ADOL - Markovian approximation of rough lognormal model

TL;DR

This paper introduces a Markovian approximation of the fractional Brownian motion using the Dobric-Ojeda process for fractional stochastic volatility modeling, enabling closed-form characteristic functions and efficient option pricing.

Contribution

It develops a semi-martingale approximation of fractional Brownian motion that simplifies pricing in rough volatility models by providing a closed-form characteristic function.

Findings

The model yields a closed-form characteristic function for g S_t

Option and variance swap pricing can be efficiently performed using FFT

The approximation works uniformly across all Hurst exponents

Abstract

In this paper we apply Markovian approximation of the fractional Brownian motion (BM), known as the Dobric-Ojeda (DO) process, to the fractional stochastic volatility model where the instantaneous variance is modelled by a lognormal process with drift and fractional diffusion. Since the DO process is a semi-martingale, it can be represented as an \Ito diffusion. It turns out that in this framework the process for the spot price $S_t$ is a geometric BM with stochastic instantaneous volatility $\sigma_t$, the process for $\sigma_t$ is also a geometric BM with stochastic speed of mean reversion and time-dependent colatility of volatility, and the supplementary process $\calV_t$ is the Ornstein-Uhlenbeck process with time-dependent coefficients, and is also a function of the Hurst exponent. We also introduce an adjusted DO process which provides a uniformly good approximation of the fractional BM for all Hurst exponents $H \in [0,1]$ but requires a complex measure. Finally, the characteristic function (CF) of $\log S_t$ in our model can be found in closed form by using asymptotic expansion. Therefore, pricing options and variance swaps (by using a forward CF) can be done via FFT, which is much easier than in rough volatility models.