| Peer-Reviewed

Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change

Received: 31 October 2016     Accepted: 10 December 2016     Published: 7 January 2017
Views:       Downloads:
Abstract

This paper focuses on Numéraire change technique for pricing financial assets with stochastic interest rate. It makes sense to introduce the notion of stochastic interest rate when dealing with long term option pricing problems rather than constant interest rate addressed in the past by most papers. We consider the application of Numéraire Change technique to pricing of Investor’s Savings Plan (ISP) with swaption between two interest rates, inflation and exchange rates of two different countries. The result shows that it is better to incorporate stochastic interest rate into long term option pricing problems and Numéraire technique is better applied if faced with several risks factors.

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)
DOI 10.11648/j.ijtam.20170301.14
Page(s) 25-29
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Numéraire Change, Stochastic Interest Rate, Investor Savings Plan (ISP)

References
[1] Michael, J. B. and Yihong X. (2000), Stochastic Interest Rates and the Bond-Stock Mix, Kluwer Academic Publishers, European Finance Review 4: 197–210.
[2] Eun-Jung N., Jeong-Hoon, K. (2011), An Optimal Portfolio model with stochastic volatility and stochastic interest rate, Journal of Mathematical Analysis and Applications 375 (2) 510-522.
[3] Mi-Hyun K., Jeong-Hoon, K. and Ji-Hoon, Y. (2015), Optimal Portfolio selection under stochastic volatility and stochastic interest rates. DOI: 10.12941/jksiam.2015.19.417
[4] Benninga, S., Björk, T. and Winner, Z., (2002), On the Use of Numéraire for pricing Futures, Forwards and Options, The Review of Financial Studies Winter, Vol. 12, No. 5, pp. 1143-1163.
[5] Carr, P. and Chesney, M., (1996), American Put Call Symmetry, Working Paper, pp. 1-25.
[6] Dalang, R. C., and Morton, W. Willinger, (1990), Equivalent Martingale Measures and No-Arbitrage in Stochastic Securities Market Models. Stochastic and Stochastic Reports, Vol. 29, pp. 185-201.
[7] Damiano, B. and Fabio, M., (2001), Interest Rate Models-Theory and Practice with Smile, Inflation and Credit, 2nd Edition, Springer Verlag.
[8] Haowen F. (2012), Option Pricing Under Stochastic Interest Rates, I. J. Engineering and Manufacturing, Published Online June 2012 in MECS (http://www.mecs-press.net) DOI: 10.5815/ijem. Pp. 82-89.
[9] Delbaen, W. and Shirakawa, H., (2002), A Note of Option Pricing for Constant Elasticity of Variance Model.
[10] Delbaen, W. and Schachermayer, (2005), The Mathematics of Arbitrage. Springer, Berlin Heidelberg New York. pp. 1-27.
[11] Andrew L. (2008), Girsanov, Numeraires, and all that, pp. 1-9.
[12] Elliott, R. J. and Kopp, P. E., (2005), (Mathematics of Financial Markets). 2nd Edition, Springer, New York.
[13] Farshid, J., (1989), An exact Bond Option Pricing Formula, The Journal of Finance , pp. 205-209.
[14] Farshid, J., Fein, M., (1990), Closed Form Solutions for Oil Futures and European Options in the Gibson Schwartz Model, Working Paper, Merrill Lynch Capital Markets. pp. 1-27.
[15] AntonioMannolini, (2014), Advanced Financial Modelling, A Crash review of the Change of Measure Theory. Lecture note 4, pg 1-9.
[16] Gawie, L. R., (2007), Applications of Change of Numéraire for Option Pricing. pp. 1-60.
[17] Geman, H. E., El Karoui, and Rouchet, J., (1995), Changes of Numeraire, Changes of Probability Measure and Option Pricing, Journal of Applied Probability, pp. 443-458.
[18] Hull, J., and White, A., (1987), The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance : 271-301.
[19] Jamshidian, F., (2008), Numéraire Invarince and Application to Option Pricing and Hedging, pp. 1-7.
[20] Jochen Wilhelm, (1986), Option Prices with Stochastic Interest Rates Black/Scholes and Ho/Lee unified. ISSN 1435-3539, pp 1-10.
[21] Black, F., Scholes, M., (1973). The Pricing of Options and Corporate Liabilities, in: Journal of Political Economy, 81 (1973), 637-654.
[22] Mark, S., (1999), Changes of Numéraire for Pricing Futures, forwards and Options, The Review of Financial Studies Winter, Vol. 12, No. 5, pp. 1143-1163.
[23] Mark, H. A. Davis (2006), Mathematical Option Pricing, Imperial College, London, pp. 33-36.
Cite This Article
  • APA Style

    Philip Ajibola Bankole, Adeniyi Adewopo. (2017). Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change. International Journal of Theoretical and Applied Mathematics, 3(1), 25-29. https://doi.org/10.11648/j.ijtam.20170301.14

    Copy | Download

    ACS Style

    Philip Ajibola Bankole; Adeniyi Adewopo. Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change. Int. J. Theor. Appl. Math. 2017, 3(1), 25-29. doi: 10.11648/j.ijtam.20170301.14

    Copy | Download

    AMA Style

    Philip Ajibola Bankole, Adeniyi Adewopo. Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change. Int J Theor Appl Math. 2017;3(1):25-29. doi: 10.11648/j.ijtam.20170301.14

    Copy | Download

  • @article{10.11648/j.ijtam.20170301.14,
      author = {Philip Ajibola Bankole and Adeniyi Adewopo},
      title = {Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {3},
      number = {1},
      pages = {25-29},
      doi = {10.11648/j.ijtam.20170301.14},
      url = {https://doi.org/10.11648/j.ijtam.20170301.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.14},
      abstract = {This paper focuses on Numéraire change technique for pricing financial assets with stochastic interest rate. It makes sense to introduce the notion of stochastic interest rate when dealing with long term option pricing problems rather than constant interest rate addressed in the past by most papers. We consider the application of Numéraire Change technique to pricing of Investor’s Savings Plan (ISP) with swaption between two interest rates, inflation and exchange rates of two different countries. The result shows that it is better to incorporate stochastic interest rate into long term option pricing problems and Numéraire technique is better applied if faced with several risks factors.},
     year = {2017}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Mathematical Modeling of Investors’ Savings Plan (ISP) with Stochastic Interest Rate via Numéraire Change
    AU  - Philip Ajibola Bankole
    AU  - Adeniyi Adewopo
    Y1  - 2017/01/07
    PY  - 2017
    N1  - https://doi.org/10.11648/j.ijtam.20170301.14
    DO  - 10.11648/j.ijtam.20170301.14
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
    SP  - 25
    EP  - 29
    PB  - Science Publishing Group
    SN  - 2575-5080
    UR  - https://doi.org/10.11648/j.ijtam.20170301.14
    AB  - This paper focuses on Numéraire change technique for pricing financial assets with stochastic interest rate. It makes sense to introduce the notion of stochastic interest rate when dealing with long term option pricing problems rather than constant interest rate addressed in the past by most papers. We consider the application of Numéraire Change technique to pricing of Investor’s Savings Plan (ISP) with swaption between two interest rates, inflation and exchange rates of two different countries. The result shows that it is better to incorporate stochastic interest rate into long term option pricing problems and Numéraire technique is better applied if faced with several risks factors.
    VL  - 3
    IS  - 1
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, University of Ibadan, Ibadan, Nigeria

  • Department of Physics, Adeniran Ogunsanya College of Education, Oto/Ijanikin, Nigeria

  • Sections