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Quantitative Finance · शब्दावली

Rough volatility क्या है?

Definition 12.5 Derivatives and Volatility · अध्याय 12 — Rough Volatility and Forward-Variance Models

Rough volatility is the property, and the class of models built on it, that the logarithm of instantaneous volatility behaves at short time scales like a fractional Brownian motion with Hurst exponent $H<\frac12$ , typically near 0.1. Its paths are rougher than any diffusion’s: the typical move over a lag $\Delta$ scales like $\Delta^H$ rather than $\Delta^{1/2}$ .

Left: the structure function of simulated daily log-volatility over eight years. The slope on log-log axes is 2H. The rough path has H=0.1; the diffusion is an Ornstein–Uhlenbeck process; the third series is the diffusion seen through a realised-variance estimate with a 0.08 error in log. Right: the published estimates for 21 indices. Data: the chapter’s code; Gatheral, Jaisson and Rosenbaum (2014), table B.1. — Figure 12.3. Left: the structure function of simulated daily log-volatility over eight years. The slope on log-log axes is $2H$ . The rough path has $H=0.1$ ; the diffusion is an Ornstein–Uhlenbeck process; the third series is the diffusion seen through a realised-variance estimate with a 0.08 error in log. Right: the published estimates for 21 indices. Data: the chapter’s code; Gatheral, Jaisson and Rosenbaum (2014), table B.1.

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