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Option bounds

Part of:Approximations and expansionsMathematical economics

Published online by Cambridge University Press:  14 July 2016

Victor H. De La Peña,
Rustam Ibragimov and
Steve Jordan
Victor H. De La Peña
Affiliation:
Department of Statistics, Columbia University, 2990 Broadway, New York, NY 10027, USA. Email address: vp@stat.columbia.edu
Rustam Ibragimov
Affiliation:
Department of Economics, Yale University, 28 Hillhouse Avenue, New Haven, CT 06511, USA. Email address: rustam.ibragimov@yale.edu
Steve Jordan
Affiliation:
Yale School of Management, Yale University, 135 Prospect Street, New Haven, CT 06520, USA. Email address: steven.jordan@yale.edu
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Abstract

In this paper, we obtain sharp estimates for the expected payoffs and prices of European call options on an asset with an absolutely continuous price in terms of the price density characteristics. These techniques and results complement other approaches to the derivative pricing problem. Exact analytical solutions to option-pricing problems and to Monte-Carlo techniques make strong assumptions on the underlying asset's distribution. In contrast, our results are semi-parametric. This allows the derivation of results without knowing the entire distribution of the underlying asset's returns. Our results can be used to test different modelling assumptions. Finally, we derive bounds in the multiperiod binomial option-pricing model with time-varying moments. Our bounds reduce the multiperiod setup to a two-period setting, which is advantageous from a computational perspective.

Keywords

Option pricingsemi-parametric bounds

MSC classification

Secondary:91B24: Price theory and market structure91B30: Risk theory, insurance41A17: Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities)

Information

Type
Part 3. Financial mathematics
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004, pp. 145 - 156
Copyright
Copyright © Applied Probability Trust 2004 

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References

Ball, C. and Roma, A. (1994). Stochastic volatility option pricing. J. Fin. Quantitative Anal 29, 584–607.Google Scholar
Barnett, N. S., Cerone, R, Dragomir, S. S. and Roumeliotis, J. (2001). Some inequalities for the dispersion of a random variable whose PDF is defined on a finite interval. J. Inequalities Pure Appl. Math. 2, 1–18.Google Scholar
Bertsimas, D. and Popescu, I. (2002). On the relation between option and stock prices: a convex optimization approach. Operat. Res. 50, 358–374.CrossRefGoogle Scholar
Boyle, P. B. and Lin, X. S. (1997). Bounds on contingent claims based on several assets. J. Fin. Econom. 46, 383–400.Google Scholar
Boyle, P. B. and Vorst, T. (1992). Option replication in discrete time with transaction costs. J. Finance 47, 271–293.Google Scholar
Britten-Jones, M. and Neuberger, A. (2000). Option prices, implied price processes, and stochastic volatility. J. Finance 55, 839–866.Google Scholar
Buraschi, A. and Jackwerth, J. (2001). The pricing of a smile: hedging and spanning in option markets. Rev. Fin. Studies 14, 495–527.Google Scholar
Constantinides, G. M. and Zariphopoulou, T. (2001), Bounds on derivative prices in an intertemporal setting with proportional transaction costs and multiple securities. Math. Finance 11, 331–346.Google Scholar
Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. J. Fin. Econom. 7, 229–263.Google Scholar
Delbaen, F. (1992). Representing martingale measures when asset prices are continuous and bounded. Math. Finance 2, 107–130.Google Scholar
Dumas, B., Fleming, J. and Whaley, R. E. (1998). Implied volatility functions: empirical tests. J. Finance 53, 2059–2106.Google Scholar
Eaton, M. L. (1974). A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2, 609–614.Google Scholar
El Karoui, N. and Quenez, M. C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optimization 33, 27–66.Google Scholar
Frey, R. (1997). Derivative asset analysis in models with level-dependent and stochastic volatility. CWI Quart. 10, 1–34.Google Scholar
Frey, R. and Sin, C. A. (1999). Bounds on European option prices under stochastic volatility. Math. Finance 9, 97–116.Google Scholar
Grundy, B. D. (1991). Option prices and the underlying asset's return distribution. J. Finance 46, 1045–1069.Google Scholar
Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721.Google Scholar
Jansen, K., Haezendonck, J. and Goovarets, M. J. (1986). Upper bounds on stop-loss premiums in case of known moments up to the fourth order. Insurance Math. Econom. 5, 315–334.Google Scholar
Kramkov, D. (1996). Optimal decomposition of supermartingales and hedging contingent claims in incomplete security markets. Prob. Theory Relat. Fields 105, 459–479.Google Scholar
Lo, A. W. (1987). Semi-parametric upper bounds for option prices and expected payoffs. J. Fin. Econom. 19, 373–387.CrossRefGoogle Scholar
Palmer, K. (2001). A note on the Boy le-Vorst discrete-time option pricing model with transaction costs. Math. Finance 11, 357–363.Google Scholar
Perrakis, S. (1986). Option bounds in discrete time: extensions and the price of the American put. J. Business 59, 119–141.Google Scholar
Perrakis, S. and Ryan, P. J. (1984). Option pricing bounds in discrete time. J. Finance 39, 519–525.Google Scholar
Scarf, H. (1958). A min-max solution of an inventory problem. In Studies in the Mathematical Theory of Inventory and Production , eds Arrow, K. J., Karlin, S. and Scarf, H., Stanford University Press, pp. 201–209.Google Scholar
Scarf, H. (2002). Inventory theory. Operat. Res. 50, 186–191.Google Scholar
Stapleton, R. C. and Subrahmanyam, M. G. (1984). The valuation of options when asset returns are generated by a binomial process. J. Finance 39, 1525–1539.Google Scholar
Sun, Q. and Yan, Y. (2003). Skewness persistence with optimal portfolio selection. J. Banking Finance 27, 1111–1121.Google Scholar