Option valuation under the VG process by a DG method

Jiří Hozman; Tomáš Tichý

Applications of Mathematics (2021)

  • Volume: 66, Issue: 6, page 857-886
  • ISSN: 0862-7940

Abstract

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The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure based on piecewise polynomial generally discontinuous approximations in the spatial domain. This technique enables a simple treatment of the American early exercise constraint by a direct encompassing it as an additional nonlinear source term to the governing equation. Special attention is paid to the proper discretization of non-local jump integral components, which is based on splitting integrals with respect to the domain according to the size of the jumps. Moreover, to preserve sparsity of resulting linear algebraic systems the pricing equation is integrated in the temporal variable by a semi-implicit Euler scheme. Finally, the numerical results demonstrate the capability of the numerical scheme presented within the reference benchmarks.

How to cite

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Hozman, Jiří, and Tichý, Tomáš. "Option valuation under the VG process by a DG method." Applications of Mathematics 66.6 (2021): 857-886. <http://eudml.org/doc/298020>.

@article{Hozman2021,
abstract = {The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure based on piecewise polynomial generally discontinuous approximations in the spatial domain. This technique enables a simple treatment of the American early exercise constraint by a direct encompassing it as an additional nonlinear source term to the governing equation. Special attention is paid to the proper discretization of non-local jump integral components, which is based on splitting integrals with respect to the domain according to the size of the jumps. Moreover, to preserve sparsity of resulting linear algebraic systems the pricing equation is integrated in the temporal variable by a semi-implicit Euler scheme. Finally, the numerical results demonstrate the capability of the numerical scheme presented within the reference benchmarks.},
author = {Hozman, Jiří, Tichý, Tomáš},
journal = {Applications of Mathematics},
keywords = {option pricing; variance gamma process; integro-differential equation; American style options; discontinuous Galerkin method; semi-implicit discretization},
language = {eng},
number = {6},
pages = {857-886},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Option valuation under the VG process by a DG method},
url = {http://eudml.org/doc/298020},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Hozman, Jiří
AU - Tichý, Tomáš
TI - Option valuation under the VG process by a DG method
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 6
SP - 857
EP - 886
AB - The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure based on piecewise polynomial generally discontinuous approximations in the spatial domain. This technique enables a simple treatment of the American early exercise constraint by a direct encompassing it as an additional nonlinear source term to the governing equation. Special attention is paid to the proper discretization of non-local jump integral components, which is based on splitting integrals with respect to the domain according to the size of the jumps. Moreover, to preserve sparsity of resulting linear algebraic systems the pricing equation is integrated in the temporal variable by a semi-implicit Euler scheme. Finally, the numerical results demonstrate the capability of the numerical scheme presented within the reference benchmarks.
LA - eng
KW - option pricing; variance gamma process; integro-differential equation; American style options; discontinuous Galerkin method; semi-implicit discretization
UR - http://eudml.org/doc/298020
ER -

