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he relationship between the S&P 500 one-day percentage return and the VIX one-day percentage change (which I assume you mean by "VIX the same return") is inherently nonlinear, reflecting the complex dynamics of market behavior, investor psychology, and the mechanics of volatility pricing. Let’s dive into this step-by-step, focusing on why and how this nonlinearity manifests.
Background
S&P 500 One-Day Return: This is the percentage change in the S&P 500 index from the close of one trading day to the next, calculated as
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