Takeaway

Empirical evidence suggests volatility is “rough” (Hurst H≈0.1), better captured by fractional stochastic processes than classical diffusions, improving fits to implied volatility surfaces.

The problem (before → after)

  • Before: Heston/Black–Scholes miss microstructure-driven roughness and term-structure of skew.
  • After: Rough fractional models reproduce short-maturity smiles and long-memory effects.

Mental model first

Volatility is like a jagged coastline—zooming in reveals more irregularity rather than smoothness. Fractional processes encode this persistence of roughness.

Just-in-time concepts

  • Fractional Brownian motion with H in (0, 1/2).
  • Rough Bergomi and related models; forward variance as primary object.
  • Calibration via characteristic functions and Monte Carlo with hybrid schemes.

First-pass solution

Model log-variance as a Volterra integral driven by fBM; price options with Fourier or simulation; calibrate to implied vol surfaces across maturities.

Iterative refinement

  1. Microstructure foundations from order flow.
  2. Efficient sampling of fBM (Cholesky, circulant embedding).
  3. Risk management: Greeks under rough dynamics.

Principles, not prescriptions

  • Match empirical scaling laws; avoid over-smoothing volatility.
  • Use forward variance for stability.

Common pitfalls

  • Naive discretizations bias dynamics.
  • Overfitting short-term smiles without robust out-of-sample checks.

Connections and contrasts

  • See also: [/blog/black-scholes], local/stochastic vol literature.

Quick checks

  1. Why “rough”? — H small implies irregular sample paths.
  2. Why better smiles? — Short-maturity skew and term structures align with data.
  3. Why forward variance? — More stable calibration target.

Further reading

  • Gatheral et al. (source above) and follow-ups