657 research outputs found
Short dated smile under Rough Volatility: asymptotics and numerics
In [Precise Asymptotics for Robust Stochastic Volatility Models; Ann. Appl.
Probab. 2021] we introduce a new methodology to analyze large classes of
(classical and rough) stochastic volatility models, with special regard to
short-time and small noise formulae for option prices, using the framework
[Bayer et al; A regularity structure for rough volatility; Math. Fin. 2020]. We
investigate here the fine structure of this expansion in large deviations and
moderate deviations regimes, together with consequences for implied volatility.
We discuss computational aspects relevant for the practical application of
these formulas. We specialize such expansions to prototypical rough volatility
examples and discuss numerical evidence
Option Pricing under Fast-varying and Rough Stochastic Volatility
Recent empirical studies suggest that the volatilities associated with
financial time series exhibit short-range correlations. This entails that the
volatility process is very rough and its autocorrelation exhibits sharp decay
at the origin. Another classic stylistic feature often assumed for the
volatility is that it is mean reverting. In this paper it is shown that the
price impact of a rapidly mean reverting rough volatility model coincides with
that associated with fast mean reverting Markov stochastic volatility models.
This reconciles the empirical observation of rough volatility paths with the
good fit of the implied volatility surface to models of fast mean reverting
Markov volatilities. Moreover, the result conforms with recent numerical
results regarding rough stochastic volatility models. It extends the scope of
models for which the asymptotic results of fast mean reverting Markov
volatilities are valid. The paper concludes with a general discussion of
fractional volatility asymptotics and their interrelation. The regimes
discussed there include fast and slow volatility factors with strong or small
volatility fluctuations and with the limits not commuting in general. The
notion of a characteristic term structure exponent is introduced, this exponent
governs the implied volatility term structure in the various asymptotic
regimes.Comment: arXiv admin note: text overlap with arXiv:1604.0010
Short-dated smile under rough volatility: asymptotics and numerics
In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small-noise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782–832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence
Log-modulated rough stochastic volatility models
We propose a new class of rough stochastic volatility models obtained by
modulating the power-law kernel defining the fractional Brownian motion (fBm)
by a logarithmic term, such that the kernel retains square integrability even
in the limit case of vanishing Hurst index . The so-obtained log-modulated
fractional Brownian motion (log-fBm) is a continuous Gaussian process even for
. As a consequence, the resulting super-rough stochastic volatility
models can be analysed over the whole range without the need of
further normalization. We obtain skew asymptotics of the form as , , so no flattening of the skew occurs as .Comment: 24 pages, 9 figure
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