Realised Volatility in Time-to-Expiry Space

Abstract

These notes document a set of tools for the empirical analysis of realised volatility in commodity futures markets, organised around the concept of time to expiry (TTE) as the primary coordinate. The motivation is the pricing and risk management of options that expire before the underlying futures contract—serial options—and in particular of accumulator strips: sequences of such options settling at successive calendar dates against the same long-dated futures. Three complementary analyses are developed. The early expiry (EE) volatility analysis addresses the central question: given that an option expires τ₀ business days before the futures delivery date, what realised volatility should that option expect to accumulate over its life? The answer is derived by pooling historical returns across all available contract years for a given delivery month, computing a cross-sectional spot volatility as a function of TTE, and then integrating this curve from expiry out to τ₀. The Samuelson effect implies that this early expiry volatility exceeds the full-term terminal variance for options with short expiries, because the high-volatility near-expiry period carries a disproportionate weight. The day-of-year (DOY) analysis re-maps the TTE-indexed volatility into calendar space, allowing seasonal patterns—crop cycles, weather-driven events, supply reporting windows—to be read directly from the chart. The monthly overlays view presents individual contract-year histories of price, returns, and realised volatility in TTE space, enabling year-on-year comparison and data quality validation. Throughout, the primary application context is agricultural and soft commodity futures (cocoa, coffee, cotton, sugar, grains), where the Samuelson effect is present, seasonal structure is important, and liquid option pricing is typically available only at the terminal expiry.