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Modern Stochastics: Theory and Applications


Option pricing in time-changed Lévy models with compound Poisson jumps
Volume 6, Issue 1 (2019), pp. 81–107
Pub. online: 27 November 2018      Type: Research Article      Open accessOpen Access
Received
14 June 2018
Revised
3 September 2018
Accepted
7 November 2018
Published
27 November 2018

Abstract

The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.

References

[1] 
Aas, K., Hobaek, H.I., Dimakos, X.: Risk estimation using the multivariate normal inverse Gaussian distribution. J. Risk 8, 39–60 (2005)
[2] 
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004). MR2072890. https://doi.org/10.1017/CBO9780511755323
[3] 
Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. 353, 410–419 (1977)
[4] 
Barndorff-Nielsen, O.E.: Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997). MR1436619. https://doi.org/10.1111/1467-9469.00045
[5] 
Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Finance Stoch. 2(1), 41–68 (1998). MR1804664. https://doi.org/10.1007/s007800050032
[6] 
Barndorff-Nielsen, O.E., Shiryaev, A.N.: Change of Time and Change of Measure. World Scientific, Singapore (2010). MR2779876. https://doi.org/10.1142/7928
[7] 
Bateman, H., Erdélyi, A.: Higher Transcendental Functions, Vol. I. McGraw-Hill, New York (1953)
[8] 
Bianchi, M.L., Rachev, S.T., Kim, Y.S., Fabozzi, F.J.: Tempered infinitely divisible distributions and processes. Theory Probab. Appl. 55(1), 2–26 (2011). MR2768518. https://doi.org/10.1137/S0040585X97984632
[9] 
Cont, R., Tankov, P.: Financial Modeling with Jump Processes. CRC Press, Boca Raton (2004). MR2042661
[10] 
Daal, E.A., Madan, D.B.: An empirical examination of the variance-gamma model for foreign currency options. J. Bus. 78(6), 2121–2152 (2005)
[11] 
Eberlein, E.: Fourier based valuation methods in mathematical finance. In: Benth, F., Kholodnyi, V., Laurence, P. (eds.) Quantitative Energy Finance. Springer, Berlin (2014). MR3184349. https://doi.org/10.1007/978-1-4614-7248-3_3
[12] 
Eberlein, E., Papapantoleon, A., Shiryaev, A.N.: Esscher transform and the duality principle for multidimensional semimartingales. Ann. Appl. Probab. 19, 1944–1971 (2009). MR2569813. https://doi.org/10.1214/09-AAP600
[13] 
Figueroa-Lopez, J.E., Lancette, S.R., Lee, K., Mi, Y.: Estimation of nig and vg models for high frequency financial data. In: Viens, F., Mariani, M.C., Florescu, I. (eds.) Handbook of Modeling High-Frequency Data in Finance. Wiley, New Jersey (2011)
[14] 
Finlay, R., Seneta, E.: Stationary-increment Student and variance-gamma processes. J. Appl. Probab. 43, 441–453 (2006). MR2248575. https://doi.org/10.1239/jap/1152413733
[15] 
Göncü, A., Karahan, M.O., Kuzubaş, T.U.: A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns. N. Am. J. Econ. Finance 36, 69–83 (2016)
[16] 
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, New York (1980). MR0669666
[17] 
Ivanov, R.V.: Closed form pricing of European options for a family of normal-inverse Gaussian processes. Stoch. Models 29(4), 435–450 (2013). MR3175852. https://doi.org/10.1080/15326349.2013.838509
[18] 
Ivanov, R.V.: On risk measuring in the variance-gamma model. Stat. Risk. Model. 35(1–2), 23–33 (2018). MR3739384. https://doi.org/10.1515/strm-2017-0008
[19] 
Ivanov, R.V.: Option pricing in the variance-gamma model under the drift jump. Int. J. Theor. Appl. Finance 21(4), 1–19 (2018). MR3817272. https://doi.org/10.1142/S0219024918500188
[20] 
Ivanov, R.V., Ano, K.: On exact pricing of fx options in multivariate time-changed Lévy models. Rev. Deriv. Res. 19(3), 201–216 (2016). MR3611537. https://doi.org/10.1007/s11009-015-9461-8
[21] 
Ivanov, R.V., Temnov, G.: Truncated moment-generating functions of the nig process and their applications. Stoch. Dyn. 17(5), 1–12 (2017). MR3669415. https://doi.org/10.1142/S0219493717500393
[22] 
Kallsen, J., Shiryaev, A.N.: The cumulant process and Esscher’s change of measure. Finance Stoch. 6, 397–428 (2002). MR1932378. https://doi.org/10.1007/s007800200069
