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We study actuarial methods of option pricing in a fractional Black–Scholes model with time-dependent volatility. We interpret the option as a potential loss and we show that the fair premium needed to insure this loss coincides with the expectation of the discounted claim payoff under the average risk neutral measure.
Wydawca
Rocznik
Tom
Strony
181--193
Opis fizyczny
Bibliogr. 16 poz.
Twórcy
autor
- Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland
Bibliografia
- [1] M. Bladt and T. H. Rydberg, An actuarial approach to option pricing under the physical measure and without market assumptions, Insurance Math. Econom. 22 (1998), 65–73.
- [2] R. M. Dudley and R. Norvaiša, An Introduction to p-Variation and Young Integrals, Maphysto Lecture Notes 1, Aarhus, 1998.
- [3] H. Gu, J.-R. Liang, J.Wang, L.-J. Lu, W.-Y. Qiu and F.-Y. Ren, Fractional Fokker–Planck equation and Black–Scholes formula in composite-diffusive regime, J. Statist. Phys. 146 (2012), 205–216.
- [4] H. Gu, J.-R. Liang and Y.-X. Zhang, On a time-changed geometric Brownian motion and its application in financial market, Acta Phys. Polon. B 43 (2012), 1667–1681.
- [5] M. L. Kleptsyna, A. Le Breton and M. C. Roubaud, Parameter estimation and optimal filtering for fractional type stochastic systems, Statist. Inference Stoch. Process 3 (2000), 173–182.
- [6] J. Mémin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51(2001), 197–206.
- [7] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008.
- [8] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), 571–587.
- [9] V. Pipiras, V. Murad and S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli 7 (2001), 873–897.
- [10] A. A. Ruzmaikina, Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion, J. Statist. Phys. 100 (2000), 1049–1069.
- [11] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987.
- [12] L. Słominski and B. Ziemkiewicz, Inequalities for the Lp norms of integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 73 (2005), 79–90.
- [13] T. Sottinen and E. Valkeila, Fractional Brownian motion as a model in finance, preprint 302, Univ. of Helsinki Dept. Math., 2001.
- [14] T. Sottinen and E. Valkeila, On arbitrage and replication in the fractional Black–Scholes pricing model, Statistics and Decisions 21 (2003), 137–151.
- [15] E. Valkeila, On some properties of geometric fractional Brownian motions, preprint 224, Univ. of Helsinki, Dept. Math., 1999.
- [16] H. Xue and Q. Y. Li, An actuarial approach to the minimum or maximum option pricing in fractional Brownian motion environment, in: 2nd IEEE Int. Conf. Inf. and Financial Engineering, 2010, 216–219.
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Bibliografia
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