Precise asymptotics: Robust stochastic volatility models

A regularity structure for rough volatility

“…P. K. Friz & Hairer, 2014, Section 9.3), which, in turn, invites to revisit Davis and Mataix-Pastor (2007) in a setting of "rough" interest rate models. Another concrete application, content of the recent P. K. Friz, Gassiat, and Pigato (2018), concerns precise asymptotics, allowing for considerable refinement of large deviations. (Translated to financial terms, this improvement leads to higher order implied volatility expansions.…”

Asymptotics for multifactor Volterra type stochastic volatility models

“…Jacquier, Pakkanen, and Stone [22] prove a large deviations principle for a scaled version of the log stock price process. In this same direction, Bayer, Friz, Gulisashvili, Horvath and Stemper [4], Forde and Zhang [12], Horvath, Jacquier and Lacombe [20] and most recently Friz, Gassiat and Pigato [13] (to name a few) prove large deviations principles for a wide range of one-dimensional stochastic volatility models. Some results concerning asymptotic in the multifactor setup can be found, for example, in Lacombe, Muguruza and Stone [23].…”

“…Thanks to Remark 4.1 (ii) and Assumption 4.2 we can suppose det(σ • ϕ) = 0 and the rate function above simplifies to J(•|ϕ) defined in (13).…”

“…Suppose σ, µ and σ satisfy Assumptions 4.2 and 5.1. Then, for every m ≥ 1, the family ((ε n B, Bn ), Z n,m ) n∈N satisfies a WLDP with the speed ε −2 n and the rate function 5) and (13), respectively. Moreover, the family of processes (Z n,m ) n∈N satisfies a LDP with the speed ε −2 n and the rate function…”

Edgeworth expansions for volatility models

“…[23]. To address these problems, a prominent and quite successful approach is to employ more complex models and use (Edgeworth) expansions to salvage comparatively simple and easy to evaluate formulas, see for instance [1], [2], [13], [19], [21], [22], [28]. To illustrate this further, consider the standard model…”