[23] 
Koponen, I.: Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)
[24] 
Küchler, U., Tappe, S.: Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl. 118(2), 261–283 (2008). MR2376902. https://doi.org/10.1016/j.spa.2007.04.006
[25] 
Küchler, U., Tappe, S.: Tempered stable distributions and processes. Stoch. Process. Appl. 123(12), 4256–4293 (2013). MR3096354. https://doi.org/10.1016/j.spa.2013.06.012
[26] 
Linders, D., Stassen, B.: The multivariate variance gamma model: basket option pricing and calibration. Quant. Finance 16(4), 555–572 (2016). MR3473973. https://doi.org/10.1080/14697688.2015.1043934
[27] 
Luciano, E., Schoutens, W.: A multivariate jump-driven financial asset model. Quant. Finance 6(5), 385–402 (2016). MR2261218. https://doi.org/10.1080/14697680600806275
[28] 
Luciano, E., Semeraro, P.: Multivariate time changes for Lévy asset models: Characterization and calibration. J. Comput. Appl. Math. 233, 1937–1953 (2010). MR2564029. https://doi.org/10.1016/j.cam.2009.08.119
[29] 
Luciano, E., Marena, M., Semeraro, P.: Dependence calibration and portfolio fit with factor-based subordinators. Quant. Finance 16(7), 1–16 (2016). MR3516138. https://doi.org/10.1080/14697688.2015.1114661
[30] 
Madan, D., Milne, F.: Option pricing with vg martingale components. Math. Finance 1(4), 39–55 (1991)
[31] 
Madan, D., Seneta, E.: The variance gamma (v.g.) model for share market returns. J. Bus. 63, 511–524 (1991)
[32] 
Madan, D., Yor, M.: Representing the cgmy and Meixner Lévy processes as time changed Brownian motions. J. Comput. Finance 12(1), 27–47 (2008). MR2504899. https://doi.org/10.21314/JCF.2008.181
[33] 
Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998)
[34] 
Moosbrucker, T.: Explaining the correlation smile using variance gamma distributions. J. Fixed Income 16(1), 71–87 (2006)
[35] 
Mozumder, S., Sorwar, G., Dowd, K.: Revisiting variance gamma pricing: An application to S&P500 index options. Int. J. Financ. Eng. 2(2), 1–24 (2015). MR3454654. https://doi.org/10.1142/S242478631550022X
[36] 
Rathgeber, A., Stadler, J., Stöckl, S.: Modeling share returns – an empirical study on the variance gamma model. J. Econ. Finance 40(4), 653–682 (2016)
[37] 
Rosinski, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 667–707 (2007). MR2327834. https://doi.org/10.1016/j.spa.2006.10.003
[38] 
Rydberg, T.H.: The normal-inverse Gaussian process: simulation and approximation. Stoch. Models 13, 887–910 (2013). MR1482297. https://doi.org/10.1080/15326349708807456
[39] 
Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New Jersey (2003)
[40] 
Seneta, E.: The early years of the variance–gamma process. In: Fu, M.C., Jarrow, R., Yen, J.Y., Elliott, R. (eds.) Advances in Mathematical Finance. Birkhauser, Boston (2007). MR2359359. https://doi.org/10.1007/978-0-8176-4545-8_1
[41] 
Shiryaev, A.N.: Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, Singapore (1999). MR1695318. https://doi.org/10.1142/9789812385192
[42] 
Teneng, D.: Modeling and forecasting foreign exchange daily closing prices with normal inverse Gaussian. AIP Conf. Proc. 9, 444–448 (2013)
[43] 
Wallmeier, M., Diethelm, M.: Multivariate downside risk: normal versus variance gamma. J. Futures Mark. 32, 431–458 (2012)
[44] 
Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, 4th Edition. Cambridge University Press, Cambridge (1990). MR1424469. https://doi.org/10.1017/CBO9780511608759
[45] 
Yor, M.: Some remarkable properties of gamma processes. In: Fu, M.C., Jarrow, R., Yen, J.Y., Elliott, R. (eds.) Advances in Mathematical Finance. Birkhauser, Boston (2007). MR2359361. https://doi.org/10.1007/978-0-8176-4545-8_3

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© 2019 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Lévy process change of time compound Poisson process digital option variance-gamma process hypergeometric function

MSC2010
6008 60G44 60G51 60H30 60J75

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