Asia-Pacific Financial Markets

, Volume 11, Issue 1, pp 55–77 | Cite as

Understanding the Implied Volatility Surface for Options on a Diversified Index

  • David Heath
  • Eckhard Platen


This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts.

Key Words

index derivatives minimal market model random scaling growth optimal portfolio fair pricing binary options 


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  1. Bakshi, G., Cao, C. and Chen, Z. (1997), Empirical performance of alternative option pricing models, J. Finance LII, 2003–2049.CrossRefGoogle Scholar
  2. Balland, P. (2002), Deterministic implied volatility surfaces, Quant. Finance. 2, 31–44.CrossRefGoogle Scholar
  3. Barndorff-Nielsen, O. and Shephard, N. (2002), Economic analysis of realised volatility and its use in estimating stochastic volatility models, J. Roy. Statist. Soc. Ser. B 64(2), 253–280.CrossRefGoogle Scholar
  4. Black, F. (1976), Studies in stock price volatility changes. In Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section, American Statistical Association, pp. 177–181.Google Scholar
  5. Breymann, W., Kelly, L. and Platen, E. (2004), Intraday empirical analysis and modeling of diversified world stock indices. Technical report, University of Technology, Sydney. QFRC Research Paper 125.Google Scholar
  6. Brigo, D. and Mercurio, F. (2001), Interest Rate Models - Theory and Practice. Springer Finance.Google Scholar
  7. Brigo, D., Mercurio, F. and Rapisarda (2004), Smile at the uncertainty, Risk 17(5), 97–101.Google Scholar
  8. Carr, P. and Wu, L. (2003), Finite moment log stable processes and option pricing, J. Finance 58, 753–777.CrossRefGoogle Scholar
  9. Cont, R. and da Fonseca, J. (2002), Dynamics of implied volatility surfaces, Quant. Finance. 2, 45–60.CrossRefGoogle Scholar
  10. Das, S. R. and Sundaram, R. K. (1999), Of smiles and smirks: A term-structure perspective. 34, 211–240.Google Scholar
  11. Derman, E. and Kani, I. (1994), The volatility smile and its implied tree, Goldman Sachs Quantitative Strategies Research Notes.Google Scholar
  12. Dumas, B., Fleming, J. and Whaley, R. (1997), Implied volatility functions: Empirical tests, J. Finance 53, 2059–2106.CrossRefGoogle Scholar
  13. Dupire, B. (1992), Arbitrage pricing with stochastic volatility. In Proceedings of AFFI Conference, Paris.Google Scholar
  14. Dupire, B. (1993), Model art, Risk 6, 118–124.Google Scholar
  15. Dupire, B. (1994), Pricing with a smile, Risk 7, 18–20.Google Scholar
  16. Fouque, J. P., Papanicolau, G. and Sircar, K. R. (2000), Derivatives in Markets with Stochastic Volatility. Cambridge University Press.Google Scholar
  17. Franks, J. R. and Schwartz, E. J. (1991), The stochastic behaviour of market variance implied in the price of index options, Econometrica 101, 1460–1475.Google Scholar
  18. Heath, D., Hurst, S. R. and Platen, E. (2001), Modelling the stochastic dynamics of volatility for equity indices, Asia-Pacific Financial Markets 8, 179–195.CrossRefGoogle Scholar
  19. Heath, D. and Platen, E. (2004), Local volatility function models under a benchmark approach. Technical report, University of Technology, Sydney. QFRC Research Paper 124, 2004.Google Scholar
  20. Heynen, R. (1993), An empirical investigation of observed smile patterns, Rev. Futures Markets 13, 317–353.Google Scholar
  21. Heynen, R., Kemna, A. and Vorst, T. (1994), Analysis of the term structure of implied volatilities, J. Financial and Quantitative Analysis 29, 31–56.CrossRefGoogle Scholar
  22. Karatzas, I. and Shreve, S. E. (1991), Brownian Motion and Stochastic Calculus (2nd ed.). Springer.Google Scholar
  23. Karatzas, I. and Shreve, S. E. (1998), Methods of Mathematical Finance, Volume 39 of Appl. Math. Springer.Google Scholar
  24. Kelly, J. R. (1956), A new interpretation of information rate, Bell Syst. Techn. J. 35, 917–926.Google Scholar
  25. Kou, S. G. (2002), A jump diffusion model for option pricing, Management Science 48, 1086–1101.CrossRefGoogle Scholar
  26. Lewis, A. L. (2000), Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach.Google Scholar
  27. Long, J. B. (1990), The numeraire portfolio, J. Financial Economics 26, 29–69.CrossRefGoogle Scholar
  28. Musiela, M. and Rutkowski, M. (1997), Martingale Methods in Financial Modelling. Theory and Applications, Volume 36 of Appl. Math. Springer.Google Scholar
  29. Platen, E. (2001), A minimal financial market model. In Trends in Mathematics, pp. 293–301. Birkhäuser.Google Scholar
  30. Platen, E. (2002), Arbitrage in continuous complete markets, Adv. in Appl. Probab. 34(3), 540–558.CrossRefGoogle Scholar
  31. Platen, E. (2004), Modeling the volatility and expected value of a diversified world index, Int. J. Theor. Appl. Finance 7(4), 511–529.CrossRefGoogle Scholar
  32. Protter, P. (1990), Stochastic Integration and Differential Equations. Springer.Google Scholar
  33. Rebonato, R. (1999), Volatility and Correlation in the Pricing of Equity, FX and Interest Rate Options. New York: Wiley.Google Scholar
  34. Renault, E. and Touzi, N. (1996), Option hedging and implied volatilities in a stochastic volatility model, Math. Finance 6, 279–302.Google Scholar
  35. Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion (3rd ed.). Springer.Google Scholar
  36. Rosenberg, J. V. (2000), Implied volatility functions: A reprise, J. Derivatives 7.Google Scholar
  37. Rubinstein, M. (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, J. Finance 11, 455–480.CrossRefGoogle Scholar
  38. Rubinstein, M. (2000), Comments on the 1987 stock market crash: Eleven years later. In Risks in Investment Accumulation Products of Financial Institutions. Society of Actuaries. Chapter 2.Google Scholar
  39. Rubinstein, M. and Reiner, E. (1991), Unscrambling the binary code, Risk 4(8), 75–83.Google Scholar
  40. Schönbucher, P. J. (1999), A market model for stochastic implied volatility, Philos. Trans. Roy. Soc. London Ser. A 357, 2071–2092.CrossRefGoogle Scholar
  41. Skiadopolous, G., Hodges, S. and Clewlow, L. (2000), Dynamics of the S&P500 implied volatility surface, Rev. Derivatives Res. 2, 263–282.CrossRefGoogle Scholar
  42. Tompkins, R. (2001), Stock index futures markets: Stochastic volatility models and smiles, J. Futures Markets 21, 4378.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, School of Finance and EconomicsUniversity of Technology SydneyBroadwayAustralia

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