scholarly journals Asymptotic behaviour of randomised fractional volatility models

2019 ◽  
Vol 56 (2) ◽  
pp. 496-523 ◽  
Author(s):  
Blanka Horvath ◽  
Antoine Jacquier ◽  
Chloé Lacombe
Keyword(s):  
Asymptotic Behaviour ◽  
Implied Volatility ◽  
Asymptotic Properties ◽  
Sample Path ◽  
Mathematical Finance ◽  
Small Time ◽  
Small Noise ◽  
Volatility Models ◽  
Rough Volatility ◽  
Sample Path Large Deviations

AbstractWe study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

scholarly journals An Intuitive Introduction to Fractional and Rough Volatilities

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 994
Author(s):  
Elisa Alòs ◽  
Jorge A. León
Keyword(s):  
Malliavin Calculus ◽  
Numerical Experiments ◽  
Implied Volatility ◽  
Natural Approach ◽  
Volatility Models ◽  
Implied Volatility Surface ◽  
Short Memory ◽  
White Type ◽  
Volatility Surface ◽  
Theoretical Results

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

2021 ◽  
pp. 1-19
Author(s):  
XUHUI WANG ◽  
SHENG-JHIH WU ◽  
XINGYE YUE
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Option Price ◽  
Second Order ◽  
Approximation Formula ◽  
Stochastic Volatility Models ◽  
Regular Perturbation ◽  
Volatility Models ◽  
Order Expansion ◽  
High Level

Abstract We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

CALIBRATION OF STOCHASTIC VOLATILITY MODELS VIA SECOND-ORDER APPROXIMATION: THE HESTON CASE

2015 ◽  
Vol 18 (06) ◽  
pp. 1550036 ◽  
Author(s):  
ELISA ALÒS ◽  
RAFAEL DE SANTIAGO ◽  
JOSEP VIVES
Keyword(s):  
Term Structure ◽  
Implied Volatility ◽  
Order Approximation ◽  
Calibration Procedure ◽  
Computational Effort ◽  
Heston Model ◽  
Option Prices ◽  
Volatility Models ◽  
Short And Long Term ◽  
Long Term Behavior

In this paper, we present a new, simple and efficient calibration procedure that uses both the short and long-term behavior of the Heston model in a coherent fashion. Using a suitable Hull and White-type formula, we develop a methodology to obtain an approximation to the implied volatility. Using this approximation, we calibrate the full set of parameters of the Heston model. One of the reasons that makes our calibration for short times to maturity so accurate is that we take into account the term structure for large times to maturity: We may thus say that calibration is not "memoryless," in the sense that the option's behavior far away from maturity does influence calibration when the option gets close to expiration. Our results provide a way to perform a quick calibration of a closed-form approximation to vanilla option prices, which may then be used to price exotic derivatives. The methodology is simple, accurate, fast and it requires a minimal computational effort.

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