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The Simulation of the Implied Distribution and Other Smile Consistent Stochastic Volatility Models: An Overview

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Financial Engineering, E-commerce and Supply Chain

Part of the book series: Applied Optimization ((APOP,volume 70))

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Abstract

This review paper focuses on the smile-consistent stochastic volatility models. Smile-consistent stochastic volatility models take the European options’ market prices as given, and they try to explain the stochastic evolution of implied volatilities over time across strikes and maturities. The main ideas behind the models by Derman and Kani (1998), Ledoit and Santa-Clara (1999), and Britten-Jones and Neuberger (1999) are highlighted. In addition, the concept and the applications of a new methodology for smile-consistent stochastic volatility pricing, that of the simulation of the implied distribution, are discussed. The simulation model by Skiadopoulos and Hodges (2001) is explained.

JEL Classification:G13.

I would like to thank the participants at FEES 2001 conference and especially Panos Pardalos, and William Ziemba who motivated me with their comments and discussions to write this paper. Because of space limitations, some contributions have not been given the attention that they deserve. Any remaining errors are my responsibility alone.

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References

  1. Alexander, C. (2001): “Principal Component Analysis of Volatility Smiles and Skews”, ISMA Centre, University of Reading.

    Google Scholar 

  2. Andersen, L. (1996): The Equity Option Volatility Smile: an Implicit Finite Difference Approach, Five Essays on the Pricing of Contingent Claims, Ph.D. Thesis, The Aarhus School of Business.

    Google Scholar 

  3. Andreasen, J. (1996): Implied Modelling: Stable Implementation, Hedging and Duality, Working Paper, The Aarhus School of Business.

    Google Scholar 

  4. Bahra, B. (1997): “Implied Risk-Neutral Probability Density Functions From Option Prices: Theory and Application”, Bank of England Working Paper Series.

    Google Scholar 

  5. Barle, S., and Cakici N. (1995): “Growing a Smiling Tree”, Risk 8, pp. 76–81.

    Google Scholar 

  6. Bates, D. (1988): “Pricing Options under Jump-Diffusion Processes ”, Working Paper, Rodney L. White Center.

    Google Scholar 

  7. Black, F., and Scholes, M. (1973): “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81, pp. 637–654.

    Article  Google Scholar 

  8. Breeden, D., and Litzenberger, R. (1978): “Prices of State-Contingent Claims Implicit in Option prices”, Journal of Business 51, pp. 621–651.

    Article  Google Scholar 

  9. Britten-Jones, M., and Neuberger, A. (1999): “Option Prices, Implied Price Processes, and Stochastic Volatility ”, Journal of Finance, (forthcoming).

    Google Scholar 

  10. Buraschi, A., and Jackwerth, J. (1998): “Explaining Option Prices: Deterministic vs. Stochastic Models”, Working Paper, London Business School.

    Google Scholar 

  11. Cont, R. (1997): “Beyond Implied Volatility, in Econphysics ”, eds. J. Kertesz and I. Kondor, Dordrecht, Kluwer.

    Google Scholar 

  12. Cox, J., Ross, S., and Rubinstein, M. (1979): “Option Pricing: A Simplified Approach”, Journal of Financial Economics 7, pp. 229–263.

    Article  Google Scholar 

  13. Cox, J., Ingersoll, J., and Ross, S. (1985): “A Theory of the Term Structure of Interest Rates, Econometrica 53, pp. 385–407.

    Google Scholar 

  14. Cox, J., and Rubinstein, M. (1985): Options Markets, Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  15. Das, S., and Sundaram, R. (1999): “Of Smiles and Smirks: A Term-Structure Perspective”, Journal of Financial and Quantitative Analysis 34, pp. 211–239.

    Article  Google Scholar 

  16. Derman, E., and Kani, I. (1994): “Riding on a Smile ”, Risk 7, pp. 32–39.

    Google Scholar 

  17. Derman, E., Ergener, D., and Kani, I. (1995): “Static Options Replication”, Journal of Derivatives 2, pp. 78–95.

    Article  Google Scholar 

  18. Derman, E. and Kani, I. and Chriss, N. (1996): “Implied Trinomial Trees of the Volatility Smile ”, Journal of. Derivatives 3, pp. 7–22.

