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INTEGRAL EQUATION FORMULATION FOR SHOUT OPTIONS

Part of: Mathematical finance Representations of solutions

Published online by Cambridge University Press:  08 August 2018

R. MALLIER  and
J. GOARD Open the ORCID record for J. GOARD [Opens in a new window]
R. MALLIER
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada email [email protected]
J. GOARD*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, Australia email [email protected]
*
✉[email protected]
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Abstract

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We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.

Keywords

shout optionsfree boundary problemsintegral equations

MSC classification

Primary: 91G20: Derivative securities
Secondary: 35C10: Series solutions
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 1 , July 2018 , pp. 65 - 85
Copyright
© 2018 Australian Mathematical Society 

References

Alobaidi, G. and Mallier, R., “Laplace transforms and the American straddle”, J. Appl. Math. 2 (2002) 121–129; doi:10.1155/S1110757X02110011.Google Scholar
Alobaidi, G. and Mallier, R., “The American straddle close to expiry”, Bound. Value Probl. 2006 (2006) article ID 32835; doi:10.1155/BVP/2006/32835.Google Scholar
Alobaidi, G., Mallier, R. and Haslam, M. C., “Integral transforms and American options: Laplace and Mellin go green”, Acta Math. Univ. Comenian. (N.S.) 83(2) (2014) 245–266; MR3267258.Google Scholar
Alobaidi, G., Mallier, R. and Mansi, S., “Laplace transforms and shout options”, Acta Math. Univ. Comenian. (N.S.) 80 (2011) 79–102; MR2784846.Google Scholar
Barles, G., Burdeau, J., Romano, M. and Samsoen, N., “Critical stock price near expiration”, Math. Finance 5 (1995) 77–95; doi:10.1111/j.1467-9965.1995.tb00103.x.Google Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–659; http://www.jstor.org/stable/1831029.Google Scholar
Boyle, P. P., Kolkiewicz, A. W. and Tan, K. S., “Valuation of the reset options embedded in some equity-linked insurance products”, N. Am. Actuar. J. 5(3) (2001) 1–18; MR1989757.Google Scholar
Carr, P., Jarrow, R. and Myneni, R., “Alternative characterizations of the American put option”, Math. Finance 2 (1992) 87–106; doi:10.1111/j.1467-9965.1992.tb00040.x.Google Scholar
Chesney, M. and Gibson, R., “State space symmetry and two-factor option pricing models”, Adv. Futures Options Res. 8 (1993) 85–112; https://hal-hec.archives-ouvertes.fr/hal-00607989.Google Scholar
Chiarella, C., Kucera, A. and Ziogas, A., “A survey of the integral representation of American option prices”, Quantitative Finance Research Centre Research Paper 118, University of Technology, Sydney, Australia, 2004.Google Scholar
Dai, M., Kwok, Y. K. and Wu, L., “Optimal shouting policies of options with strike reset right”, Math. Finance 14(3) (2004) 383–401; doi:10.1111/j.0960-1627.2004.00196.x.Google Scholar
Dewynne, J. N., Howison, S. D., Rupf, I. and Wilmott, P., “Some mathematical results in the pricing of American options”, European J. Appl. Math. 4 (1993) 381–398; doi:10.1017/S0956792500001194.Google Scholar
Duffie, D., Dynamic asset pricing theory (Princeton University Press, Princeton, NJ, 1992).Google Scholar
Evans, J. D., Kuske, R. and Keller, J. B., “American options on assets with dividends near expiry”, Math. Finance 12(3) (2002) 219–237; doi:10.1111/1467-9965.02008.Google Scholar
Friedman, A., “Analyticity of the free boundary for the Stefan problem”, Arch. Ration. Mech. Anal. 61 (1976) 97–125; doi:10.1007/BF00249700.Google Scholar
