🤖 AI Summary
This study addresses the optimal joint allocation of put options and trend-following strategies for tail risk management under multifaceted adverse scenarios—including market crashes, volatility repricing, and prolonged drawdowns. The authors develop a continuous-time Conditional Value-at-Risk (CVaR) framework that unifies both mechanisms within a single optimization objective. By modeling wealth, spot price, stochastic variance, and an exponentially weighted log-return signal as Markovian state variables, they derive the viscosity solution to the associated Hamilton–Jacobi–Bellman equation. A key innovation is the temporal decoupling of protective mechanisms: put options deliver immediate convexity-based protection, while trend following enhances resilience during sustained drawdowns. The framework further incorporates a four-dimensional diagnostic layer assessing conditional convexity, tail-event reliability, holding costs, and drawdown persistence. Monte Carlo simulations demonstrate that the hybrid strategy substantially reduces terminal CVaR, with optimal allocations highly sensitive to parameter calibration.
📝 Abstract
Tail-risk management is not only an instrument-selection problem. It is an allocation problem across loss mechanisms: abrupt crash states, volatility repricing, and persistent drawdowns require different forms of protection. This paper develops a continuous-time CVaR framework that places two common protection sleeves -- long out-of-the-money put options and systematic trend-following overlays -- inside one coherent tail-risk mandate. The option sleeve is modeled as a marked-to-market traded asset, so premium drag, diffusion exposure, and jump repricing enter through its physical return process rather than through inconsistent terminal-payoff accounting. The resulting Markov state contains wealth, spot, stochastic variance, and an exponentially weighted log-return signal, and we derive the associated Hamilton--Jacobi--Bellman equation in viscosity form. The main analytical separation is temporal: convex insurance reprices immediately on jump impact, whereas trend following is late on the first shock because its signal must cross zero, but becomes increasingly defensive during persistent drawdowns without requiring fresh option premium. We then give sufficient and local conditions for an interior hybrid allocation, derive a CVaR policy-gradient identity, and introduce a four-axis diagnostic layer separating conditional convexity, tail-event reliability, non-stress carry, and drawdown persistence. Stylized Monte Carlo experiments illustrate the mechanism: fixed equal-weight hybrids and grid-optimized hybrids reduce terminal CVaR relative to either pure sleeve in the reported regimes, while the exact weight location remains calibration-dependent. The contribution is a transparent risk-management framework for deciding how much convex crash protection and how much signal-driven drawdown protection a mandate should hold.