Model verification for Lévy-driven Ornstein–Uhlenbeck processes with estimated parameters

Abdelrazeq, Ibrahim

doi:10.1016/j.spl.2015.04.014

Cited by 6 publications

(6 citation statements)

References 8 publications

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“…On the other hand, if a < 0.4, the empirical levels increase when a is estimated. This is not particularly surprising in light of Theorem 4.3 of Abdelrazeq (), which shows that if a is small, there is greater variability in the estimated increments of L when a is replaced by

{\overset{a}{true}}_{N}^{(M)}

.…”

Section: Goodness‐of‐fit Test and Simulationsmentioning

confidence: 91%

“…As discussed previously, our theoretical results are valid when a is known. Now we investigate the performance of the bootstrap procedure when a is replaced by the estimate proposed in Abdelrazeq ():

{\overset{a}{true}}_{N}^{(M)} = \frac{\sum_{n = 1}^{N M} true(Y_{\frac{n - 1}{M}} - Y_{\frac{n}{M}} true) true(Y_{\frac{n - 1}{M}} - \bar{Y} true)}{\frac{1}{M} \sum_{n = 1}^{N M} true(Y_{\frac{n - 1}{M}} - \bar{Y} true)},

where

\bar{Y} = \frac{\sum_{n = 1}^{N M} Y_{\frac{n}{M}}}{N M} .

In all of our simulations, we compare the performance of our test statistics calculated using the true increments of L with those using estimated increments of L , calculated as in (2.3) and (2.4), but replacing a with the estimate above. We test the null hypothesis that the driving process is gamma with

μ > 1

.…”

Section: Goodness‐of‐fit Test and Simulationsmentioning

confidence: 99%

“…If a is unknown, it has to be estimated. Abdelrazeq () showed that the asymptotic behaviour of the sample covariances changes when a is replaced with an estimator. Nevertheless, in the present context we expect that estimation of a will not influence the validity of the bootstrap techniques used in the proposed goodness‐of‐fit‐tests, and we support this conjecture with a small simulation study.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, in the present context we expect that estimation of a will not influence the validity of the bootstrap techniques used in the proposed goodness‐of‐fit‐tests, and we support this conjecture with a small simulation study. However, given the results in Abdelrazeq (), the theoretical picture is not completely clear and a rigorous analysis is not straightforward. This question is beyond the scope of this article and will be the subject of a separate publication, currently in preparation.…”

Section: Introductionmentioning

confidence: 99%

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Goodness‐of‐fit tests for Lévy‐driven Ornstein‐Uhlenbeck processes

2018

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Lévy‐Driven Ornstein‐Uhlenbeck (or CAR(1)) processes have been introduced in the literature as a model for stochastic volatility. A general formula to recover the unobserved driving process from a continuously observed CAR(1) was developed. When the CAR(1) process is observed at discrete times, the driving process must be approximated. Approximated increments of the driving process are used to test the hypothesis that the CAR(1) belongs to a specified class of Lévy processes. Two goodness‐of‐fit tests are proposed. The performance of the tests is illustrated through simulation. The Canadian Journal of Statistics 46: 355–376; 2018 © 2018 Statistical Society of Canada

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{\overset{a}{true}}_{N}^{(M)}

.…”

Section: Goodness‐of‐fit Test and Simulationsmentioning

confidence: 91%

{\overset{a}{true}}_{N}^{(M)} = \frac{\sum_{n = 1}^{N M} true(Y_{\frac{n - 1}{M}} - Y_{\frac{n}{M}} true) true(Y_{\frac{n - 1}{M}} - \bar{Y} true)}{\frac{1}{M} \sum_{n = 1}^{N M} true(Y_{\frac{n - 1}{M}} - \bar{Y} true)},

where

\bar{Y} = \frac{\sum_{n = 1}^{N M} Y_{\frac{n}{M}}}{N M} .

μ > 1

.…”

Section: Goodness‐of‐fit Test and Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Goodness‐of‐fit tests for Lévy‐driven Ornstein‐Uhlenbeck processes

2018

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“…It has been demonstrated by Ait-Sahalia and Jacod [3], Klingler [4] that stochastic volatility and jumps are inherent components of the stock price dynamics that play important roles in the explanation of the implied volatility smile in options. Many alternative models are developed to reflect the intrinsic characteristics of asset returns and volatility smile effects of option prices, as shown by Kou [5], Mozumder [6], Shi [7], and Abdelrazeq [8]. Therefore, when constructing models to price options, it is necessary to incorporate both stochastic volatility and jumps.…”

Section: Introductionmentioning

confidence: 99%

Option pricing for stochastic volatility model with infinite activity Lévy jumps

Gong

Xin-tian

2016

Physica A: Statistical Mechanics and its Applications

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The purpose of this paper is to apply the stochastic volatility model driven by infinite activity Lévy processes to option pricing which displays infinite activity jumps behaviors and time varying volatility that is consistent with the phenomenon observed in underlying asset dynamics. We specially pay attention to three typical Lévy processes that replace the compound Poisson jumps in Bates model, aiming to capture the leptokurtic feature in asset returns and volatility clustering effect in returns variance. By utilizing the analytical characteristic function and fast Fourier transform technique, the closed form formula of option pricing can be derived. The intelligent global optimization search algorithm called Differential Evolution is introduced into the above highly dimensional models for parameters calibration so as to improve the calibration quality of fitted option models. Finally, we perform empirical researches using both time series data and options data on financial markets to illustrate the effectiveness and superiority of the proposed method.

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Option pricing and hedging for optimized Lévy driven stochastic volatility models

Gong

Xin-tian

2016

Chaos, Solitons & Fractals

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This paper pays attention to Ornstein-Uhlenbeck (OU) based stochastic volatility models with marginal law given by Classical Tempered Stable (CTS) distribution and Normal Inverse Gaussian (NIG) distribution, which are subclasses of infinite activity Lévy processes and are compared to finite activity Barndorff-Nielsen and Shephard (BNS) model. They are applied to option pricing and hedging in capturing leptokurtic features in asset returns and clustering effect in volatility that are consistently observed phenomena in underlying asset dynamics. The analytical formula of option pricing can be obtained through use of characteristic functions and Fast Fourier Transform (FFT) technique. Additionally, we introduce two hybrid optimization techniques such as hybrid Particle Swarm optimization (PSO) algorithm and hybrid Differential Evolution (DE) algorithm into parameters calibration schemes to improve the calibration quality for newly constructed models. Finally, we conduct experiments on Chinese emerging option markets to examine the performance of proposed models exploiting hybrid optimization techniques.

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Model verification for Lévy-driven Ornstein–Uhlenbeck processes with estimated parameters

Cited by 6 publications

References 8 publications

Goodness‐of‐fit tests for Lévy‐driven Ornstein‐Uhlenbeck processes

Goodness‐of‐fit tests for Lévy‐driven Ornstein‐Uhlenbeck processes

Option pricing for stochastic volatility model with infinite activity Lévy jumps

Option pricing and hedging for optimized Lévy driven stochastic volatility models

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