Long memory in continuous‐time stochastic volatility models

TLDR

The paper extends the Black–Scholes framework to the Hull–White stochastic volatility model. The authors model volatility as a stochastic long‑memory process, analyze its effects on both volatility and asset price dynamics, compare it to discrete‑time approximations, and study option pricing through theoretical formulas, implied volatility properties, and statistical inference, supported by simulation experiments. They derive theoretical option pricing formulas, characterize implied volatility behavior, demonstrate tractable statistical inference, and confirm these results with simulation studies.

Abstract

This paper studies a classical extension of the Black and Scholes model for option pricing, often known as the Hull and White model. Our specification is that the volatility process is assumed not only to be stochastic, but also to have long‐memory features and properties. We study here the implications of this continuous‐time long‐memory model, both for the volatility process itself as well as for the global asset price process. We also compare our model with some discrete time approximations. Then the issue of option pricing is addressed by looking at theoretical formulas and properties of the implicit volatilities as well as statistical inference tractability. Lastly, we provide a few simulation experiments to illustrate our results.