Limit theorems for prices of options written on semi-Markov processes

Scalas, E.; Toaldo, B.

doi:10.1090/tpms/1153

Authors: E. Scalas and B. Toaldo
Journal: Theor. Probability and Math. Statist. 105 (2021), 3-33
MSC (2020): Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/tpms/1153
Published electronically: December 7, 2021
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Abstract: We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

References

David Applebaum, Lévy processes and stochastic calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. MR 2512800, DOI 10.1017/CBO9780511809781
Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103, DOI 10.1007/978-3-0348-0087-7
Giacomo Ascione, Abstract Cauchy problems for the generalized fractional calculus, Nonlinear Anal. 209 (2021), Paper No. 112339, 22. MR 4236481, DOI 10.1016/j.na.2021.112339
Giacomo Ascione, Yuliya Mishura, and Enrica Pirozzi, Time-changed fractional Ornstein-Uhlenbeck process, Fract. Calc. Appl. Anal. 23 (2020), no. 2, 450–483. MR 4098657, DOI 10.1515/fca-2020-0022
Peter Becker-Kern, Mark M. Meerschaert, and Hans-Peter Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004), no. 1B, 730–756. MR 2039941, DOI 10.1214/aop/1079021462
Jean Bertoin, Subordinators: examples and applications, Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1–91. MR 1746300, DOI 10.1007/978-3-540-48115-7_{1}
Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989. MR 1015093
Álvaro Cartea, Derivatives pricing with marked point processes using tick-by-tick data, Quant. Finance 13 (2013), no. 1, 111–123. MR 3005353, DOI 10.1080/14697688.2012.661447
J. L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948), 422–438. MR 25098, DOI 10.1090/S0002-9947-1948-0025098-8
Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. II, Die Grundlehren der mathematischen Wissenschaften, Band 218, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by Samuel Kotz. MR 0375463
Jean Jacod and Philip Protter, Options prices in incomplete markets, Enlargement of filtrations, ESAIM Proc. Surveys, vol. 56, EDP Sci., Les Ulis, 2017, pp. 72–87. MR 3720064, DOI 10.1051/proc/201756072
Antoine Jacquier and Lorenzo Torricelli, Anomalous diffusions in option prices: connecting trade duration and the volatility term structure, SIAM J. Financial Math. 11 (2020), no. 4, 1137–1167. MR 4173215, DOI 10.1137/19M1289832
Olav Kallenberg, Foundations of modern probability, Probability and its Applications (New York), Springer-Verlag, New York, 1997. MR 1464694
Anatoly N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Operator Theory 71 (2011), no. 4, 583–600. MR 2854867, DOI 10.1007/s00020-011-1918-8
Francesco Mainardi, On some properties of the Mittag-Leffler function $E_\alpha (-t^\alpha )$, completely monotone for $t>0$ with $0<\alpha <1$, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 7, 2267–2278. MR 3253257, DOI 10.3934/dcdsb.2014.19.2267
Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), no. 59, 1600–1620. MR 2835248, DOI 10.1214/EJP.v16-920
Mark M. Meerschaert and Hans-Peter Scheffler, Triangular array limits for continuous time random walks, Stochastic Process. Appl. 118 (2008), no. 9, 1606–1633. MR 2442372, DOI 10.1016/j.spa.2007.10.005
Mark M. Meerschaert and Alla Sikorskii, Stochastic models for fractional calculus, De Gruyter Studies in Mathematics, vol. 43, Walter de Gruyter & Co., Berlin, 2012. MR 2884383
Mark M. Meerschaert and Peter Straka, Semi-Markov approach to continuous time random walk limit processes, Ann. Probab. 42 (2014), no. 4, 1699–1723. MR 3262490, DOI 10.1214/13-AOP905
Mark M. Meerschaert and Bruno Toaldo, Relaxation patterns and semi-Markov dynamics, Stochastic Process. Appl. 129 (2019), no. 8, 2850–2879. MR 3980146, DOI 10.1016/j.spa.2018.08.004
R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics 3 (1976), 125–144.
M. Montero, Renewal equations for option pricing, The European Journal of Physics B 65 (2008), 295–306.
M. O’Hara, Market Microstructure Theory, Wiley, New York, 1997.
Pierre Patie and Anna Srapionyan, Self-similar Cauchy problems and generalized Mittag-Leffler functions, Fract. Calc. Appl. Anal. 24 (2021), no. 2, 447–482. MR 4254314, DOI 10.1515/fca-2021-0020
K. A. Penson and K. Górska, Exact and explicit probability densities for one-sided Lévy stable distributions, Phys. Rev. Lett. 105 (2010), no. 21, 210604, 4. MR 2740992, DOI 10.1103/PhysRevLett.105.210604
M. Politi, T. Kaizoji, and E. Scalas, Full characterization of the fractional Poisson process, Europhysics Letters 96 (2011), 20004.
L. Ponta, M. Trinh, M. Raberto, E. Scalas, and S. Cincotti, Modeling non-stationarities in high-frequency financial time series, Physica A 521 (2019), 173–196.
Mladen Savov and Bruno Toaldo, Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation, Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 4, 2640–2671 (English, with English and French summaries). MR 4164851, DOI 10.1214/20-AIHP1053
E. Scalas and M. Politi, A parsimonious model for intraday European option pricing, Preprint (2012), arXiv:1202.4332.
Bruno Toaldo, Lévy mixing related to distributed order calculus, subordinators and slow diffusions, J. Math. Anal. Appl. 430 (2015), no. 2, 1009–1036. MR 3351994, DOI 10.1016/j.jmaa.2015.05.024
René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, 2nd ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. Theory and applications. MR 2978140, DOI 10.1515/9783110269338
Richard L. Wheeden and Antoni Zygmund, Measure and integral, 2nd ed., Pure and Applied Mathematics (Boca Raton), CRC Press, Boca Raton, FL, 2015. An introduction to real analysis. MR 3381284, DOI 10.1201/b18361