2019/04/19 by Carr, Peter, Itkin, Andrey
#Computational Finance (q-fin.CP) #FOS: Economics and business #Mathematical Finance (q-fin.MF) #Pricing of Securities (q-fin.PR)
paper · doi:10.48550/arxiv.1904.09240
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 St is a geometric BM with stochastic instantaneous volatility σt, the process for σt is also a geometric BM with stochastic speed of mean reversion and time-dependent colatility of volatility, and the supplementary process \calVt 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 log St 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.