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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

Jiao Song 1,2,, , Jiang-Lun Wu 2, and  Fangzhou Huang 3,

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

* Corresponding author

Received  September 2019 Revised  March 2020 Published  May 2020

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 $-stable white noise. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934/cpaa.2020184
References:
[1] D. Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511755323.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

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O. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), 1-13.  doi: 10.1111/1467-9469.00045.  Google Scholar

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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.   Google Scholar

[6] J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15] A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993.   Google Scholar
[16]

S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. Google Scholar

[17]

R. Jarrow, Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198.   Google Scholar

[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.   Google Scholar

[19]

P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. Google Scholar

[20]

P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. Google Scholar

[21]

B. Mandelbrot, The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106.   Google Scholar

[22] G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[25]

O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188.   Google Scholar

[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.  Google Scholar

[27]

V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986.  Google Scholar

[28]

C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013. Google Scholar

show all references

References:
[1] D. Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511755323.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Campbell, S. Giglio, C. Polk and R. Turley, An intertemporal CAPM with stochastic volatility, J. Financ. Econ., 128 (2018), 207-233.   Google Scholar

[6] J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15] A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993.   Google Scholar
[16]

S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. Google Scholar

[17]

R. Jarrow, Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198.   Google Scholar

[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.   Google Scholar

[19]

P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. Google Scholar

[20]

P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. Google Scholar

[21]

B. Mandelbrot, The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106.   Google Scholar

[22] G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[25]

O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188.   Google Scholar

[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.  Google Scholar

[27]

V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986.  Google Scholar

[28]

C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013. Google Scholar

Figure 1.  $ \alpha_1<1 $ and $ \alpha_2>1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 1 and $ \mu_2 $ = 10
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Figure 2.  $ \alpha_1<1 $ and $ \alpha_2>1 $: $ \lambda $ changes when $ \alpha_1 $ = 0.5 and $ \alpha_2 $ = 1.5
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Figure 3.  $ \alpha_1<1 $ and $ \alpha_2>1 $: $ \mu_2 $ changes when $ \alpha_1 $ = 0.25 and $ \alpha_2 $ = 1.75
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Figure 4.  $ \alpha_1>1 $ and $ \alpha_2<1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 1 and $ \mu_2 $ = 10
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Figure 5.  $ \alpha_1>1 $ and $ \alpha_2<1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 10 and $ \mu_2 $ = 1
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Figure 6.  $ \alpha_1>1 $ and $ \alpha_2<1 $: $ \mu_2 $ changes when $ \alpha_1 $ = 1.5 and $ \alpha_2 $ = 0.5
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Table 1.  Data processed for sample trajectories when $ \alpha_1<1 $ and $ \alpha_2>1 $
$ \lambda $ $ \mu_1 $ $ \mu_2 $ $ \alpha_1 $ $ \alpha_2 $ t $ X^\alpha_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
Table Options

Download as excel

$ \lambda $ $ \mu_1 $ $ \mu_2 $ $ \alpha_1 $ $ \alpha_2 $ t $ X^\alpha_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
Table 2.  Data processed for sample trajectories when $ \alpha_1>1 $ and $ \alpha_2<1 $
$ \lambda $ $ \mu_1 $ $ \mu_2 $ $ \alpha_1 $ $ \alpha_2 $ t $ X^\alpha_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
Table Options

Download as excel

$ \lambda $ $ \mu_1 $ $ \mu_2 $ $ \alpha_1 $ $ \alpha_2 $ t $ X^\alpha_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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