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Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model.
Rocznik
Strony
99--116
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
autor
  • Taras Shevchenko National University of Kyiv Faculty of Mechanics and Mathematics Department of Probability, Statistics and Actuarial Mathematics Volodymyrska 64, 01601 Kyiv, Ukraine
Bibliografia
  • [1] M. Broadie, O. Glafferman, S.J. Kou, Connecting discrete continuous path-dependent options, Finance Stochast. 3 (1999), 55-82.
  • [2] L.-B. Chang, K. Palmer, Smooth convergence in the binomial model, Finance Stochast. 11 (2007), 91-105.
  • [3] N.J. Cutland, E. Kopp, W. Willinger, From discrete to continuous financial models: new convergence results for option pricing, Math. Finance 2 (1993), 101-123.
  • [4] G. Deelstra, F. Delbaen, Long-term returns in stochastic interest rate models: differ­ent convergence results, Applied Stochastic Models and Data Analysis 13 (1997) 34, 401-407.
  • [5] D. Duffie, P. Protter, From discrete- to continuous- time finance: weak convergence of the financial gain process, Math. Finance 2 (1992), 1-15.
  • [6] H.J. Engelbert, W. Schmidt, On the behaviour of certain functionals of the Wiener process and applications to stochastic differential equations, [in:] Stochastic differential systems, Lecture Notes in Control and Inform. Sci. 36 (1981), 47-55.
  • [7] H.J. Engelbert, W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, I, II, III. Math. Nachr. 143 (1989) 1, 167-184, 144 (1989) 1, 241-281, 151 (1991) 1, 149-197.
  • [8] H. Follmer, M. Schweizer, A microeconomic approach to diffusion models for stock prices, Math. Finance 3 (1993), 1-23.
  • [9] R. Gibson, E.S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance 45 (1990), 959-976.
  • [10] I.I. Gikhman, A.V. Skorokhod, The Theory of Stochastic Processes III, Springer-Verlag, Berlin, Heidelberg, 2007.
  • [11] S. Heston, G. Zhou, On the rate of convergence of discrete-time contingent claims, Math. Finance 10 (2000), 53-75.
  • [12] F. Hubalek, W. Schachermayer, When does convergence of asset price processes imply convergence of option prices, Math. Finance 4 (1998), 385-403.
  • [13] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Springer--Verlag, Berlin Heidelberg, 2003.
  • [14] F. Jamshidian, An exact bond option formula, Journal of Finance 44 (1989), 205-209.
  • [15] T. Kato, An optimal execution problem in geometric Ornstein-Uhlenbeck price process, arXiv:1107.1787 (2011).
  • [16] R.S. Liptser, A.N. Shiryaev, Statistics of Random Processes I. General Theory (Appli­cations of Mathematics) Springer-Verlag, Berlin, Heidelberg, New York, 2001.
  • [17] N.V. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, Heidelberg, 2009.
  • [18] R.S. Liptser, A.N. Shiryaev, Theory of Martingales, Mathematics and its Applications, Kluwer, Dordrecht, 1989.
  • [19] J.-L. Prigent, Weak Convergence of Financial Markets, Springer, 2003.
  • [20] D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, New York, 1979.
  • [21] M. Schroeder, Discrete-time approximations of functionals in models of Ornstein--Uhlenbeck type, with applications to finance, Methodol. Comput. Appl. Probab. (2013), 1-29.
  • [22] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics 5 (1977), 177-188.
  • [23] X. Su, W. Wang, Pricing options with credit risk in a reduced form model, J. Korean Statist. Soc. 41 (2012), 437-444.
  • [24] J.B. Walsh, O.D. Walsh, Embedding and the convergence of the binomial and trinomial tree schemes, [in:] T.J. Lyons, T.S. Salisbury (eds.), Numerical Methods and Stochastics, Fields Institute Communications 34 (2002), 101-123.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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