References

top
  1. Abramowitz, M., (eds.), I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, U. S. Department of Commerce, Washington (1964),9999MR99999 0167642 . (1964) Zbl0171.38503
  2. Almendral, A., Oosterlee, C. W., 10.1080/13504860600724885, Appl. Math. Finance 14 (2007), 131-152. (2007) Zbl1160.91346MR2323277DOI10.1080/13504860600724885
  3. Bensoussan, A., Lions, J. L., Contrôle impulsionnel et inéquations quasi variationnelles, Gauthier-Villars, Paris (1982), French. (1982) Zbl0491.93002MR0673169
  4. Black, F., Scholes, M., 10.1086/260062, J. Polit. Econ. 81 (1973), 637-654. (1973) Zbl1092.91524MR3363443DOI10.1086/260062
  5. Cantarutti, N., Guerra, J., 10.1080/00207160.2018.1427853, Int. J. Comput. Math. 96 (2019), 1087-1106. (2019) MR3927357DOI10.1080/00207160.2018.1427853
  6. Carr, P., Jarrow, R., Myneni, R., 10.1111/j.1467-9965.1992.tb00040.x, Math. Finance 2 (1992), 87-106. (1992) Zbl0900.90004DOI10.1111/j.1467-9965.1992.tb00040.x
  7. Cont, R., Tankov, P., 10.1201/9780203485217, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton (2004). (2004) Zbl1052.91043MR2042661DOI10.1201/9780203485217
  8. Cont, R., Voltchkova, E., A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal. 43 (2005), 1596-1626 9999DOI99999 10.1137/S0036142903436186 . (2005) Zbl1101.47059MR2182141
  9. Cont, R., Voltchkova, E., 10.1007/s00780-005-0153-z, Finance Stoch. 9 (2005), 299-325. (2005) Zbl1096.91023MR2211710DOI10.1007/s00780-005-0153-z
  10. Delbaen, F., Schachermayer, W., 10.1007/BF01450498, Math. Ann. 300 (1994), 463-520. (1994) Zbl0865.90014MR1304434DOI10.1007/BF01450498
  11. Dolejší, V., Feistauer, M., 10.1007/978-3-319-19267-3, Springer Series in Computational Mathematics 48. Springer, Cham (2015). (2015) Zbl1401.76003MR3363720DOI10.1007/978-3-319-19267-3
  12. Eberlein, E., 10.1007/978-3-540-71297-8_19, Handbook of Financial Time Series Springer, Berlin (2009), 439-455. (2009) Zbl1186.60042DOI10.1007/978-3-540-71297-8_19
  13. Feng, L., Linetsky, V., 10.1287/opre.1070.0419, Oper. Res. 56 (2008), 304-325. (2008) Zbl1167.91367MR2410308DOI10.1287/opre.1070.0419
  14. Fu, M. C., 10.1007/978-0-8176-4545-8_2, Advances in Mathematical Finance Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2007), 21-35. (2007) Zbl1159.62069MR2359360DOI10.1007/978-0-8176-4545-8_2
  15. Haug, E. G., The Complete Guide to Option Pricing Formulas, McGraw-Hill, New York (2007). (2007) 
  16. Hecht, F., 10.1515/jnum-2012-0013, J. Numer. Math. 20 (2012), 251-265. (2012) Zbl1266.68090MR3043640DOI10.1515/jnum-2012-0013
  17. Hirsa, A., Madan, D. B., 10.21314/JCF.2003.112, J. Comput. Finance 7 (2004), 63-80. (2004) DOI10.21314/JCF.2003.112
  18. Hozman, J., Tichý, T., 10.2298/FIL1615253H, Filomat 30 (2016), 4253-4263. (2016) Zbl1461.91351MR3601917DOI10.2298/FIL1615253H
  19. Hozman, J., Tichý, T., DG framework for pricing European options under one-factor stochastic volatility models, J. Comput. Appl. Math. 344 (2018), 585-600 9999DOI99999 10.1016/j.cam.2018.05.064 . (2018) Zbl1394.65099MR3825537
  20. Hozman, J., Tichý, T., Vlasák, M., 10.21136/AM.2019.0305-18, Appl. Math., Praha 64 (2019), 501-530. (2019) Zbl07144726MR4022161DOI10.21136/AM.2019.0305-18
  21. Ikonen, S., Toivanen, J., 10.1016/j.aml.2004.06.010, Appl. Math. Lett. 17 (2004), 809-814. (2004) Zbl1063.65081MR2072839DOI10.1016/j.aml.2004.06.010
  22. Itkin, A., 10.1007/978-1-4939-6792-6, Pseudo-Differential Operators. Theory and Applications 12. Birkhäuser, Basel (2017). (2017) Zbl1419.91002MR3618292DOI10.1007/978-1-4939-6792-6
  23. Jacod, J., Shiryaev, A. N., 10.1007/978-3-662-02514-7, Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin (1987). (1987) Zbl0635.60021MR0959133DOI10.1007/978-3-662-02514-7
  24. nar, S. Kozpı, Uzunca, M., Karasözen, B., 10.1016/j.matcom.2020.05.022, Math. Comput. Simul. 177 (2020), 568-587. (2020) Zbl07318117MR4103885DOI10.1016/j.matcom.2020.05.022
  25. Kufner, A., John, O., Fučík, S., Function Spaces, Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis 3. Noordhoff International Publishing, Leyden (1977). (1977) Zbl0364.46022MR0482102
  26. Kwon, Y., Lee, Y., 10.1137/090777529, SIAM J. Numer. Anal. 49 (2011), 2598-2617. (2011) Zbl1232.91712MR2873249DOI10.1137/090777529
  27. Madan, D. B., Carr, P. P., Chang, E. C., 10.1023/A:1009703431535, Eur. Finance Rev. 2 (1998), 79-105. (1998) Zbl0937.91052DOI10.1023/A:1009703431535
  28. Madan, D. B., Milne, F., 10.1111/j.1467-9965.1991.tb00018.x, Math. Finance 1 (1991), 39-55. (1991) Zbl0900.90105DOI10.1111/j.1467-9965.1991.tb00018.x
  29. Madan, D. B., Seneta, E., 10.1086/296519, J. Business 63 (1990), 511-524. (1990) DOI10.1086/296519
  30. Matache, A.-M., Petersdorff, T. von, Schwab, C., 10.1051/m2an:2004003, M2AN, Math. Model. Numer. Anal. 38 (2004), 37-71. (2004) Zbl1072.60052MR2073930DOI10.1051/m2an:2004003
  31. Merton, R. C., 10.2307/3003143, Bell J. Econ. Manage. Sci. 4 (1973), 141-183. (1973) Zbl1257.91043MR0496534DOI10.2307/3003143
  32. Nicholls, D. P., Sward, A., 10.4208/cicp.190513.131114a, Commun. Comput. Phys. 17 (2015), 761-778. (2015) Zbl1375.91243MR3371520DOI10.4208/cicp.190513.131114a
  33. Wong, H. Y., Zhao, J., 10.1137/060671541, SIAM J. Numer. Anal. 46 (2008), 2183-2209. (2008) Zbl1178.35363MR2399414DOI10.1137/060671541
  34. Zvan, R., Forsyth, P. A., Vetzal, K. R., 10.1016/S0377-0427(98)00037-5, J. Comput. Appl. Math. 91 (1998), 199-218. (1998) Zbl0945.65005MR1628686DOI10.1016/S0377-0427(98)00037-5

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