The Algorithm for the Bermuda Option Pricing Using Deep Learning

doi:10.22716/sckt.2019.7.4.058

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2019 Vol.7, Issue 4 Preview Page Next Page

Research Article

The Algorithm for the Bermuda Option Pricing Using Deep Learning 딥러닝을 활용한 버뮤다옵션 가격 결정의 알고리듬: Kil-Joo Kim¹, Chang-Ho An², Kang-Jin Hwang³
김길주¹, 안창호², 황강진³; ¹Ph.D. in business administration at Korea University, former lecturer at Sangmyung University
²Department of Financial Information Engineering, Seo Kyeong University
³Department of Management Information Systems, Gangneung Yeongdong University

¹고려대학교 경영학박사 (전)상명대학교 강사
²서경대학교 금융정보공학과
³강릉영동대학교 경영정보과

31 December 2019. pp. 125-131

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Abstract

버뮤다 옵션의 가격을 계산하기 위한, 좀더 엄밀하게 말하면 시장가격에 가장 근사하다고 의견의 일치를 보는 방법은 아직 없다. 버뮤다옵션의 가격결정은 동적 프로그래밍 원칙을 해결하는 것과 같으며, 특히 큰 차원에 있어서 주된 어려움은 지속 가치와 관련된 조건부 기대치들의 계산에서 비롯된다. 이러한 조건부 기대치들은 유한 차원 벡터 공간상에서 전형적인 회귀 기법에 의해 계산된다. 본 연구에서는 조건부 기대치들의 신경망근사치를 연구한다. 또한 표준 최소자승회귀분석을 신경망 근사치로 대체함으로써 잘 알려진 Longstaff-Schwartz 알고리듬과 합치됨을 보인다.

There is no consensus on Bermuda option pricing, more precisely, the closest approximation to market prices. The pricing of Bermuda options is like solving dynamic programming principles, and the main difficulty, especially on a larger scale, comes from the calculation of conditional expectations related to continuation value. These conditional expectations are computed by a typical regression technique on finite dimensional vector spaces. In this study, we study the neural network approximation of conditional expectations. It is also shown that it replaces the standard least squares regression with a neural network approximation, which is consistent with the well-known Longstaff-Schwartz algorithm.

Keywords

Deep learning

Bermuda option

Neural network approximation

Regression

Convergence

References

F. Longstaff and R. Schwartz “Valuing American options by simulation : A simple least-square approach”, Review of Financial Studies, Vol. 14, pp. 113-147, 2001.10.1093/rfs/14.1.113
A. Ibanez “Valuation by Simulation of Contingent Claims with Multiple Early Exercise Opportunities”, Mathematical Finance, Vol. 13, pp. 223-248, 2004.10.1111/j.0960-1627.2004.00190.x
L C. G. Rogers “Monte Carlo Valuation of American Options”, Mathematical Finance, Vol. 12, pp. 271-286, 2002.10.1111/1467-9965.02010
M. B. Haugh and L. Kogan “Pricing American Options: A Duality Approach.”, Operations Research, Vol. 52, pp. 258-270, 2004.10.1287/opre.1030.0070
J. A. Tilley “Valuing american options in a path simulation model”, Transactions of the Society of Actuaries, Vol. 45, No. 83, p. 104, 1993.
J. F. Carriere “Valuation of the early-exercise price for options using simulations and nonparametric regression”, Insurance: Mathematics and Economics, Vol. 19, No. 1, pp. 19-30, 1996.10.1016/S0167-6687(96)00004-2
J. Tsitsiklis and B. V. Roy “Regression methods for pricing complex American-style options”, IEEE Trans. Neural Netw., Vol. 12, No. 4, pp. 694-703, 2001.10.1109/72.935083 18249905
M. Broadie and P. Glasserman “A stochastic mesh method for pricing high-dimensional american options”, Journal of Computational Finance, Vol., No. 7, pp. 35-72, 2004.10.21314/JCF.2004.117
V. Bally and G. Pages “A quantization algorithm for solving multidimensional discrete-time optimal stopping problems”, Bernoulli, Vol. 9, No. 6, pp. 1003-1049, 2003.10.3150/bj/1072215199
A. L. Bronstein, G. Pages, and J. Portes “Multi-asset american options and parallel quantization”, Methodology and Computing in Applied Probability, Vol. 15, No. 3, pp. 547-561, 2013.10.1007/s11009-011-9265-4
B. Bouchard and X.Warin “Monte-carlo valuation of american options: Facts and new algorithms to improve existing methods”, In R. A. Carmona, P. Del Moral, P. Hu, and N. Oudjane, editors, Numerical Methods in Finance, vol.12 of Springer Proceedings in Mathematics, pp. 215-255. Springer Berlin Heidelberg, 2012.10.1007/978-3-642-25746-9_7
G. Pages. Numerical Probability: An Introduction with Applications to Finance, Springer, pp. 204-227, 2018.10.1007/978-3-319-90276-0
안창호 “비상장기업 경영성과예측모형 추정에 관한 연구”, 융복합지식학회논문지, 제4권, 제2호, pp. 1-8, 2016.
E. Gobet, J. Lemor, and X. Warin “A regression-based Monte Carlo method to solve backward stochastic differential equations”, Annals of Applied Probability, Vol. 15, No. 3, pp. 2172-2202, 2005.10.1214/105051605000000412
M. Kohler, A. Krzyzak, and N. Todorovic “Pricing of high-dimensional american options by neural networks”, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, Vol. 20, No. 3, pp. 383-410, 2010.10.1111/j.1467-9965.2010.00404.x
S. Becker, P. Cheridito, and A. Jentzen. Deep optimal stopping, arXiv preprint arXiv:1804.05394, 2018.
V. I. Arnold. On functions of three variables. Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957-1965, pp. 5-8, 2009.10.1007/978-3-642-01742-1_2
K. Hornik “Approximation capabilities of multilayer feedforward networks”, Neural Networks, Vol. 4, No. 2, pp. 251-257, 1991.10.1016/0893-6080(91)90009-T
G. Cybenko “Approximation by superpositions of a sigmoidal function. Mathematics of Control”, Signals and Systems, Vol. 2, No. 4, pp. 303-314, 1989.10.1007/BF02551274
A. Kolmogorov “On the representation of continuous functions of several variables as superpositions of functions of smaller number of variables”, Soviet. Math. Dokl, Vol. 108, pp. 179-182, 1956.

