5 results found
Horvath B, Jacquier A, Lacombe C, Asymptotic behaviour of randomised fractional volatility models, Journal of Applied Probability, ISSN: 0021-9002
We study the asymptotic behaviour of a class of small-noise diffusions drivenby fractional Brownian motion, with random starting points. Different scalingsallow for different asymptotic properties of the process (small-time and tailbehaviours in particular). In order to do so, we extend some results on samplepath large deviations for such diffusions. As an application, we show how theseresults characterise the small-time and tail estimates of the impliedvolatility for rough volatility models, recently proposed in mathematicalfinance.
Gulisashvili A, Horvath B, Jacquier A, 2018, Mass at zero in the uncorrelated SABR model and implied volatility asymptotics, Quantitative Finance, Vol: 18, Pages: 1753-1765, ISSN: 1469-7688
We study the mass at the origin in the uncorrelated SABR stochasticvolatility model, and derive several tractable expressions, in particular whentime becomes small or large. As an application--in fact the original motivationfor this paper--we derive small-strike expansions for the implied volatilitywhen the maturity becomes short or large. These formulae, by definitionarbitrage free, allow us to quantify the impact of the mass at zero on existingimplied volatility approximations, and in particular how correct/erroneousthese approximations become.
Gulisashvili A, Horvath B, Jacquier A, 2018, Mass at Zero in the Uncorrelated SABR Model and Implied Volatility Asymptotics, Quantitative Finance, ISSN: 1469-7688
Döring L, Horvath B, Teichmann J, 2017, Functional Analytic (Ir-)Regularity Properties of SABR-type Processes
Gulisashvili AG, Horvath BH, Jacquier A, 2016, On the probability of hitting the boundary for Brownian motions on the SABR plane, Electronic Communications in Probability, Vol: 21, Pages: 1-13, ISSN: 1083-589X
Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.
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