Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T07:53:32.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Of Smiles and Smirks: A Term Structure Perspective

Published online by Cambridge University Press:  06 April 2009

Sanjiv Ranjan Das  and
Rangarajan K. Sundaram
Sanjiv Ranjan Das
Affiliation:
Department of Finance, Harvard Business School, Harvard University, Boston, MA 02163
Rangarajan K. Sundaram
Affiliation:
Department of Finance, Stern School of Business, New York University, New York, NY 10012
Get access Rights & Permissions [Opens in a new window]

Abstract

An extensive empirical literature in finance has documented not only the presence of anomalies in the Black-Scholes model, but also the term structures of these anomalies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical efforts in the literature at addressing these anomalies have largely focused on two extensions of the Black-Scholes model: introducing jumps into the return process, and allowing volatility to be stochastic. We employ commonly used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anomalies. We find that each model exhibits some term structure patterns that are fundamentally inconsistent with those observed in the data. As a consequence, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare somewhat better than jumps.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 34 , Issue 2 , June 1999 , pp. 211 - 239
Copyright
Copyright © School of Business Administration, University of Washington 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahn, C. M. “Option Pricing when Jump Risk is Systematic.” Mathematical Finance, 2 (1992), 299–308.CrossRefGoogle Scholar
Ait-Sahalia, Y. “Do Interest Rates Really follow Continuous-Time Markov Diffusions?” Working Paper, Univ. of Chicago, Graduate School of Business (1996).Google Scholar
Amin, K. I. “Jump Diffusion Option Valuation in Discrete Time.” Journal of Finance, 48 (1993), 1833–1863.CrossRefGoogle Scholar
Amin, K. I., and Ng, V.. “Option Valuation with Systematic Stochastic Volatility.” Journal of Finance, 48 (1993), 881–910.CrossRefGoogle Scholar
Backus, D.; Foresi, S.; Li, K.; and Wu, L.. “Accounting for Biases in Black-Scholes.” Working Paper, NYU Stem School of Business (1997).Google Scholar
Ball, C., and Torous, W.. “On Jumps in Common Stock Prices and Their Impact on Call Option Pricing.” Journal of Finance, 40 (1985), 155–173.CrossRefGoogle Scholar
Bates, D. S. “The Crash of '87: Was it Expected? The Evidence from the Options Markets.” Journal of Finance, 40 (1991), 1009–1044.Google Scholar
Bates, D. S. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies, 9 (1996), 69–107.CrossRefGoogle Scholar
Black, F. “Fact and Fantasy in the Use of Options.” Financial Analysts Journal, 31 (1975), 36–72.CrossRefGoogle Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (1973), 637–654.CrossRefGoogle Scholar
Blattberg, R., and Gonedes, N.. “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Mees.” Journal of Business, 47 (1974), 244–280.CrossRefGoogle Scholar
Bodurtha, J., and Courtadon, G.. “Tests of the American Option Pricing Model in the Foreign Currency Options Market.” Journal of Financial and Quantitative Analysis, 22 (1987), 153–167.CrossRefGoogle Scholar
Bollerslev, T.; Chou, R. Y.; and Kroner, K. F.. “ARCH Modelling in Finance.” Journal of Econometrics, 52 (1992), 5–59.CrossRefGoogle Scholar
Campa, J., and Chang, K. H.. “Testing the Expectations Hypothesis on the Term Structure of Volatilities.” Journal of Finance, 50 (1995), 529–547.CrossRefGoogle Scholar
Campa, J.; Chang, K. H.; and Rieder, R.. “Implied Exchange Rate Distributions: Evidence from OTC Options Markets.” Mimeo, New York Univ. (1997).CrossRefGoogle Scholar
Das, S. R., and Foresi, S.. “Exact Prices for Bond and Option Prices with Systematic Jump Risk.” Review of Derivatives Research, 1 (1996), 7–24.CrossRefGoogle Scholar
Das, S. R., and Sundaram, R. K.. “Of Smiles and Smirks: A Term-Structure Perspective.” Working Paper, Stem School of Business (1998).CrossRefGoogle Scholar
