Research article

Non-extensive minimal entropy martingale measures and semi-Markov regime switching interest rate modeling

  • Received: 15 August 2019 Accepted: 04 November 2019 Published: 13 November 2019
  • MSC : 60Gxx, 60Jxx, 90Bxx

  • A minimal entropy martingale measure problem is studied to investigate risk-neutral densities and interest rate modelling. Hunt & Devolder focused on the method of Shannon minimal entropy martingale measure to select the best measure among all the equivalent martingale measures and, proposed a generalization of the Ho & Lee model in the semi-Markov regime-switching framework [1]. We formulate and solve the optimization problem of Hunt & Devolder for deriving risk-neutral densities using a new non-extensive entropy measure [2]. We use the Lambert function and a new type of approach to obtain results without depending on stochastic calculus techniques.

    Citation: Muhammad Sheraz, Vasile Preda, Silvia Dedu. Non-extensive minimal entropy martingale measures and semi-Markov regime switching interest rate modeling[J]. AIMS Mathematics, 2020, 5(1): 300-310. doi: 10.3934/math.2020020

    Related Papers:

  • A minimal entropy martingale measure problem is studied to investigate risk-neutral densities and interest rate modelling. Hunt & Devolder focused on the method of Shannon minimal entropy martingale measure to select the best measure among all the equivalent martingale measures and, proposed a generalization of the Ho & Lee model in the semi-Markov regime-switching framework [1]. We formulate and solve the optimization problem of Hunt & Devolder for deriving risk-neutral densities using a new non-extensive entropy measure [2]. We use the Lambert function and a new type of approach to obtain results without depending on stochastic calculus techniques.


    加载中


    [1] J. Hunt, P. Devolder, semi-Markov regime switching interest rate models and minimal entropy measure, Phys. A Stat. Mech. Appl., 390 (2011), 3767-3778.
    [2] F. Shafee, Lambert function and a new non-extensive form of entropy, J. App. Math., 72 (2007), 785-800.
    [3] C. E. Shannon, W. Weaver, The Mathematical Theory of Communication, Urbana: The University of Illinois Press, 1998.
    [4] G. Kaniadakis, Non-linear kinetics underlying generalized statistics, Phys. A Stat. Mech. Appl., 296 (2001), 405-425.
    [5] A. Renyi, On measures of entropy and information, In: Proceed. 4th. Berkely. Symp. Math Stat. Prob., 1 (1961), 547-561.
    [6] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52 (1988), 479-487.
    [7] M. R. Ubriaco, Entropies based on fractional calculus, Phys. Lett. A Stat. Mech. Appl., 373 (2009), 2516-2519.
    [8] T. S. Y. Ho, S. B. Lee, Term structure movements and pricing interest rate contingent claims, J. Financ., 41 (1986), 1011-1029.
    [9] J. C. Cox, J. E. Ingersoll, S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-408.
    [10] J. Hull, A. White, Pricing interest-rate derivative securities, Rev. Financ. Stud., 3 (1990), 573-592.
    [11] O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 5 (1977), 177-188.
    [12] P. J. Hunt, J. E. Kennedy, Financial Derivatives in Theory and Practice, John Wiley and Sons, 2004.
    [13] G. C. Philippatos, C. J. Wilson, Entropy, market risks and the selection of efficient portfolios, Appl. Econ., 4 (1972), 209-220.
    [14] L. Gulko, The entropy theory of stock option pricing, Int. J. Theo. Appl. Financ., 2 (1999), 331-355.
    [15] B. Trivellato, Deformed exponentials and applications to finance, Entropy, 15 (2013), 3471-3489.
    [16] B. Trivellato, The minimal k-entropy martingale measure, Int. J. Theo. Appl. Financ., 15 (2012), 1250038.
    [17] V. Preda, S. Dedu, M. Sheraz, New measure selection for Hunt and Devolder semi-Markov regime switching interest rate model, Phys. A Stat. Mech. Appl., 407 (2014), 350-359.
    [18] V. Preda, S. Dedu, C. Gheorghe, New classes of Lorenz curves by maximizing Tsallis entropy under mean and Gini equality and inequality constraints, Phys. A Stat. Mech. Appl., 436 (2015), 925-932.
    [19] V. Preda, M. Sheraz, Risk-neutral densities in entropy theory of stock options using Lambert function and a new approach, P. Romanian Acad. A, 4 (2015), 20-27.
    [20] M. Sheraz, V. Preda, S. Dedu, Tsallis and Kaniadakis entropy measures for risk neutral densities, International Conference on Computer Aided Systems Theory, Springer, 2017.
    [21] T. Bjö rk, Arbitrage Theory in Continuous Time, New York: Oxford University Press, 2009.
    [22] J. C. Cox, S. A. Ross, M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.
    [23] S. R. Valluri, R. M. Corrless, D. J. Jeffrey, Some applications of Lambert W function to physics, Canadian J. Phys., 78 (2000), 823-831.
    [24] J. Borwein, R. Choksi, P. Maréchal, Probability distributions of assets inferred from option prices via the principle of maximum entropy, SIAM J. Optimiz., 14 (2003), 464-478.
    [25] D. G. Luenberger, Optimization by Vector Space Methods, New York: Wiely, 1969.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3202) PDF downloads(428) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog