New Challenges in (Rough) Volatility Modelling
The Minisymposium New Challenges in (Rough) Volatility Modelling is a part of the Conference on Mathematical Modelling in Finance.
On the relationship between implied volatilities and volatility swaps:
a Malliavin calculus approach
Abstract
This work is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility(ATMI) of a European call option. It is well-known that the difference between these two quantities converges to zero as the time to maturity decreases. We make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of the convergence is different in the correlated and in the uncorrelated case, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we will see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H. Moreover, in the case H ≥ 1/2, we develop a model-free approximation formula for the of the volatility swap, in terms of the ATMI and its skew. (joint work with Kenichiro Shiraya, University of Tokyo)
M. Fukasawa (Osaka University)
Abstract
In this talk, first the LAN property of fractional Gaussian noises under high-frequency observations (joint work with A. Brouste) and the asymptotic efficiency of a novel Whittle-type estimator (joint work with T. Takabatake) are discussed. Second, a modification of the Whittle-type estimator for rough volatility models is given. Our empirical analysis estimates H = 0.05 or so (joint work with T. Takabatake and R. Westphal).
A. Antonov (Numerix LLC)
Exactly solvable cases in the SABR model
Abstract
In this talk we present certain cases when the SABR (option price and/or the density) can be expressed in terms of low dimensional integrals of elementary functions. They include: zero correlation case for all beta-powers for both classic and free boundary SABR's (option and pdf), normal SABR with the free boundary (option and pdf) log-normal case (pdf only).
G. Livieri (Pisa Scuola Normale Superiore) A discrete-time stochastic volatility framework for pricing options with realized measures
Abstract
Motivated by the fact that realized measures of volatility are affected by measuremeant errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both the observed returns and realized measures to the latent conditional variance. A fully analytical option pricing framework is developed for this new discrete-time stochastic volatility class of models. In addition, we provide analytical filtering and smoothing recursions for the basic version of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the efficacy of the filtering and smoothing of realized measures in enhancing the latent volatility persistence (the crucial parameter for the effective pricing of Standard and Poor’s 500 Index options).
N. Marie (Université Paris 10 and ESME Sudria)
Singularities, invariance theorem and fractional stochastic volatility models
Abstract
In Itô’s calculus framework, there exist several ways to constrain the solution of a SDE to stay in a subset of Rd: Skorokhod problem, invariance condition, singularities of the vector field, etc. Our purpose is to present some extensions of these techniques to fractional SDE and to apply them to constrain the volatility, at least to stay positive, in a fractional stochastic volatility model.
M. Podolskij (Aarhus University)
Statistical inference for fractional models
Abstract
In this talk we present some estimation methods for fractional models of different types. In recent years fractional Brownian motion (fBm) and related processes became a popular tool in volatility modelling. We will review some classical estimation methods for fBm and present some recent results for multifractional Brownian motion and fractional stable motion, which constitute a natural extension of fBm. From statistical point of view the new theory provides a robustness analysis when the fBm model is not correctly specified.
A. Muguruza (Imperial College London)
Functional central limit theorems for rough volatily models
Abstract
We extend Donsker’s approximation of Brownian motion to fractional Brownian motion with any Hurst exponent (including the ’rough’ case H < 1/2), and Volterra-like processes. Some of the most relevant consequences of our ‘rough Donsker (rDonsker) Theorem’ are convergence results (with rates) for discrete approximations of a large class of rough models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme of Bennedsen, Lunde, and Pakkanen and find remarkable agreement (for a large range of values of H). This rDonsker Theorem further provides a weak convergence proof for the hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan (joint work with Blanka Horvath and Antoine Jacquier).
Slides of the presentations: