Monte Carlo method for pricing lookback options in Lévy models

O. E. Kudryavtsev   ; A. S. Grechko   ; I. E. Mamadov

Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 2, pp. 305-334

Résumé

We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener–Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.

Détail

Keywords: Monte Carlo method, Wiener–Hopf factorization, integral transformations, option pricing.
Mots-clés : Lévy processes

Bibliographie

[1] J. Abate, W. Whitt, “A unified framework for numerically inverting Laplace transforms”, INFORMS J. Comput., 18:4 (2006), 408–421 | DOI | MR | Zbl

[2] L. Andersen, J. Andreasen, “Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing”, Rev. Deriv. Res., 4:3 (2000), 231–262 | DOI | Zbl

[3] O. E. Barndorff-Nielsen, “Processes of normal inverse Gaussian type”, Finance Stoch., 2:1 (1998), 41–68 | DOI | MR | Zbl

[4] O. E. Barndorff-Nielsen, S. Z. Levendorskii, “Feller processes of normal inverse Gaussian type”, Quant. Finance, 1:3 (2001), 318–331 | DOI | MR | Zbl

[5] M. Boyarchenko, M. de Innocentis, S. Z. Levendorskiĭ, “Prices of barrier and first-touch digital options in Lévy-driven models, near barrier”, Int. J. Theor. Appl. Finance, 14:7 (2011), 1045–1090 | DOI | MR | Zbl

[6] S. I. Boyarchenko, S. Z. Levendorskiĭ, Non-Gaussian Merton–Black–Scholes theory, Adv. Ser. Stat. Sci. Appl. Probab., 9, World Sci. Publ., River Edge, NJ, 2002, xxii+398 pp. | DOI | MR | Zbl

[7] S. Boyarchenko, S. Levendorskiĭ, “American options: the EPV pricing model”, Ann. Finance, 1:3 (2005), 267–292 | DOI | Zbl

[8] S. Boyarchenko, S. Levendorskiĭ, “Efficient Laplace inversion, Wiener–Hopf factorization and pricing lookbacks”, Int. J. Theor. Appl. Finance, 16:3 (2013), 1350011, 40 pp. | DOI | MR | Zbl

[9] P. Carr, H. Geman, D. B. Madan, M. Yor, “The fine structure of asset returns: an empirical investigation”, J. Bus., 75:2 (2002), 305–332 | DOI

[10] R. Cont, P. Tankov, Financial modelling with jump processes, Chapman Hall/CRC Financ. Math. Ser., Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+535 pp. | DOI | MR | Zbl

[11] R. Cont, E. Voltchkova, “A finite difference scheme for option pricing in jump diffusion and exponential Lévy models”, SIAM J. Numer. Anal., 43:4 (2005), 1596–1626 | DOI | MR | Zbl

[12] G. Cybenko, “Approximation by superpositions of a sigmoidal function”, Math. Control Signals Systems, 2:4 (1989), 303–314 | DOI | MR | Zbl

[13] M. F. Dixon, I. Halperin, P. Bilokon, Machine learning in finance. From theory to practice, Springer, Cham, 2020, xxiv+548 pp. | DOI | MR | Zbl

[14] G. Doetsch, Introduction to the theory and application of the Laplace transformation, Springer-Verlag, New York–Heidelberg, 1974, vii+326 pp. | DOI | MR | Zbl

[15] E. Eberlein, K. Glau, A. Papapantoleon, “Analyticity of the Wiener–Hopf factors and valuation of exotic options in Lévy models”, Advanced mathematical methods for finance, Springer, Heidelberg, 2011, 223–245 | DOI | MR | Zbl

[16] E. A. Feinberg, A. N. Shiryaev, “Kolmogorov's equations for jump Markov processes and their applications to control problems”, Theory Probab. Appl., 66:4 (2022), 582–600 | DOI | DOI | MR | Zbl

[17] W. Feller, “On the integro-differential equations of purely discontinuous Markoff processes”, Trans. Amer. Math. Soc., 48:3 (1940), 488–515 | DOI | MR | Zbl

[18] A. Ferreiro-Castilla, A. E. Kyprianou, R. Scheichl, G. Suryanarayana, “Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation”, Stochastic Process. Appl., 124:2 (2014), 985–1010 | DOI | MR | Zbl

[19] A. Ferreiro-Castilla, K. Van Schaik, “Applying the Wiener–Hopf Monte Carlo simulation technique for Lévy processes to path functionals”, J. Appl. Probab., 52:1 (2015), 129–148 | DOI | MR | Zbl

[20] K. Glau, “Classification of Lévy processes with parabolic Kolmogorov backward equations”, Theory Probab. Appl., 60:3 (2016), 383–406 | DOI | DOI | MR | Zbl

[21] B. V. Gnedenko, A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, Inc., Cambridge, MA, 1954, ix+264 pp. | MR | MR | Zbl

[22] L. Goudenège, A. Molent, A. Zanette, “Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models”, Quant. Finance, 20:4 (2020), 573–591 | DOI | MR | Zbl

[23] Jeonggyu Huh, “Pricing options with exponential Lévy neural network”, Expert Syst. Appl., 127 (2019), 128–140 | DOI

[24] A. Itkin, Pricing derivatives under Lévy models. Modern finite-difference and pseudo-differential operators approach, Pseudo Diff. Oper., 12, Birkhäuser/Springer, New York, 2017, xx+308 pp. | DOI | MR | Zbl

