IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v70y2016icp19-37.html
   My bibliography  Save this article

Discrete sums of geometric Brownian motions, annuities and Asian options

Author

Listed:
  • Pirjol, Dan
  • Zhu, Lingjiong

Abstract

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.

Suggested Citation

  • Pirjol, Dan & Zhu, Lingjiong, 2016. "Discrete sums of geometric Brownian motions, annuities and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 19-37.
    • Handle: RePEc:eee:insuma:v:70:y:2016:i:c:p:19-37
    DOI: 10.1016/j.insmatheco.2016.05.020
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668716300075
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2016.05.020?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. M. Schroder & P. Carr, 2003. "Bessel processes, the integral of geometric Brownian motion, and Asian options," Papers math/0311280, arXiv.org.
    2. Moshe Arye Milevsky & Steven E. Posner, 1999. "Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 7, pages 203-218, World Scientific Publishing Co. Pte. Ltd..
    3. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    4. Lingjiong Zhu, 2015. "Options with Extreme Strikes," Risks, MDPI, vol. 3(3), pages 1-16, July.
    5. Daniel Dufresne, 2007. "Fitting combinations of exponentials to probability distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 23-48, January.
    6. Levy, Edmond, 1992. "Pricing European average rate currency options," Journal of International Money and Finance, Elsevier, vol. 11(5), pages 474-491, October.
    7. Asmussen, Søren & Rojas-Nandayapa, Leonardo, 2008. "Asymptotics of sums of lognormal random variables with Gaussian copula," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2709-2714, November.
    8. De Schepper, A. & Goovaerts, M. & Delbaen, F., 1992. "The Laplace transform of annuities certain with exponential time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 11(4), pages 291-294, December.
    9. Feng, Runhuan & Volkmer, Hans W., 2012. "Analytical calculation of risk measures for variable annuity guaranteed benefits," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 636-648.
    10. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2015. "Geometric stopping of a random walk and its applications to valuing equity-linked death benefits," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 313-325.
    11. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    12. Daniel Dufresne, 2007. "Stochastic Life Annuities," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(1), pages 136-157.
    13. N/A, 2007. "Statistical Appendix," National Institute Economic Review, National Institute of Economic and Social Research, vol. 199(1), pages 126-126, January.
    14. N/A, 2007. "Statistical Appendix," National Institute Economic Review, National Institute of Economic and Social Research, vol. 199(1), pages 126-135, January.
    15. Vanduffel, Steven & Shang, Zhaoning & Henrard, Luc & Dhaene, Jan & Valdez, Emiliano A., 2008. "Analytic bounds and approximations for annuities and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1109-1117, June.
    16. N/A, 2007. "Statistical Appendix," National Institute Economic Review, National Institute of Economic and Social Research, vol. 200(1), pages 118-127, April.
    17. Gjessing, Håkon K. & Paulsen, Jostein, 1997. "Present value distributions with applications to ruin theory and stochastic equations," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 123-144, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    2. Seiji Harikae & James S. Dyer & Tianyang Wang, 2021. "Valuing Real Options in the Volatile Real World," Production and Operations Management, Production and Operations Management Society, vol. 30(1), pages 171-189, January.
    3. Dan Pirjol & Lingjiong Zhu, 2020. "Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface," Papers 2001.09850, arXiv.org, revised Mar 2020.
    4. Zadourian, Rubina & Klümper, Andreas, 2018. "Exact probability distribution function for the volatility of cumulative production," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 495(C), pages 59-66.
    5. Dan Pirjol & Lingjiong Zhu, 2023. "Asymptotics for Short Maturity Asian Options in Jump-Diffusion models with Local Volatility," Papers 2308.15672, arXiv.org, revised Feb 2024.
    6. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    7. Brignone, Riccardo & Kyriakou, Ioannis & Fusai, Gianluca, 2021. "Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 232-247.
    8. Ravi Kashyap, 2016. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Papers 1609.01274, arXiv.org, revised Mar 2022.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dan Pirjol & Lingjiong Zhu, 2016. "Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options," Papers 1609.07558, arXiv.org.
    2. Liang, Xiaoqing & Tsai, Cary Chi-Liang & Lu, Yi, 2016. "Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 150-161.
    3. Xing-Fang Huang & Ting Zhang & Yang Yang & Tao Jiang, 2017. "Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks," Risks, MDPI, vol. 5(1), pages 1-14, March.
    4. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    5. Brignone, Riccardo & Kyriakou, Ioannis & Fusai, Gianluca, 2021. "Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 232-247.
    6. Runhuan Feng & Hans W. Volkmer, 2013. "An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit," Papers 1307.7070, arXiv.org.
    7. García Nieto, P.J. & García-Gonzalo, E. & Sánchez Lasheras, F. & de Cos Juez, F.J., 2015. "Hybrid PSO–SVM-based method for forecasting of the remaining useful life for aircraft engines and evaluation of its reliability," Reliability Engineering and System Safety, Elsevier, vol. 138(C), pages 219-231.
    8. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," Papers 1505.06946, arXiv.org.
    9. Chueh-Yung Tsao & Chao-Ching Liu, 2012. "Asian Options with Credit Risks: Pricing and Sensitivity Analysis," Emerging Markets Finance and Trade, Taylor & Francis Journals, vol. 48(S3), pages 96-115, September.
    10. Feng, Runhuan & Jing, Xiaochen, 2017. "Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 36-48.
    11. Lemmens, D. & Liang, L.Z.J. & Tempere, J. & De Schepper, A., 2010. "Pricing bounds for discrete arithmetic Asian options under Lévy models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(22), pages 5193-5207.
    12. Manuel Moreno & Javier F. Navas, 2008. "Australian Options," Australian Journal of Management, Australian School of Business, vol. 33(1), pages 69-93, June.
    13. Ching-Sung Wu & Chih-Sheng Hsu, 2013. "Linking International High-Tech New Ventures' Firm Life Cycle To Internationalization, Organizational Learning, And Alliance Networks," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 175-197.
    14. Nane, Gabriela F. & Larivière, Vincent & Costas, Rodrigo, 2017. "Predicting the age of researchers using bibliometric data," Journal of Informetrics, Elsevier, vol. 11(3), pages 713-729.
    15. De Schepper, Ann & Goovaerts, Marc & Dhaene, Jan & Kaas, Rob & Vyncke, David, 2002. "Bounds for present value functions with stochastic interest rates and stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 87-103, August.
    16. Shi, Li., 2008. "Rural migrant workers in China : scenario, challenges and public policy," ILO Working Papers 994204063402676, International Labour Organization.
    17. Keng‐Hsin Lo & Kehluh Wang & Ming‐Feng Hsu, 2008. "Pricing European Asian options with skewness and kurtosis in the underlying distribution," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 28(6), pages 598-616, June.
    18. Jaehyuk Choi, 2018. "Sum of all Black–Scholes–Merton models: An efficient pricing method for spread, basket, and Asian options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(6), pages 627-644, June.
    19. Jinke Zhou & Xiaolu Wang, 2008. "Accurate closed‐form approximation for pricing Asian and basket options," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(4), pages 343-358, July.
    20. Feng, Runhuan & Volkmer, Hans W., 2012. "Analytical calculation of risk measures for variable annuity guaranteed benefits," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 636-648.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:70:y:2016:i:c:p:19-37. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.