    Article  Google Scholar 

  19. Derman, E., and Kani, I., (1997): “Stochastic Implied Trees: Arbitrage Pricing With Stochastic Term and Strike Structure of Volatility”, International Journal of Theoretical and Applied Finance 1, pp. 61–110.

    Article  Google Scholar 

  20. Derman, E., Kani, I., and Zou, J., (1996): “The Local Volatility Surface: Unlocking the Information in Index Option Prices ”, Financial Analysts Journal 52, pp. 25–36.

    Google Scholar 

  21. Dumas B., Fleming, J, and Whaley, R. (1998): “Implied Volatility functions: Empirical Tests”, Journal of Finance 53, pp. 2059–2106.

    Article  Google Scholar 

  22. Dupire, B. (1992): “Arbitrage Pricing with Stochastic Volatility”, Proceedings of French Finance Association Meetings, Paris.

    Google Scholar 

  23. Dupire, B (1994): “A Unified Theory of Volatility”, Working Paper, Paribas Capital Markets.

    Google Scholar 

  24. Gemmill, G. (1996): “Did Option Traders Anticipate the Crash? Evidence from Volatility Smiles in the U.K., with U.S. Comparisons”, Journal of Futures Markets 16, pp. 881–897.

    Article  Google Scholar 

  25. Heath, D., Jarrow, R. and Morton, A., (1992): “Bond Pricing and the Term Structure of Interest Rates: A New Methodology For Contingent Claims Valuation ”, Econometrica 60, pp. 77–105.

    Article  Google Scholar 

  26. Ho, L., and Lee, S. (1986): “Term Structure Movements and Pricing Interest Rate Contingent Claims ”, Journal of Finance 41, pp. 1011–1029.

    Article  Google Scholar 

  27. Hull, J., and White, A. (1987): “The Pricing of Options on Assets with Stochastic Volatilities ”, Journal of Finance 3, pp. 281–300.

    Article  Google Scholar 

  28. Hull, J., and White, A. (1992): “One-Factor Interest-Rate Models and the Valuation of Interest-Rate Contingent Claims ”, Journal of Financial and Quantitative Analysis 28, pp. 235–254.

    Article  Google Scholar 

  29. Jamshidian, F. (1991): “Forward Induction and Construction of Yield Curve Diffusion Models”, Journal of Fixed Income 1, pp. 62–74.

    Article  Google Scholar 

  30. Jamshidian, F. (1993): “Option and Futures Evaluation with Deterministic Volatilities ”, Mathematical Finance 3, pp. 149–159.

    Article  Google Scholar 

  31. Johnson, H., and Shanno, D. (1987): “Option Pricing when the Variance is Changing ”, Journal of Financial and Quantitative Analysis 22, pp. 143151.

    Google Scholar 

  32. Kamal, M., and Derman, E. (1997): ‘The Patterns of Change in Implied Index Volatilities “, Goldman Sachs, Quantitative Strategies Notes.

    Google Scholar 

  33. Kani, I., Derman, E., and Kamal, M. (1997): “Trading and Hedging Local Volatility ”, Journal of Financial Engineering 6, pp. 1233–1268.

    Google Scholar 

  34. Kon, S. (1984): “Models of Stock Returns-A Comparison”, Journal of Finance 39, pp. 147–165.

    Google Scholar 

  35. Mayhew, S., (1995): “Implied Volatility ”, Financial Analysts Journal 51, pp. 8–13.

    Article  Google Scholar 

  36. Mayhew, S. (1995). “On Estimating The Risk-Neutral Probability Distribution Implied by Option Prices”, Working Paper, University of California, Berkeley.

    Google Scholar 

  37. Ledoit, O., and Santa-Clara, P. (1998): “Relative Pricing of Options with Stochastic Volatility”, Working Paper, University of California, Los Angeles.

    Google Scholar 

  38. Merton, R., (1976): “Option Pricing when Underlying Stock Returns are Discontinuous ”, Journal of Financial Economics 3, pp. 125–144.

