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Proxy-Reliance Control in Conformal Recalibration of One-Sided Value-at-Risk

Tenghan Zhong

Abstract

We introduce a proxy-reliance-controlled conformal recalibration framework for one-sided Value-at-Risk (VaR), and study a question that existing state-aware methods do not usually isolate: how strongly should the recalibration adjustment depend on an imperfect volatility proxy? We formalize this through a proxy-reliance parameter that continuously interpolates between an approximately constant-shift correction and a fully proxy-scaled correction. This makes proxy reliance a distinct and practically interpretable design choice in one-sided VaR recalibration. We show theoretically that larger proxy reliance increases the responsiveness of the tail adjustment to proxy scale, but also increases stressed-state fragility when the proxy underreacts. Empirically, in rolling out-of-sample tests on a six-ETF panel with VIX-linked state variables, and with supporting evidence from SPY, we find that the empirical value of proxy-reliance control lies in improved stressed-state robustness rather than uniform overall dominance. In particular, when the baseline forecast remains exposed to proxy imperfection in stressed states, lower or intermediate proxy reliance can outperform fully proxy-scaled recalibration in stressed left-tail VaR control.

Proxy-Reliance Control in Conformal Recalibration of One-Sided Value-at-Risk

Abstract

We introduce a proxy-reliance-controlled conformal recalibration framework for one-sided Value-at-Risk (VaR), and study a question that existing state-aware methods do not usually isolate: how strongly should the recalibration adjustment depend on an imperfect volatility proxy? We formalize this through a proxy-reliance parameter that continuously interpolates between an approximately constant-shift correction and a fully proxy-scaled correction. This makes proxy reliance a distinct and practically interpretable design choice in one-sided VaR recalibration. We show theoretically that larger proxy reliance increases the responsiveness of the tail adjustment to proxy scale, but also increases stressed-state fragility when the proxy underreacts. Empirically, in rolling out-of-sample tests on a six-ETF panel with VIX-linked state variables, and with supporting evidence from SPY, we find that the empirical value of proxy-reliance control lies in improved stressed-state robustness rather than uniform overall dominance. In particular, when the baseline forecast remains exposed to proxy imperfection in stressed states, lower or intermediate proxy reliance can outperform fully proxy-scaled recalibration in stressed left-tail VaR control.
Paper Structure (32 sections, 6 theorems, 117 equations, 4 figures, 9 tables)

This paper contains 32 sections, 6 theorems, 117 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. VaR Forecasting, Dynamic Quantiles, and Backtesting
  4. Conformal Calibration under Dependence and Recent Financial Applications
  5. Volatility Proxies, Regime Dependence, and the Gap Addressed Here
  6. Data
  7. Sample and Assets
  8. Variables and Features
  9. Volatility Proxy and Stress-State Definition
  10. Out-of-Sample Design
  11. Methodology
  12. Baseline Forecasts and Proxy-Reliance-Controlled Recalibration
  13. Formal Properties of Proxy Reliance
  14. Proxy-Reliance Selection and Regime-Aware Extension
  15. Empirical Implementation and Evaluation
  16. ...and 17 more sections

Key Result

Proposition 4.1

Fix $\rho\in[0,1]$. For any $\eta>0$, Hence the sensitivity of the adjustment magnitude to proportional changes in proxy scale is governed exactly by $\rho$. Moreover, if the proxy is uniformly rescaled by the same positive multiplicative factor at all calibration points and at the forecast point, while the baseline forecast is held fix

Figures (4)

  • Figure 1: Pooled overall exceedance across the six-asset panel under the clean proxy specification. The dashed horizontal line marks the nominal 5% target.
  • Figure 2: Cross-asset exceedance heatmaps under the clean proxy specification.
  • Figure 3: Cross-asset exceedance--capital trade-offs for two representative baseline families.
  • Figure 4: Cross-asset exceedance heatmaps under clean and underreacting-stress proxy specifications.

Theorems & Definitions (12)

  • Proposition 4.1: Elasticity and invariance under uniform proxy rescaling
  • Proposition 4.2: Cross-state contrast induced by $\rho$
  • Proposition 4.3: Stress-state exceedance distortion under proxy underreaction
  • Corollary 4.4: Ordering under matched clean-proxy stress adjustment
  • Proposition A.1: Heterogeneous proxy distortion and directional forecast shift
  • Proposition A.2: Selection implication under stress screening
  • proof : Proof of Proposition \ref{['prop:uniform_scaling']}
  • proof : Proof of Proposition \ref{['prop:state_contrast']}
  • proof : Proof of Proposition \ref{['prop:heterogeneous_distortion']}
  • proof : Proof of Proposition \ref{['prop:stress_distortion']}
  • ...and 2 more