On the martingale property in the rough Bergomi model

Paul Gassiat

doi:10.1214/19-ecp239

Abstract

We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation $\rho $ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each $\rho \frac{1} {{1-\rho ^{2}}}$, the $m$-th moment of the stock price is infinite at each positive time.

Highlights

Introduction and main resultsWe are interested in fractional stochastic volatility models where the dynamics of stock price under a risk-neutral measure take the form (1.1)dSt/St = σ(t, Yt)dWt t (1.2)Yt = K(t, s)dZs where Zt = ρWt + ρWt, W, Ware two independent Brownian motions on a filtered probability space (Ω, (Ft)t≥0, P), and ρ2 + ρ2 = 1.A specific example we have in mind is the so-called Rough Bergomi model introduced in [2]
We are interested in fractional stochastic volatility models where the dynamics of stock price under a risk-neutral measure take the form dSt/St = σ(t, Yt)dWt t
The first question we consider in this note is whether the price process S, which is obviously a local martingale is a true martingale

Summary

Introduction and main results

We are interested in fractional stochastic volatility models where the dynamics of (discounted) stock price under a risk-neutral measure take the form (1.1). Yt = K(t, s)dZs where Zt = ρWt + ρWt, W, Ware two independent Brownian motions on a filtered probability space (Ω, (Ft)t≥0, P), and ρ2 + ρ2 = 1. A specific example we have in mind is the so-called Rough Bergomi model introduced in [2]. In that model Y is a Riemann-Liouville fractional Brownian motion of Hurst parameter H ∈ (0, 1), i.e. 1 2 and the volatility function takes the form σ(t, y) = ζ(t) exp (ηy) where η > 0 and ζ is a continuous function of t

Bergomi model with

Assume in addition that there exists

Assume that

Then for each