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Pricing American Options under the Constant Elasticity of Variance Model and Subject to Bankruptcy

Published online by Cambridge University Press:  01 October 2009

João Pedro Vidal Nunes*
Affiliation:
ISCTE Business School, Complexo INDEG/ISCTE, Av. Prof. Aníbal Bettencourt, 1600-189 Lisboa, Portugal. [email protected]

Abstract

This paper proposes an alternative characterization of the early exercise premium that is valid for any Markovian and diffusion underlying price process as well as for any parameterization of the exercise boundary. This new representation is shown to provide the best pricing alternative available in the literature for medium- and long-term American option contracts, under the constant elasticity of variance model. Moreover, the proposed pricing methodology is also extended easily to the valuation of American options on defaultable equity and possesses appropriate asymptotic properties.

Type
Research Articles
Copyright
Copyright © Michael G. Foster School of Business, University of Washington 2009

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References

Abramowitz, M., and Stegun, I.. Handbook of Mathematical Functions. New York: Dover Publications (1972).Google Scholar
Barone-Adesi, G. “The Saga of the American Put.” Journal of Banking and Finance, 29 (2005), 29092918.CrossRefGoogle Scholar
Barone-Adesi, G., and Whaley, R.. “Efficient Analytic Approximation of American Option Values.” Journal of Finance, 42 (1987), 301320.CrossRefGoogle Scholar
Bekaert, G., and Wu, G.. “Asymmetric Volatility and Risk in Equity Markets.” Review of Financial Studies, 13 (2000), 142.CrossRefGoogle Scholar
Bensoussan, A. “On the Theory of Option Pricing.” Acta Applicandae Mathematicae, 2 (1984), 139158.CrossRefGoogle Scholar
Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (1973), 637654.CrossRefGoogle Scholar
Boyle, P., and Tian, Y.. “Pricing Lookback and Barrier Options under the CEV Process.” Journal of Financial and Quantitative Analysis, 34 (1999), 241264.CrossRefGoogle Scholar
Brennan, M., and Schwartz, E.. “The Valuation of American Put Options.” Journal of Finance, 32 (1977), 449462.CrossRefGoogle Scholar
Broadie, M., and Detemple, J.. “American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods.” Review of Financial Studies, 9 (1996), 12111250.CrossRefGoogle Scholar
Brown, B.; Lovato, J.; and Russell, K.. “Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters.” Working Paper, University of Texas (1997).Google Scholar
Bunch, D., and Johnson, H.. “The American Put Option and Its Critical Stock Price.” Journal of Finance, 55 (2000), 23332356.CrossRefGoogle Scholar
Buonocore, A.; Nobile, A.; and Ricciardi, L.. “A New Integral Equation for the Evaluation of First-Passage-Time Probability Densities.” Advances in Applied Probability, 19 (1987), 784800.CrossRefGoogle Scholar
Campbell, J., and Taksler, G.. “Equity Volatility and Corporate Bond Yields.” Journal of Finance, 58 (2003), 23212349.CrossRefGoogle Scholar
Carr, P. “Randomization and the American Put.” Review of Financial Studies, 11 (1998), 597626.CrossRefGoogle Scholar
Carr, P.; Jarrow, R.; and Myneni, R.. “Alternative Characterizations of American Put Options.” Mathematical Finance, 2 (1992), 87106.CrossRefGoogle Scholar
Carr, P., and Linetsky, V.. “A Jump to Default Extended CEV Model: An Application of Bessel Processes.” Finance and Stochastics, 10 (2006), 303330.CrossRefGoogle Scholar
Chung, S., and Shackleton, M.. “Generalised Geske-Johnson Interpolation of Option Prices.” Journal of Business Finance and Accounting, 34 (2007), 9761001.CrossRefGoogle Scholar
Cox, J. “Notes on Option Pricing I: Constant Elasticity of Variance Diffusions.” Working Paper, Stanford University (1975). Reprinted in Journal of Portfolio Management, 23(1996), 15–17.Google Scholar
Cox, J.; Ross, S.; and Rubinstein, M.. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7 (1979), 229263.CrossRefGoogle Scholar
Davydov, D., and Linetsky, V.. “Pricing and Hedging Path-Dependent Options under the CEV Process.” Management Science, 47 (2001), 949965.CrossRefGoogle Scholar
Delbaen, F., and Shirakawa, H.. “A Note on Option Pricing for the Constant Elasticity of Variance Model.” Asia-Pacific Financial Markets, 9 (2002), 8599.CrossRefGoogle Scholar
Dennis, P., and Mayhew, S.. “Risk-Neutral Skewness: Evidence from Stock Options.” Journal of Financial and Quantitative Analysis, 37 (2002), 471493.CrossRefGoogle Scholar
Detemple, J., and Tian, W.. “The Valuation of American Options for a Class of Diffusion Processes.” Management Science, 48 (2002), 917937.CrossRefGoogle Scholar
