| 1. |
- Albuhayri, Mohammed, et al.
(author)
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An Improved Asymptotics of Implied Volatility in the Gatheral Model
- 2022
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In: <em>Springer Proceedings in Mathematics and Statistics</em>. - Cham : Springer Nature. - 9783031178191 - 9783031178207 ; , s. 3-13
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Conference paper (peer-reviewed)abstract
- We study the double-mean-reverting model by Gatheral. Our previous results concerning the asymptotic expansion of the implied volatility of a European call option, are improved up to order 3, that is, the error of the approximation is ultimately smaller that the 1.5th power of time to maturity plus the cube of the absolute value of the difference between the logarithmic security price and the logarithmic strike price.
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| 2. |
- Albuhayri, Mohammed, et al.
(author)
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Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model
- 2019
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In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019. - : ISAST: International Society for the Advancement of Science and Technology. - 9786185180331 ; , s. 81-90
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Conference paper (peer-reviewed)abstract
- 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.
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| 3. |
- Albuhayri, Mohammed, et al.
(author)
-
Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model
- 2021
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In: Applied Modeling Techniques and Data Analysis 2. - Hoboken, NJ, USA : John Wiley & Sons. - 9781786306746 ; , s. 27-38
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Book chapter (peer-reviewed)abstract
- The double-mean-reverting model by Gatheral 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, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.
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| 4. |
- Albuhayri, Mohammed
(author)
-
Asymptotics of implied volatility in the Gatheral double stochastic volatility model
- 2022
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Doctoral thesis (other academic/artistic)abstract
- 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.
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| 5. |
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| 6. |
- Albuhayri, Mohammed, et al.
(author)
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Numerical Studies of the Implied Volatility Expansions up to Third Order under the Gatheral Model
- 2022
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Conference paper (other academic/artistic)abstract
- The Gatheral double stochastic volatility model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. Our previous paper investigated the performance of the first and second-order implied volatilities expansions under this model. Moreover, a simple partial calibration method has been proposed. This paper reviews and extends previous results to the third-order implied volatility expansions under the same model. Using Monte-Carlo simulation as the benchmark method, extensive numerical studies are conducted to investigate the accuracy and properties of the third-order expansion.
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| 7. |
- Dimitrov, Marko, 1993-, et al.
(author)
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Numerical Studies of Implied Volatility Expansions Under the Gatheral Model
- 2022
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In: Data Analysis and Related Applications 1. - London : ISTE Ltd. - 9781394165513 - 9781394165506 ; , s. 135-148
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Book chapter (other academic/artistic)abstract
- We calculate the price of the European call option in the Gatheral double stochastic volatility model by two independent methods. The first one is Monte Carlo simulation. For the second one, we use asymptotic expansions up to order 3 of the implied volatility in the above model calculated in our previous papers. We substitute the approximate value of the implied volatility to the Black--Scholes pricing formula. The results showing the accuracy of our approximation are presented.
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