Geske, R. and Johnson, H., “The American put option valued analytically”, J. Finance 39 (1984) 1511–1524; doi:10.1111/j.1540-6261.1984.tb04921.x.Google Scholar
Goard, J., “Exact solutions for a strike reset put option and a shout call option”, Math. Comput. Modelling 55 (2012) 1787–1797; doi:10.1016/j.mcm.2011.11.033.Google Scholar
Huang, J.-Z., Subrahmanyan, M. G. and Yu, G. G., “Pricing and hedging American options: A recursive integration method”, Rev. Financ. Stud. 9(3) (1996) 277–300; doi:10.1093/rfs/9.1.277.Google Scholar
Jacka, S. D., “Optimal stopping and the American put”, Math. Finance 1 (1991) 1–14; doi:10.1111/j.1467-9965.1991.tb00007.x.Google Scholar
Ju, N., “Pricing an American option by approximating its early exercise boundary as a multipiece exponential function”, Rev. Financ. Stud. 11 (1998) 627–646; doi:10.1093/rfs/11.3.627.Google Scholar
Karatzas, I., “On the pricing of American options”, Appl. Math. Optim. 17 (1988) 37–60; doi:10.1007/BF01448358.Google Scholar
Kim, I. J., “The analytic valuation of American options”, Rev. Financ. Stud. 3 (1990) 547–552; doi:10.1093/rfs/3.4.547.Google Scholar
Knessl, C., “A note on a moving boundary problem arising in the American put option”, Stud. Appl. Math. 107 (2001) 157–183; doi:10.1111/1467-9590.00183.Google Scholar
Kolodner, I. I., “Free boundary problems for the heat conduction equation with applications to problems of change of phase”, Comm. Pure Appl. Math. 9 (1956) 1–31; doi:10.1002/cpa.3160090102.Google Scholar
Kuske, R. and Keller, J. B., “Optimal exercise boundary for an American put option”, Appl. Math. Finance 5 (1998) 107–116; doi:10.1080/135048698334673.Google Scholar
Kwok, Y. K., Mathematical models of financial derivatives (Springer, Singapore, 1998).Google Scholar
Maple user manual (Maplesoft, a division of Waterloo Maple Inc., Toronto, 2005–2015).Google Scholar
McDonald, R. and Schroder, M., “A parity result for American options”, J. Comput. Finance 1 (1998) 5–13; doi:10.21314/JCF.1998.010.Google Scholar
McKean, H. P. Jr., “Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics”, Ind. Manage. Rev. 6 (1965) 32–39.Google Scholar
Merton, R. C., “The theory of rational option pricing”, Bell J. Econ. 4 (1973) 141–183; doi:10.2307/3003143.Google Scholar
Merton, R. C., “On the problem of corporate debt: The risk structure of interest rates”, J. Finance 29 (1974) 449–470; doi:10.1111/j.1540-6261.1974.tb03058.x.Google Scholar
Samuelson, P. A., “Rational theory of warrant pricing”, Ind. Manage. Rev. 6 (1965) 13–31; http://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/ReferencesPapers.aspx?ReferenceID=1346056.Google Scholar
Tao, L. N., “The analyticity of solutions of the Stefan problem”, Arch. Ration. Mech. Anal. 72 (1980) 285–301; doi:10.1007/BF00281593.Google Scholar
Tao, L. N., “The Cauchy–Stefan problem”, Acta Mech. 45 (1982) 49–64; doi:10.1007/BF01295570.Google Scholar
Thomas, B., “Something to shout about”, Risk 6 (1993) 56–58.Google Scholar
Wilmott, P., Paul Wilmott on quantitative finance (Wiley, Chichester, 2000).Google Scholar
Windcliff, H., Forsyth, P. A. and Vetzal, K. R., “Shout options: A framework for pricing contracts which can be modified by the investor”, J. Comput. Appl. Math. 134 (2001) 213–241; doi:10.1016/S0377-0427(00)00551-3.Google Scholar
Zhang, J. E. and Li, T., “Pricing and hedging American options analytically: A perturbation method”, Math. Finance 20(1) (2010) 59–87; doi:10.1111/j.1467-9965.2009.00389.x.Google Scholar