Emanuel, D., and MacBeth, J.. “Further Results on the Constant Elasticity of Variance Call Option Pricing Model.” Journal of Financial and Quantitative Analysis, 17 (1982), 533554.CrossRefGoogle Scholar
Fortet, R. “Les Fonctions Aléatoires du Type de Markov Associées à Certaines Équations Linéaires aux Dérivées Partielles du Type Parabolique.” Journal des Mathématiques Pures et Appliquées, 22 (1943), 177243.Google Scholar
Geske, R., and Johnson, H.. “The American Put Option Valued Analytically.” Journal of Finance, 39 (1984), 15111524.CrossRefGoogle Scholar
Huang, J.; Subrahmanyam, M.; and Yu, G. G.. “Pricing and Hedging American Options: A Recursive Integration Method.” Review of Financial Studies, 9 (1996), 277300.CrossRefGoogle Scholar
Ingersoll, J. “Approximating American Options and Other Financial Contracts Using Barrier Derivatives.” Journal of Computational Finance, 2 (1998), 85112.CrossRefGoogle Scholar
Jacka, S. “Optimal Stopping and the American Put.” Mathematical Finance, 1 (1991), 114.CrossRefGoogle Scholar
Jamshidian, F. “An Analysis of American Options.” Review of Futures Markets, 11 (1992), 7280.Google Scholar
Johnson, H. “An Analytical Approximation for the American Put Price.” Journal of Financial and Quantitative Analysis, 18 (1983), 141148.CrossRefGoogle Scholar
Ju, N. “Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function.” Review of Financial Studies, 11 (1998), 627646.CrossRefGoogle Scholar
Ju, N., and Zhong, R.. “An Approximate Formula for Pricing American Options.” Journal of Derivatives, 7 (1999), 3140.CrossRefGoogle Scholar
Karatzas, I. “On the Pricing of American Options.” Applied Mathematics and Optimization, 17 (1988), 3760.CrossRefGoogle Scholar
Kim, J. “The Analytic Valuation of American Options.” Review of Financial Studies, 3 (1990), 547572.CrossRefGoogle Scholar
Kim, J., and Yu, G. G.. “An Alternative Approach to the Valuation of American Options and Applications.” Review of Derivatives Research, 1 (1996), 6185.CrossRefGoogle Scholar
Kuan, G., and Webber, N.. “Pricing Barrier Options with One-Factor Interest Rate Models.” Journal of Derivatives, 10 (2003), 3350.CrossRefGoogle Scholar
Linetsky, V. “Pricing Equity Derivatives Subject to Bankruptcy.” Mathematical Finance, 16 (2006), 255282.CrossRefGoogle Scholar
Longstaff, F., and Schwartz, E.. “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt.” Journal of Finance, 50 (1995), 789819.CrossRefGoogle Scholar
MacMillan, L. “An Analytical Approximation for the American Put Price.” Advances in Futures and Options Research, 1 (1986), 119139.Google Scholar
Madan, D., and Unal, H.. “Pricing the Risks of Default.” Review of Derivatives Research, 2 (1998), 121160.CrossRefGoogle Scholar
McKean, H. “Appendix: A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics.” Industrial Management Review, 6 (1965), 3239.Google Scholar
Merton, R. “The Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science, 4 (1973), 141183.Google Scholar
Nelson, D., and Ramaswamy, K.. “Simple Binomial Processes as Diffusion Approximations in Financial Models.” Review of Financial Studies, 3 (1990), 393430.CrossRefGoogle Scholar
Park, C., and Schuurmann, F.. “Evaluations of Barrier-Crossing Probabilities of Wiener Paths.” Journal of Applied Probability, 13 (1976), 267275.CrossRefGoogle Scholar
Penev, S., and Raykov, T.. “A Wiener Germ Approximation of the Noncentral Chi Square Distribution and of Its Quantiles.” Working Paper, University of New South Wales and Melbourne University (1997).Google Scholar
Press, W.; Flannery, B.; Teukolsky, S.; and Vetterling, W.. Numerical Recipes in Pascal: The Art of Scientific Computing. New York, NY: Cambridge University Press (1994).Google Scholar
Sbuelz, A. “Analytical American Option Pricing: The Flat-Barrier Lower Bound.” Economic Notes, 33 (2004), 399413.CrossRefGoogle Scholar
Schroder, M. “Computing the Constant Elasticity of Variance Option Pricing Formula.” Journal of Finance, 44 (1989), 211219.CrossRefGoogle Scholar
Shreve, S. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer (2004).CrossRefGoogle Scholar
Sullivan, M. “Valuing American Put Options Using Gaussian Quadrature.” Review of Financial Studies, 13 (2000), 7594.CrossRefGoogle Scholar
Temme, N. “A Set of Algorithms for the Incomplete Gamma Functions.” Probability in the Engineering and Informational Sciences, 8 (1994), 291307.CrossRefGoogle Scholar
Van Moerbeke, P. “On Optimal Stopping and Free Boundary Problems.” Archive for Rational Mechanics and Analysis, 60 (1976), 101148.CrossRefGoogle Scholar