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August
2020, 19(8): 4127-4142.
doi: 10.3934/cpaa.2020184
First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $ \alpha $-stable white noise
| 1. | Research and Evaluation Division, Public Health Wales, Cardiff, UK |
| 2. | Department of Mathematics, Computational Foundry, Swansea University, Swansea, UK |
| 3. | Accounting and Finance Department, School of Management, Swansea University, Swansea, UK |
The aim of this paper is twofold. Firstly, we derive an explicit expression of the (theoretical) solutions of stochastic differential equations with affine coefficients driven by $ \alpha $-stable white noise. This is done by means of Itô formula. Secondly, we develop a detection algorithm for the first jump time in simulation of sampling trajectories which are described by the solutions. The algorithm is carried out through a multivariate Lagrange interpolation approach. To this end, we utilise a computer simulation algorithm in MATLAB to visualise the sampling trajectories of the jump-diffusions for two combinations of parameters arising in the modelling structure of stochastic differential equations with affine coefficients.
Keywords:
Stochastic differential equations,
affine coefficients,
$ \alpha $-stable processes,
simulation,
multivariate Lagrange interpolation.
Mathematics Subject Classification:
65C99, 68U20, 60E07, 60G17.
Citation:
Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $ \alpha $-stable white noise. Communications on Pure and Applied Analysis,
2020, 19
(8)
: 4127-4142.
doi: 10.3934/cpaa.2020184
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Inferring volatility dynamics and risk premia from the S & P 500 and VIX markets, J. Financ. Econ., 131 (2019), 593-618.
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A. Barletta, P. Magistris and D. Sloth,
It only takes a few moments to hedge options, J. Econ. Dyn. Control, 100 (2019), 251-269.
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O. Barndorff-Nielsen,
Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), 1-13.
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J. Campbell, S. Giglio, C. Polk and R. Turley,
An intertemporal CAPM with stochastic volatility, J. Financ. Econ., 128 (2018), 207-233.
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J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997.
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J. C. Cox, J. E. Ingersoll and S. A. Ross,
A Theory of the Term Structure of Interest Rates, Econometrica, 53 (1985), 385-407.
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M. Dror, P. L$\prime$ecuyer and F. Szidarovszky, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, Springer Science & Business Media, 2002. |
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H. Du, J. Wu and W. Yang,
On the mechanism of CDOs behind the current financial crisis and mathematical modeling with L$\acute{e}$vy distributions, Intel. Inform. Manag., 2 (2010), 149-158.
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D. Duffie, D. Filipović and W. Schachermayer,
Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.
doi: 10.1214/aoap/1060202833. |
| [11] |
D. Duffie, J. Pan and K. Singleton,
Transform Analysis and Asset Pricing for Affine Jump-diffusions, Econometrica, 68 (2000), 1343-1376.
doi: 10.1111/1468-0262.00164. |
| [12] |
A. Fiche, J. C. Cexus, A. Martin and A. Khenchaf,
Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions, Inform. Fusion, 14 (2013), 504-520.
|
| [13] |
R. Giacometti, M. Bertocchi, S. T. Rachev and F. J. Fabozzi,
Stable distributions in the Black-Litterman approach to asset allocation, Quant. Finance, 7 (2007), 423-433.
doi: 10.1080/14697680701442731. |
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M. Hain, M. Uhrig-Homburg and N. Unger,
Risk factors and their associated risk premia: An empirical analysis of the crude oil market, J. Bank. Finance, 95 (2018), 44-63.
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A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993.
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S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. |
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Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198.
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W. E. Leland, M. S. Taqqu, W. Willinger and D. W. Wilson,
On the self-similar nature of Ethernet traffic, ACM SIGCOMM Comput. Commun. Rev., 23 (1993), 183-193.
