scholarly journals TIME-CHANGED FAST MEAN-REVERTING STOCHASTIC VOLATILITY MODELS

2011 ◽  
Vol 14 (08) ◽  
pp. 1355-1383 ◽  
Author(s):  
MATTHEW LORIG
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Perturbation Techniques ◽  
European Option ◽  
Stochastic Volatility Models ◽  
Multiple Factors ◽  
Price Process ◽  
Volatility Models ◽  
Time Changes ◽  
Implied Volatility Surfaces

We introduce a class of randomly time-changed fast mean-reverting stochastic volatility (TC-FMR-SV) models. Using spectral theory and singular perturbation techniques, we derive an approximation for the price of any European option in the TC-FMR-SV setting. Three examples of random time-changes are provided and are shown to induce distinct implied volatility surfaces. The key features of the TC-FMR-SV framework are that (i) it is able to incorporate jumps into the price process of the underlying asset (ii) it allows for the leverage effect and (iii) it can accommodate multiple factors of volatility, which operate on different time-scales.

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.

scholarly journals Inference in Lévy-type stochastic volatility models

2007 ◽  
Vol 39 (02) ◽  
pp. 531-549 ◽  
Author(s):  
Jeannette H. C. Woerner
Keyword(s):  
Stochastic Volatility ◽  
Market Microstructure ◽  
Inverse Gaussian ◽  
Robust Estimators ◽  
Stochastic Volatility Models ◽  
Integrated Volatility ◽  
Volatility Models ◽  
Time Changes ◽  
Normal Inverse Gaussian

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox-Ingersoll-Ross or Ornstein-Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates.

Functional Relationships Between Price and Volatility Jumps and Their Consequences for Discretely Observed Data

2012 ◽  
Vol 49 (04) ◽  
pp. 901-914
Author(s):  
Jean Jacod ◽  
Claudia Klüppelberg ◽  
Gernot Müller
Keyword(s):  
Asymptotic Behaviour ◽  
Stochastic Volatility ◽  
Continuous Time ◽  
Discrete Observations ◽  
Stochastic Volatility Models ◽  
Functional Relationships ◽  
Price Process ◽  
Functional Relations ◽  
Volatility Models ◽  
Volatility Jumps

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

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