[25] K. R. Jackson, S. Jaimungal, V. Surkov, “Fourier space time-stepping for option pricing with Lévy models”, J. Comput. Finance, 12:2 (2008), 1–29 | DOI | MR | Zbl

[26] J. L. Kirkby, “Robust barrier option pricing by frame projection under exponential Lévy dynamics”, Appl. Math. Finance, 24:4 (2017), 337–386 | DOI | MR | Zbl

[27] J. L. Kirkby, “Robust option pricing with characteristic functions and the B-spline order of density projection”, J. Comput. Finance, 21:2 (2017), 61–100 | DOI

[28] A. N. Kolmogorov, “Ob analiticheskikh metodakh v teorii veroyatnostei”, UMN, 1938, no. 5, 5–41 ; A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung”, Math. Ann., 104:1 (1931), 415–458 ; A. N. Kolmogorov, “On analytical methods in probability theory”, Selected works of A. N. Kolmogorov, т. II, Math. Appl. (Soviet Ser.), 26, Kluwer Acad. Publ., Dordrecht, 1992, 62–108 | DOI | MR | Zbl | MR | Zbl

[29] A. N. Kolmogorov, “On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition”, Amer. Math. Soc. Transl. Ser. 2, 28, Amer. Math. Soc., Providence, RI, 1963, 55–59 | DOI | MR | MR | Zbl | Zbl

[30] S. G. Kou, Hui Wang, “First passage times of a jump diffusion process”, Adv. in Appl. Probab., 35:2 (2003), 504–531 | DOI | MR | Zbl

[31] O. Ye. Kudryavtsev, “An efficient numerical method to solve a special class of integro-differential equations relating to the Levy models”, Math. Models Comput. Simul., 3:6 (2011), 706–711 | DOI | MR | Zbl

[32] O. Kudryavtsev, “Finite difference methods for option pricing under Lévy processes: Wiener–Hopf factorization approach”, Sci. World J., 2013 (2013), 963625, 12 pp. | DOI | Zbl

[33] O. Kudryavtsev, “Advantages of the Laplace transform approach in pricing first touch digital options in Lévy-driven models”, Bol. Soc. Mat. Mex. (3), 22:2 (2016), 711–731 | DOI | MR | Zbl

[34] O. E. Kudryavtsev, “Approximate Wiener–Hopf factorization and Monte Carlo methods for Lévy processes”, Theory Probab. Appl., 64:2 (2019), 186–208 | DOI | DOI | MR | Zbl

[35] O. Kudryavtsev, S. Levendorskiĭ, “Fast and accurate pricing of barrier options under Lévy processes”, Finance Stoch., 13:4 (2009), 531–562 | DOI | MR | Zbl

[36] O. Kudryavtsev, S. Levendorskiĭ, Efficient pricing options with barrier and lookback features under Lévy processes, Working paper, 2011, 29 pp. https://ssrn.com/abstract=1857943

[37] O. Kudryavtsev, P. Luzhetskaya, “The Wiener–Hopf factorization for pricing options made easy”, Eng. Lett., 28:4 (2020), 1310–1317

[38] O. Kudryavtsev, A. Zanette, “Efficient pricing of swing options in Lévy-driven models”, Quant. Finance, 13:4 (2013), 627–635 | DOI | MR | Zbl

[39] O. Kudryavtsev, A. Zanette, Applications of Lévy Processes, Math. Res. Dev., Nova Sci. Publ., New York, 2021, 259 pp.

[40] A. Kuznetsov, A. E. Kyprianou, J. C. Pardo, K. van Schaik, “A Wiener–Hopf Monte Carlo simulation technique for Lévy processes”, Ann. Appl. Probab., 21:6 (2011), 2171–2190 | DOI | MR | Zbl

[41] C. Lemieux, Monte Carlo and quasi-Monte Carlo sampling, Springer Ser. Statist., Springer, New York, 2009, xvi+373 pp. | DOI | MR | Zbl

[42] S. Z. Levendorskiĭ, “Method of paired contours and pricing barrier options and CDSs of long maturities”, Int. J. Theor. Appl. Finance, 17:5 (2014), 1450033, 58 pp. | DOI | MR | Zbl

[43] S. Levendorskiĭ, “Ultra-fast pricing barrier options and CDSs”, Int. J. Theor. Appl. Finance, 20:5 (2017), 1750033, 27 pp. | DOI | MR | Zbl

[44] D. B. Madan, E. Seneta, “The variance gamma (V.G.) model for share market returns”, J. Bus., 63:4 (1990), 511–524 | DOI

[45] C. E. Phelan, D. Marazzina, G. Fusai, G. Germano, “Fluctuation identities with continuous monitoring and their application to the pricing of barrier options”, European J. Oper. Res., 271:1 (2018), 210–223 | DOI | MR | Zbl

[46] K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math., 68, Cambridge Univ. Press, Cambridge, 1999, xii+486 pp. | MR | Zbl

[47] A. N. Shiryaev, “Part 1. Facts. Models”, Essentials of stochastic finance, Adv. Ser. Stat. Sci. Appl. Probab., 3, World Sci. Publ., River Edge, NJ, 1999, 2–379 | DOI | MR | Zbl