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Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0139-0747
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0003-4554-6528
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)ORCID iD: 0000-0002-0835-7536
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2019 (English)In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 81-90Conference paper, Published paper (Refereed)
Abstract [en]

The double-mean-reverting model by Gatheral [1] is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the- money. Using the method by Pagliarani and Pascucci [6], we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

Place, publisher, year, edition, pages
ISAST: International Society for the Advancement of Science and Technology , 2019. p. 81-90
Keywords [en]
Double-mean-reverting, European option, implied volatility, asymptotic expansion
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-47084ISBN: 978-618-5180-33-1 (electronic)OAI: oai:DiVA.org:mdh-47084DiVA, id: diva2:1394754
Conference
ASMDA2019, 18th Applied Stochastic Models and Data Analysis International Conference
Funder
Sida - Swedish International Development Cooperation AgencyAvailable from: 2020-02-20 Created: 2020-02-20 Last updated: 2022-11-25Bibliographically approved
In thesis
1. Asymptotics of implied volatility in the Gatheral double stochastic volatility model
Open this publication in new window or tab >>Asymptotics of implied volatility in the Gatheral double stochastic volatility model
2022 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We consider a market model of financial engineering with three factors represented by three correlated Brownian motions. The volatility of the risky asset in this model is the sum of two stochastic volatilities. The dynamic of each volatility is governed by a mean-reverting process. The first stochastic volatility of mean-reversion process reverts to the second volatility at a fast rate, while the second volatility moves slowly to a constant level over time with the state of the economy.

The double mean-reverting model by Gatheral (2008) is motivated by empirical dynamics of the variance of the stock price. This model can be consistently calibrated to both the SPX options and the VIX options. However due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time consuming methods like Monte Carlo simulation or the finite difference method.

To solve the above issue, we use the method of asymptotic expansion developed by Pagliarani and Pascucci (2017). In paper A, we study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. We calculate explicitly the asymptotic expansions of implied volatility within a parabolic region up the second order. In paper B we improve the results obtain in paper A by calculating the asymptotic expansion of implied volatility under the Gatheral model up to order three. In paper C, we perform numerical studies on the asymptotic expansion up to the second order. The Monte-Carlo simulation is used as the benchmark value to check the accuracy of the expansions. We also proposed a partial calibration procedure using the expansions. The calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the COVID-19 crisis. Finally, in paper D we check the performance of the third order expansion and compare it with the previous results.

Place, publisher, year, edition, pages
Västerås: Mälardalens universitet, 2022
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 368
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-60176 (URN)978-91-7485-569-2 (ISBN)
Public defence
2022-11-29, Kappa, Mälardalens universitet, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2022-10-12 Created: 2022-10-10 Last updated: 2023-04-11Bibliographically approved

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Albuhayri, MohammedMalyarenko, AnatoliySilvestrov, SergeiNi, YingEngström, ChristopherTewolde, Finnan

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