    Article  Google Scholar 

  39. Pappalardo, L., (1996): “Option Pricing and Smile Effect when Underlying Stock Prices are Driven by a Jump Process ”, Working Paper, Financial Options Research Centre, University of Warwick.

    Google Scholar 

  40. Ramberg, J., Dudewicz, E., Tadikamalla, P., and Mykytka, E. (1979): “A Probability Distribution and its Uses in Fitting Data”, Technometrics 21, pp. 201–209.

    Article  Google Scholar 

  41. Rebonato, R. (1996): Interest-Rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-Rate Options,John Wiley and Sons.

    Google Scholar 

  42. Rubinstein, M. (1985): “Non-Parametric 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 40, pp. 455–480.

    Article  Google Scholar 

  43. Scott, L. (1987): “Option Pricing when the Variance changes Randomly: Theory, Estimation and an Application ”, Journal of Financial and Quantitative Analysis 22, pp. 419–438.

    Article  Google Scholar 

  44. Scott, L. (1997): “Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods”, Mathematical Finance 7, pp. 413–426.

    Article  Google Scholar 

  45. Skiadopoulos, G., Hodges, S., and Clewlow, L. (1999): “The Dynamics of the SP 500 Implied Volatility Surface”, Review of Derivatives Research 3, pp. 263–282.

    Article  Google Scholar 

  46. Skiadopoulos, G., Hodges, S., and Clewlow, L. (2000): “The Dynamics of Implied Volatility Surfaces”. In Decision Making: Recent Developments and Worldwide Applications,S.H. Zanakis, G. Doukidis, C. Zopounidis Editors, pp. 197–211, Kluwer Editions.

    Google Scholar 

  47. Skiadopoulos, G., and Hodges, S. (2001): “Simulating the Evolution of the Implied Distribution”, European Financial Management Journal, 7: 4, pp. 497–521.

    Article  Google Scholar 

  48. Strickland, C. (1996): “A Comparison of Diffusion Models of the Term Structure ”, European Journal of Finance 2, pp. 103–123.

    Article  Google Scholar 

  49. Taylor, S. and Xu, X. (1994): “The Magnitude of Implied Volatility Smiles: Theory and Empirical Evidence for Exchange Rates ”, Review of Futures Markets 13, pp. 355–380.

    Google Scholar 

  50. Vasicek, O. (1997): “An Equilibrium Characterization of the Term Structure”, Journal of Financial Economics 5, pp. 177–88.

    Article  Google Scholar 

  51. Wiggins, J. (1987): “Option Values under Stochastic Volatility ”, Journal of Financial Economics 19, pp. 351–372.

    Article  Google Scholar 

  52. Zou, J., and Derman, E. (1996): “Monte Carlo Valuation of Path-Dependent Options on Indexes with a Volatility Smile” Journal of Financial Engineering 6, pp. 149–168.

    Google Scholar 

  53. Timmerington, D., Smith, A., and Makov, U. (1985): Statistical Analysis of Finite Mixture Distributions, John Wiley Sons.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Financial Options Research Centre, University of Warwick, UK

    George Skiadopoulos (Associate Research Fellow, [2mm] Research Fellow)

  2. Financial Engineering Research Centre. Department of Decision Sciences, Athens University of Economics and Business, Greece

    George Skiadopoulos (Associate Research Fellow, [2mm] Research Fellow)

Authors
  1. George Skiadopoulos
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Editor information

Editors and Affiliations

  1. University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Deal-Fx, Athens, Greece

    Vassilis K. Tsitsiringos

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© 2002 Springer Science+Business Media Dordrecht

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Skiadopoulos, G. (2002). The Simulation of the Implied Distribution and Other Smile Consistent Stochastic Volatility Models: An Overview. In: Pardalos, P.M., Tsitsiringos, V.K. (eds) Financial Engineering, E-commerce and Supply Chain. Applied Optimization, vol 70. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5226-7_12

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  • DOI: https://doi.org/10.1007/978-1-4757-5226-7_12

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5222-6

  • Online ISBN: 978-1-4757-5226-7

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