Combining Put Options and Trend Following for Tail Risk Management
Summary
This paper frames tail-risk management as an allocation across protections for distinct loss patterns: sudden crashes, volatility repricing and prolonged drawdowns. It develops a continuous-time conditional value-at-risk framework that combines long out-of-the-money puts with a systematic trend-following overlay. The put sleeve is treated as a marked-to-market asset, incorporating premium cost and changing market exposure into returns. The model tracks wealth, the underlying price, stochastic variance and a smoothed return signal, and derives an associated Hamilton–Jacobi–Bellman equation.
The analysis distinguishes the timing of the two sleeves: puts can respond to a crash immediately, while a trend signal may react late to the initial shock but offer more defense if losses persist, without repeated option premiums. The paper gives conditions for hybrid allocations, a policy-gradient identity and diagnostics for comparing protection mechanisms. Stylized Monte Carlo experiments report lower terminal CVaR for hybrid allocations than for either sleeve alone in the tested regimes. The preferred weights remain calibration-dependent, and the reported simulations do not by themselves establish out-of-sample or live performance.
Key ideas
- The framework allocates tail protection between long out-of-the-money puts and trend-following overlays.
- Marked-to-market option returns account for premium drag and changing exposure.
- Put protection can respond immediately to a crash, while trend signals may require time to react but can help during persistent drawdowns.
- The paper derives conditions for interior hybrid allocations and a CVaR policy-gradient identity.
- Stylized simulations report reduced terminal CVaR for hybrid sleeves, but suitable weights depend on calibration.
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Full text
# Tail Risk Management with Puts and Trend Following: A CVaR Framework for Crashes and Drawdowns # Tail Risk Management with Puts and Trend Following: A CVaR Framework for Crashes and Drawdowns 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.
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