulating realistic financial time series is essential for stress testing, scenario generation, and decision-making under uncertainty. Despite advances in deep generative models, there is no consensus metric for their evaluation. We focus on generative AI for financial time series in decision-making applications and employ the nested optimal transport distance, a time-causal variant of optimal transport distance, which is robust to tasks such as hedging, optimal stopping, and reinforcement learning. Moreover, we propose a statistically consistent, naturally parallelizable algorithm for its computation, achieving substantial speedups over existing approaches. ...

Design and hedging of unit linked life insurance with environmental factors ArXiv ID: 2509.05676 “View on arXiv” Authors: Katia Colaneri, Alessandra Cretarola, Edoardo Lombardo, Daniele Mancinelli Abstract We study the problem of designing and hedging unit-linked life policies whose benefits depend on an investment fund that incorporates environmental criteria in its selection process. Offering these products poses two key challenges: constructing a green investment fund and developing a hedging strategy for policies written on that fund. We address these two problems separately. First, we design a portfolio selection rule driven by firms’ carbon intensity that endogenously selects assets and avoids ad hoc pre-screens based on ESG scores. The effectiveness of our new portfolio selection method is tested using real market data. Second, we adopt the perspective of an insurance company issuing unit-linked policies written on this fund. Such contracts are exposed to market, carbon, and mortality risk, which the insurer seeks to hedge. Due to market incompleteness, we address the hedging problem via a quadratic approach aimed at minimizing the tracking error. We also make a numerical analysis to assess the performance of the hedging strategy. For our simulation study, we use an efficient weak second-order scheme that allows for variance reduction. ...

Hedging with memory: shallow and deep learning with signatures ArXiv ID: 2508.02759 “View on arXiv” Authors: Eduardo Abi Jaber, Louis-Amand Gérard Abstract We investigate the use of path signatures in a machine learning context for hedging exotic derivatives under non-Markovian stochastic volatility models. In a deep learning setting, we use signatures as features in feedforward neural networks and show that they outperform LSTMs in most cases, with orders of magnitude less training compute. In a shallow learning setting, we compare two regression approaches: the first directly learns the hedging strategy from the expected signature of the price process; the second models the dynamics of volatility using a signature volatility model, calibrated on the expected signature of the volatility. Solving the hedging problem in the calibrated signature volatility model yields more accurate and stable results across different payoffs and volatility dynamics. ...

Pricing Multi-strike Quanto Call Options on Multiple Assets with Stochastic Volatility, Correlation, and Exchange Rates ArXiv ID: 2411.16617 “View on arXiv” Authors: Unknown Abstract Quanto options allow the buyer to exchange the foreign currency payoff into the domestic currency at a fixed exchange rate. We investigate quanto options with multiple underlying assets valued in different foreign currencies each with a different strike price in the payoff function. We carry out a comparative performance analysis of different stochastic volatility (SV), stochastic correlation (SC), and stochastic exchange rate (SER) models to determine the best combination of these models for Monte Carlo (MC) simulation pricing. In addition, we test the performance of all model variants with constant correlation as a benchmark. We find that a combination of GARCH-Jump SV, Weibull SC, and Ornstein Uhlenbeck (OU) SER performs best. In addition, we analyze different discretization schemes and their results. In our simulations, the Milstein scheme yields the best balance between execution times and lower standard deviations of price estimates. Furthermore, we find that incorporating mean reversion into stochastic correlation and stochastic FX rate modeling is beneficial for MC simulation pricing. We improve the accuracy of our simulations by implementing antithetic variates variance reduction. Finally, we derive the correlation risk parameters Cora and Gora in our framework so that correlation hedging of quanto options can be performed. ...

CVA Sensitivities, Hedging and Risk ArXiv ID: 2407.18583 “View on arXiv” Authors: Unknown Abstract We present a unified framework for computing CVA sensitivities, hedging the CVA, and assessing CVA risk, using probabilistic machine learning meant as refined regression tools on simulated data, validatable by low-cost companion Monte Carlo procedures. Various notions of sensitivities are introduced and benchmarked numerically. We identify the sensitivities representing the best practical trade-offs in downstream tasks including CVA hedging and risk assessment. ...

Asymptotic methods for transaction costs ArXiv ID: 2407.07100 “View on arXiv” Authors: Unknown Abstract We propose a general approximation method for determining optimal trading strategies in markets with proportional transaction costs, with a polynomial approximation of the residual value function. The method is exemplified by several problems from optimally tracking benchmarks, hedging the Log contract, to maximizing utility from terminal wealth. Strategies are also approximated by practically executable, discrete trades. We identify the necessary trade-off between trading frequency and trade sizes to have satisfactory agreement with the theoretically optimal, continuous strategies of infinite activity. ...