Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

Abstract

We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.Date: March 25, 2019.2010 Mathematics Subject Classification. Primary 60F10, 60G22; Secondary 91G20, 60G15, 91G60. ε > 0, P ε is the joint law and, for each δ > 0, the set ω :and lim sup ε↓0 h ε log P ε (Γ δ ) = −∞, where Γ δ := {(ỹ, y) : d(ỹ, y) > δ} ⊂ Y × Y. Theorem D.2. Let X and Y be topological spaces and f : X → Y a continuous function. Consider a good rate function I : X → [0, ∞]. For each y ∈ Y, define I (y) := inf{I(x) : x ∈ X , y = f (x)}. Then, if I controls the LDP associated with a family of probability measures {µ ε } on X , the I controls the LDP associated with the family of probability measures {µ ε • f −1 } on Y and I is a good rate function on Y.

Weak approximations and VIX option price expansions in forward variance curve models

“…Several recent papers have tackled the problem of VIX derivatives pricing in the rough Bergomi model (1.4). In [26], instead of expansion formulas, upper and lower bounds for VIX futures are provided, and in [28], large deviation theory is applied to derive small-maturity asymptotic formulas (covering one-and multi-factor mixed rough Bergomi models). In [1], the authors specifically focus on at-the-money (ATM) implied volatility level and skew.…”

Rough multifactor volatility for SPX and VIX options

“…Further numerical methods were developed in [13,14,41]. However, the lack of analytical tractability of rough volatility models is holding back the progress of theoretical results on the VIX, with the notable exception of large devations results from [19,35] and the small-time asymptotics of [2].…”

“…As a result, numerical computations under this model induce a linear smile, or equivalently a null curvature, unfortunately inconsistent with market observations. To remedy this, we incorporate Bergomi's and Gatheral's insights on multifactor models, (integrated by [17,35] into rough volatility models) and extend [2] to the multi-factor case; we also compute the short-time ATM implied volatility curvature, deriving a second criterion for a more accurate model choice. We gather in Section 2 our abstract framework and assumptions.…”