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Model-based pricing for financial derivatives

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  • Zhu, Ke
  • Ling, Shiqing

Abstract

Assume that S_{t} is a stock price process and Bt is a bond price process with a constant continuously compounded risk-free interest rate, where both are defined on an appropriate probability space P. Let y_{t} = log(S_{t}/S_{t-1}). y_{t} can be generally decomposed into a conditional mean plus a noise with volatility components, but the discounted St is not a martingale under P. Under a general framework, we obtain a risk-neutralized measure Q under which the discounted St is a martingale in this paper. Using this measure, we show how to derive the risk neutralized price for the derivatives. Special examples, such as NGARCH, EGARCH and GJR pricing models, are given. Simulation study reveals that these pricing models can capture the "volatility skew" of implied volatilities in the European option. A small application highlights the importance of our model-based pricing procedure.

Suggested Citation

  • Zhu, Ke & Ling, Shiqing, 2014. "Model-based pricing for financial derivatives," MPRA Paper 56623, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:56623
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    1. Monfort, Alain & Pegoraro, Fulvio, 2012. "Asset pricing with Second-Order Esscher Transforms," Journal of Banking & Finance, Elsevier, vol. 36(6), pages 1678-1687.
    2. Christophe Chorro & Dominique Guégan & Florian Ielpo, 2012. "Option pricing for GARCH-type models with generalized hyperbolic innovations," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1079-1094, April.
    3. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    4. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Christoffersen, Peter & Heston, Steve & Jacobs, Kris, 2006. "Option valuation with conditional skewness," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 253-284.
    6. Rubinstein, Mark, 1985. " Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978," Journal of Finance, American Finance Association, vol. 40(2), pages 455-480, June.
    7. Smith, Peter & Wickens, Michael, 2002. " Asset Pricing with Observable Stochastic Discount Factors," Journal of Economic Surveys, Wiley Blackwell, vol. 16(3), pages 397-446, July.
    8. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    9. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    10. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    12. Schmitt, Christian, 1996. "Option pricing using EGARCH models," ZEW Discussion Papers 96-20, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
    13. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    14. Engle, Robert F & Ng, Victor K, 1993. " Measuring and Testing the Impact of News on Volatility," Journal of Finance, American Finance Association, vol. 48(5), pages 1749-1778, December.
    15. Giovanni Barone-Adesi & Robert F. Engle & Loriano Mancini, 2008. "A GARCH Option Pricing Model with Filtered Historical Simulation," Review of Financial Studies, Society for Financial Studies, vol. 21(3), pages 1223-1258, May.
    16. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    17. Bates, David S., 2003. "Empirical option pricing: a retrospection," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 387-404.
    18. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    19. Ravi Jagannathan & Zhenyu Wang, 2002. "Empirical Evaluation of Asset-Pricing Models: A Comparison of the SDF and Beta Methods," Journal of Finance, American Finance Association, vol. 57(5), pages 2337-2367, October.
    20. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    21. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    22. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    23. Engle, Robert F. & Mustafa, Chowdhury, 1992. "Implied ARCH models from options prices," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 289-311.
    24. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    25. Sheikh, Aamir M., 1991. "Transaction Data Tests of S&P 100 Call Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(04), pages 459-475, December.
    26. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    27. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
    28. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    29. Gerber, Hans U. & Shiu, Elias S.W., 1994. "Martingale Approach to Pricing Perpetual American Options," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 24(02), pages 195-220, November.
    30. Christoffersen, Peter & Dorion, Christian & Jacobs, Kris & Wang, Yintian, 2010. "Volatility Components, Affine Restrictions, and Nonnormal Innovations," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(4), pages 483-502.
    31. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    32. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-547, August.
    33. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    34. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    35. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    36. Peter Ritchken & Rob Trevor, 1999. "Pricing Options under Generalized GARCH and Stochastic Volatility Processes," Journal of Finance, American Finance Association, vol. 54(1), pages 377-402, February.
    37. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Chang, Chia-Lin & McAleer, Michael, 2015. "Econometric analysis of financial derivatives: An overview," Journal of Econometrics, Elsevier, vol. 187(2), pages 403-407.
    2. Zhu, Ke, 2015. "Hausman tests for the error distribution in conditionally heteroskedastic models," MPRA Paper 66991, University Library of Munich, Germany.
    3. Badescu, Alexandru & Cui, Zhenyu & Ortega, Juan-Pablo, 2016. "A note on the Wang transform for stochastic volatility pricing models," Finance Research Letters, Elsevier, vol. 19(C), pages 189-196.
    4. Chang, C-L. & McAleer, M.J., 2014. "Econometric Analysis of Financial Derivatives," Econometric Institute Research Papers EI 2015-02, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.

    More about this item

    Keywords

    NGARCH; EGARCH and GJR models; Non-normal innovation; Option valuation; Risk neutralized measure; Volatility skew.;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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