Information

Publisher :The Society of Convergence Knowledge
Publisher(Ko) :융복합지식학회
Journal Title :The Society of Convergence Knowledge Transactions
Journal Title(Ko) :융복합지식학회논문지
Volume : 7
No :4
Pages :125-131
DOI :https://doi.org/10.22716/sckt.2019.7.4.058

[1] F. Longstaff and R. Schwartz “Valuing American options by simulation : A simple least-square approach”, Review of Financial Studies, Vol. 14, pp. 113-147, 2001.10.1093/rfs/14.1.113

[2] A. Ibanez “Valuation by Simulation of Contingent Claims with Multiple Early Exercise Opportunities”, Mathematical Finance, Vol. 13, pp. 223-248, 2004.10.1111/j.0960-1627.2004.00190.x

[3] L C. G. Rogers “Monte Carlo Valuation of American Options”, Mathematical Finance, Vol. 12, pp. 271-286, 2002.10.1111/1467-9965.02010

[4] M. B. Haugh and L. Kogan “Pricing American Options: A Duality Approach.”, Operations Research, Vol. 52, pp. 258-270, 2004.10.1287/opre.1030.0070

[5] J. A. Tilley “Valuing american options in a path simulation model”, Transactions of the Society of Actuaries, Vol. 45, No. 83, p. 104, 1993.

[6] J. F. Carriere “Valuation of the early-exercise price for options using simulations and nonparametric regression”, Insurance: Mathematics and Economics, Vol. 19, No. 1, pp. 19-30, 1996.10.1016/S0167-6687(96)00004-2

[7] J. Tsitsiklis and B. V. Roy “Regression methods for pricing complex American-style options”, IEEE Trans. Neural Netw., Vol. 12, No. 4, pp. 694-703, 2001.10.1109/72.935083 18249905

[8] M. Broadie and P. Glasserman “A stochastic mesh method for pricing high-dimensional american options”, Journal of Computational Finance, Vol., No. 7, pp. 35-72, 2004.10.21314/JCF.2004.117

[9] V. Bally and G. Pages “A quantization algorithm for solving multidimensional discrete-time optimal stopping problems”, Bernoulli, Vol. 9, No. 6, pp. 1003-1049, 2003.10.3150/bj/1072215199

[10] A. L. Bronstein, G. Pages, and J. Portes “Multi-asset american options and parallel quantization”, Methodology and Computing in Applied Probability, Vol. 15, No. 3, pp. 547-561, 2013.10.1007/s11009-011-9265-4

[11] B. Bouchard and X.Warin “Monte-carlo valuation of american options: Facts and new algorithms to improve existing methods”, In R. A. Carmona, P. Del Moral, P. Hu, and N. Oudjane, editors, Numerical Methods in Finance, vol.12 of Springer Proceedings in Mathematics, pp. 215-255. Springer Berlin Heidelberg, 2012.10.1007/978-3-642-25746-9_7

[12] G. Pages. Numerical Probability: An Introduction with Applications to Finance, Springer, pp. 204-227, 2018.10.1007/978-3-319-90276-0

[13] 안창호 “비상장기업 경영성과예측모형 추정에 관한 연구”, 융복합지식학회논문지, 제4권, 제2호, pp. 1-8, 2016.

[14] E. Gobet, J. Lemor, and X. Warin “A regression-based Monte Carlo method to solve backward stochastic differential equations”, Annals of Applied Probability, Vol. 15, No. 3, pp. 2172-2202, 2005.10.1214/105051605000000412

[15] M. Kohler, A. Krzyzak, and N. Todorovic “Pricing of high-dimensional american options by neural networks”, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, Vol. 20, No. 3, pp. 383-410, 2010.10.1111/j.1467-9965.2010.00404.x

[16] S. Becker, P. Cheridito, and A. Jentzen. Deep optimal stopping, arXiv preprint arXiv:1804.05394, 2018.

[17] V. I. Arnold. On functions of three variables. Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957-1965, pp. 5-8, 2009.10.1007/978-3-642-01742-1_2

[18] K. Hornik “Approximation capabilities of multilayer feedforward networks”, Neural Networks, Vol. 4, No. 2, pp. 251-257, 1991.10.1016/0893-6080(91)90009-T

[19] G. Cybenko “Approximation by superpositions of a sigmoidal function. Mathematics of Control”, Signals and Systems, Vol. 2, No. 4, pp. 303-314, 1989.10.1007/BF02551274

[20] A. Kolmogorov “On the representation of continuous functions of several variables as superpositions of functions of smaller number of variables”, Soviet. Math. Dokl, Vol. 108, pp. 179-182, 1956.

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