ADOL - Markovian approximation of rough lognormal model

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 diffusion. It turns out that in this framework the process for the spot price S_t is a geometric BM with stochastic instantaneous volatility _t, the process for _t is also a geometric BM with stochastic speed of mean reversion and time-dependent colatility of volatility, and the supplementary process _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 [0,1] but requires a complex measure. Finally, the characteristic function (CF) of 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.

Tasks

Reproductions