Theory of Probability and Mathematical Statistics
Limit theorems for option prices written on semi-Markov processes
E. Scalas, B. Toaldo
Link
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.
Keywords: Differential geometry, algebraic geometry
Bibliography: 1. D. Applebaum, L\'{e}vy Processes and Stochastic Calculus, second edition, Cambridge University Press, New York, 2009.
2. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector valued Laplace transform and Cauchy problem, second edition, Birkh\"{a}user, Berlin, 2010.
3. G. Ascione, Abstract Cauchy problems for the generalized fractional calculus, Nonlinear Anal. 209 (2021), 112339.
4. G. Ascione, Y. Mishura, and E. Pirozzi, Convergence results for the time-changed fractional Ornstein–Uhlenbeck processes, Preprint (2020), arxiv:2011.02733v1.
5. P. Becker Kern, M. M. Meerschaert, and H. P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004), 730–756.
6. J. Bertoin, Subordinators: examples and applications, Lectures on probability theory and statistics (Saint-Flour, 1997), pp. 1–91. Lectures Notes in Math., 1717, Springer, Berlin, 1999.
7. J. Bertoin, L\'{e}vy processes. Cambridge University Press, Cambridge, 1996.
8. P. Billingsley, Probability and Measure, Wiley, New York, 1986.
9. N. H. Bingham, C. M. Goldie, and J. F. Teugels, Regular variation, Cambridge university press, Cambridge, 1989.
10. A. Cartea, Derivatives pricing with marked point processes using tick-by-tick data, Quant. Finance 13 (2013), 111–123.
11. J. L. Doob, Renewal theory from the point of view of the theory of probability, Trans. Amer. Math. Soc. 63 (1948), 422–438.
12. I. I. Gihman and A. V. Skorohod, The theory of stochastic processes II, Springer-Verlag, 1975.
13. J. Jacod and P. Protter, Option prices in incomplete markets, ESAIM Proc. Surveys 56 (2017), 72–87.
14. A. Jacquier and L. Torricelli, Anomalous Diffusions in Option Prices: Connecting Trade Duration and the Volatility Term Structure, SIAM J. Financial Math. 11 (2020), 1137–1167.
15. O. Kallenberg, Foundations of Modern Probability, Springer, 1997.
16. A. N. Kochubei, General fractional calculus, evolution equations and renewal processes, Integral Equations Operator Theory 71 (2011), 583–600.
17. F. 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 9 (2014), 2267–2278.
18. M. M. Meerschaert, E. Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), 1600–1620.
19. M. M. Meerschaert and H. P. Scheffler, Triangular array limits for continuous time random walks, Stochastic Process. Appl. 118 (2008), 1606–1633.
20. M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Stud. Math., vol. 43, 2012.
21. M. M. Meerschaert and P. Straka, Semi-Markov approach to continuous time random walk limit processes, Ann. Probab. 42 (2014), 1699–1723.
22. M. M. Meerschaert and B. Toaldo, Relaxation patterns and semi-Markov dynamics, Stochastic Process. Appl. 129 (2019), 2850–2879.
23. R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics 3 (1976), 125–144.
24. M. Montero, Renewal equations for option pricing, The European Journal of Physics B 65 (2008), 295–306.
25. M. O'Hara, Market Microstructure Theory, Wiley, New York, 1997.
26. P. Patie and A. Srapionyan, Self-similar Cauchy problems and generalized Mittag-Leffler functions, Fract. Calc. Appl. Anal. 24 (2021), 447–482.
27. K. A. Penson and K. G\'{o}rska, Exact and explicit probability densities for one-sided L\'{e}vy stable distributions, Physical Review Letters 105 (2010), 210604.
28. M. Politi, T. Kaizoji, and E. Scalas, Full characterization of the fractional Poisson process, Europhysics Letters 96 (2011), 20004.
29. 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.
30. M. Savov and B. Toaldo, Semi-Markov processes, integro-differential equations and anomalous diffusion–aggregation, Ann. Inst. Henri Poincar\'{e} Probab. Stat. 56 (2020), 2640–2671.
31. E. Scalas and M. Politi, A parsimonious model for intraday European option pricing, Preprint (2012), arXiv:1202.4332.
32. B. Toaldo, L\'{e}vy mixing related to distributed order calculus, subordinators and slow diffusions, J. Math. Anal. Appl. 430 (2015), 1009–1036.
33. R. L. Schilling, R. Song, and Z. Vondra\v{c}ek, Bernstein functions: theory and applications, De Gruyter Stud. Math., vol. 37, Walter de Gruyter GmbH & Company KG, 2010.
34. R. L. Wheeden and A. Zygmund, Measure and integral: an introduction to real analysis, CRC Press vol. 308, 2015.