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| [19] |
P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. |
| [20] |
P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. |
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B. Mandelbrot,
The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106.
|
| [22] |
G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994.
|
| [23] |
M.F.Shlesinger, G.M.Zaslavsky and U.Frisch, L$\acute{e}$vy flights and related topics in physics, in Lecture notes in physics, vol. 450, (1995), Springer-Verlag.
doi: 10.1007/3-540-59222-9. |
| [24] |
J. Song and J. Wu,
A detection algorithm for the first jump time in sample trajectories of jump-diffusions driven by $\alpha$-stable white noise, Commun. Statist. Theory Meth., 48 (2019), 4888-4902.
doi: 10.1080/03610926.2018.1500602. |
| [25] |
O. Vasicek,
An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188.
|
| [26] |
J. Wu and W. Yang,
Valuation of synthetic CDOs with affine jump-diffusion processes involving Lévy stable distributions, Math. Comput. Model., 57 (2013), 570-583.
doi: 10.1016/j.mcm.2012.06.038. |
| [27] |
V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986. |
| [28] |
C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013. |
show all references
References:
| [1] |
D. Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus, Cambridge University Press, 1994.
doi: 10.1017/CBO9780511755323.
|
| [2] |
C. Bardgett, E. Gourier and M. Leippold,
Inferring volatility dynamics and risk premia from the S & P 500 and VIX markets, J. Financ. Econ., 131 (2019), 593-618.
|
| [3] |
A. Barletta, P. Magistris and D. Sloth,
It only takes a few moments to hedge options, J. Econ. Dyn. Control, 100 (2019), 251-269.
doi: 10.1016/j.jedc.2018.11.008. |
| [4] |
O. Barndorff-Nielsen,
Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), 1-13.
doi: 10.1111/1467-9469.00045. |
| [5] |
J. Campbell, S. Giglio, C. Polk and R. Turley,
An intertemporal CAPM with stochastic volatility, J. Financ. Econ., 128 (2018), 207-233.
|
| [6] |
J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997.
|
| [7] |
J. C. Cox, J. E. Ingersoll and S. A. Ross,
A Theory of the Term Structure of Interest Rates, Econometrica, 53 (1985), 385-407.
doi: 10.2307/1911242. |
| [8] |
M. Dror, P. L$\prime$ecuyer and F. Szidarovszky, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, Springer Science & Business Media, 2002. |
| [9] |
H. Du, J. Wu and W. Yang,
On the mechanism of CDOs behind the current financial crisis and mathematical modeling with L$\acute{e}$vy distributions, Intel. Inform. Manag., 2 (2010), 149-158.
|
| [10] |
D. Duffie, D. Filipović and W. Schachermayer,
Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.
doi: 10.1214/aoap/1060202833. |
| [11] |
D. Duffie, J. Pan and K. Singleton,
Transform Analysis and Asset Pricing for Affine Jump-diffusions, Econometrica, 68 (2000), 1343-1376.
doi: 10.1111/1468-0262.00164. |
| [12] |
A. Fiche, J. C. Cexus, A. Martin and A. Khenchaf,
Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions, Inform. Fusion, 14 (2013), 504-520.
|
| [13] |
R. Giacometti, M. Bertocchi, S. T. Rachev and F. J. Fabozzi,
Stable distributions in the Black-Litterman approach to asset allocation, Quant. Finance, 7 (2007), 423-433.
doi: 10.1080/14697680701442731. |
| [14] |
M. Hain, M. Uhrig-Homburg and N. Unger,
Risk factors and their associated risk premia: An empirical analysis of the crude oil market, J. Bank. Finance, 95 (2018), 44-63.
|
| [15] |
A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993.
|
| [16] |
S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. |
| [17] |
R. Jarrow,
Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198.
|
| [18] |
W. E. Leland, M. S. Taqqu, W. Willinger and D. W. Wilson,
On the self-similar nature of Ethernet traffic, ACM SIGCOMM Comput. Commun. Rev., 23 (1993), 183-193.
|
| [19] |
P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. |
| [20] |
P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. |
| [21] |
B. Mandelbrot,
The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106.
|
| [22] |
G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994.
|
| [23] |
M.F.Shlesinger, G.M.Zaslavsky and U.Frisch, L$\acute{e}$vy flights and related topics in physics, in Lecture notes in physics, vol. 450, (1995), Springer-Verlag.
doi: 10.1007/3-540-59222-9. |
| [24] |
J. Song and J. Wu,
A detection algorithm for the first jump time in sample trajectories of jump-diffusions driven by $\alpha$-stable white noise, Commun. Statist. Theory Meth., 48 (2019), 4888-4902.
doi: 10.1080/03610926.2018.1500602. |
| [25] |
O. Vasicek,
An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188.
|
| [26] |
J. Wu and W. Yang,
Valuation of synthetic CDOs with affine jump-diffusion processes involving Lévy stable distributions, Math. Comput. Model., 57 (2013), 570-583.
doi: 10.1016/j.mcm.2012.06.038. |
| [27] |
V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986. |
| [28] |
C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013. |
Figure 2.