Derman, E., and Kani, I.. “Riding on a Smile.” Risk, 7 (1994), 32–39.Google Scholar
Drost, F., and Nijman, T.. “Temporal Aggregation of GARCH Models.” Econometrica, 61 (1993), 909–927.CrossRefGoogle Scholar
Drost, F.; Nijman, T.; and Werker, B.. “Estimation and Testing in Models Containing Both Jumps and Conditional Heteroskedasticity.” Mimeo, Department of Economics, Tilburg Univ. (1995).Google Scholar
Duffie, D.Dynamic Asset Pricing Theory. Second Ed. Princeton, NJ: Princeton Univ. Press (1996).Google Scholar
Duffie, D., and Pan, J.. “An Overview of Value at Risk.” Journal of Derivatives, 4 (Spring 1997), 7–49.CrossRefGoogle Scholar
Fama, E. “The Behavior of Stock Mees.” Journal of Business, 47 (1965), 244–280.Google Scholar
Heston, S. L. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327–343.CrossRefGoogle Scholar
Heynen, R.; Kemna, A. G. Z.; and Vorst, T.. “Analysis of the Term Structure of Implied Volatilities.” Journal of Financial and Quantitative Analysis, 29 (1994), 31–56.CrossRefGoogle Scholar
Hull, J., and White, A.. “The Pricing of Options on Assets with Stochastic Volatilities.” Journal of Finance, 42 (1987), 281–300.CrossRefGoogle Scholar
Jackwerth, J., and Rubinstein, M.. “Recovering Probability Distributions from Options Prices.” Journal of Finance, 51 (1996), 1611–1631.CrossRefGoogle Scholar
Jackwerth, J., and Rubinstein, M.. “Recovering Stochastic Processes from Options Prices.” Working Paper, London Business School and Univ. of California, Berkeley (1998).Google Scholar
Jarrow, R., and Rosenfeld, E.. “Jump Risks and the Intertemporal Capital Asset Pricing Model.” Journal of Business, 57 (1984), 337–351.CrossRefGoogle Scholar
Jarrow, R., and Rudd, A.. “Approximate Option Valuation for Arbitrary Stochastic Processes.” Journal of Financial Economics, 10 (1982), 347–369.CrossRefGoogle Scholar
Jorion, P. “On Jump Processes in the Foreign Exchange and Stock Markets.” Review of Financial Studies, 1 (1988), 427–445.CrossRefGoogle Scholar
Kon, S. “Models of Stock Returns: A Comparison.” Journal of Finance, 39 (1984), 147–165.Google Scholar
Melino, A. “Estimation of Continuous-Time Models in Finance.” In Advances in Econometrics: Sixth World Congress, Vol. II, Sims, C., ed. Cambridge and New York: Cambridge Univ. Press (1994).Google Scholar
Melino, A., and Turnbull, S. M.. “Pricing Foreign Currency Options with Stochastic Volatility.” Journal of Econometrics, 45 (1990), 239–265.CrossRefGoogle Scholar
Merton, R. C. “Option Pricing when the Underlying Process for Stock Returns is Discontinuous.” Journal of Financial Economics, 3 (1976), 124–144.CrossRefGoogle Scholar
Nandi, S. “How Important is the Correlation between Returns and Volatility in a Stochastic Volatility Model? Empirical Evidence from Pricing and Hedging in the S&P 500 Index Options Market.” Journal of Banking and Finance, 22 (1998), 589–610.CrossRefGoogle Scholar
Rosenberg, J. “Pricing Multivariate Contingent Claims Using Estimated Risk-Neutral Density Functions.” Mimeo, New York Univ. (1996).Google Scholar
Rubinstein, M. “Implied Binomial Trees.” Journal of Finance, 49 (1994), 393–440.CrossRefGoogle Scholar
Skiadopolous, G.; Hodges, S.; and Clewlow, L.. “The Dynamics of Smiles.” Working Paper, Univ. of Warwick (1998).Google Scholar
Stein, J. “Overreaction in the Options Market.” Journal of Finance, 44 (1989), 1011–1023.CrossRefGoogle Scholar
Stein, E. M., and Stein, J. C.. “Stock Rice Distributions with Stochastic Volatility: An Analytic Approach.” Review of Financial Studies, 4 (1991), 727–752.CrossRefGoogle Scholar
Wiggins, J. B. “Option Values under Stochastic Volatility: Theory and Empirical Estimates.” Journal of Financial Economics, 19 (1987), 351–372.CrossRefGoogle Scholar
Xu, X., and Taylor, S. J.. “The Term Structure of Volatility Implied by Foreign Exchange Options.” Journal of Financial and Quantitative Analysis, 29 (1994), 57–74.CrossRefGoogle Scholar
Zhu, Y. “Three Essays in Mathematical Finance.” Doctoral Diss., Department of Mathematics, New York Univ. (1997).Google Scholar