$ \alpha_1<1 $ and $ \alpha_2>1 $ : $ \lambda $ changes when $ \alpha_1 $ = 0.5 and $ \alpha_2 $ = 1.5
Figure 3.
$ \alpha_1<1 $ and $ \alpha_2>1 $ : $ \mu_2 $ changes when $ \alpha_1 $ = 0.25 and $ \alpha_2 $ = 1.75
Figure 6.
$ \alpha_1>1 $ and $ \alpha_2<1 $ : $ \mu_2 $ changes when $ \alpha_1 $ = 1.5 and $ \alpha_2 $ = 0.5
| t | ||||||
| 10 | 1 | 100 | 0.5 | 1.5 | 0.08203 | -63.86 |
| 1 | 0.25 | 100 | 0.75 | 1.25 | 0.1855 | -303.2 |
| 1 | 100 | 1 | 0.75 | 1.75 | 0.1035 | 122.5 |
| 1 | 100 | 0.25 | 0.75 | 1.5 | 0.207 | -896.1 |
| 10 | 100 | 0.25 | 0.5 | 1.25 | 0.3301 | 252.1 |
| 100 | 100 | 1 | 0.25 | 1.75 | 0.1934 | -3028000 |
| 10 | 1 | 0.25 | 0.25 | 1.25 | 0.5762 | 533.7 |
| t | ||||||
| 10 | 1 | 100 | 0.5 | 1.5 | 0.08203 | -63.86 |
| 1 | 0.25 | 100 | 0.75 | 1.25 | 0.1855 | -303.2 |
| 1 | 100 | 1 | 0.75 | 1.75 | 0.1035 | 122.5 |
| 1 | 100 | 0.25 | 0.75 | 1.5 | 0.207 | -896.1 |
| 10 | 100 | 0.25 | 0.5 | 1.25 | 0.3301 | 252.1 |
| 100 | 100 | 1 | 0.25 | 1.75 | 0.1934 | -3028000 |
| 10 | 1 | 0.25 | 0.25 | 1.25 | 0.5762 | 533.7 |
| t | ||||||
| 10 | 0.25 | 1 | 1.5 | 0.5 | 0.1973 | -91.87 |
| 10 | 1 | 100 | 1.5 | 0.25 | 0.01953 | 95.72 |
| 1 | 100 | 1 | 1.25 | 0.5 | 0.04492 | 305.9 |
| 100 | 1 | 0.25 | 1.5 | 0.25 | 0.1211 | 346 |
| 1 | 1 | 100 | 1.75 | 0.5 | 0.09766 | 311.1 |
| 100 | 1 | 100 | 1.5 | 0.5 | 0.05273 | -242.9 |
| 10 | 1 | 0.25 | 1.75 | 0.75 | 0.5742 | -105.1 |
| t | ||||||
| 10 | 0.25 | 1 | 1.5 | 0.5 | 0.1973 | -91.87 |
| 10 | 1 | 100 | 1.5 | 0.25 | 0.01953 | 95.72 |
| 1 | 100 | 1 | 1.25 | 0.5 | 0.04492 | 305.9 |
| 100 | 1 | 0.25 | 1.5 | 0.25 | 0.1211 | 346 |
| 1 | 1 | 100 | 1.75 | 0.5 | 0.09766 | 311.1 |
| 100 | 1 | 100 | 1.5 | 0.5 | 0.05273 | -242.9 |
| 10 | 1 | 0.25 | 1.75 | 0.75 | 0.5742 | -105.1 |
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