×

Integral equation formulation for shout options. (English) Zbl 1416.91379

Summary: 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.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
45A05 Linear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alobaidi, G.; Mallier, R., Laplace transforms and the American straddle, J. Appl. Math., 2, 121-129, (2002) · Zbl 1005.91058 · doi:10.1155/S1110757X02110011
[2] Alobaidi, G.; Mallier, R., The American straddle close to expiry, Bound. Value Probl., 2006, (2006) · Zbl 1138.91417 · doi:10.1155/BVP/2006/32835
[3] Alobaidi, G.; Mallier, R.; Haslam, M. C., Integral transforms and American options: Laplace and Mellin go green, Acta Math. Univ. Comenian. (N.S.), 83, 2, 245-266, (2014) · Zbl 1349.91265
[4] Alobaidi, G.; Mallier, R.; Mansi, S., Laplace transforms and shout options, Acta Math. Univ. Comenian. (N.S.), 80, 79-102, (2011) · Zbl 1240.91161
[5] Barles, G.; Burdeau, J.; Romano, M.; Samsoen, N., Critical stock price near expiration, Math. Finance, 5, 77-95, (1995) · Zbl 0866.90029 · doi:10.1111/j.1467-9965.1995.tb00103.x
[6] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-659, (1973) · Zbl 1092.91524 · doi:10.1086/260062
[7] Boyle, P. P.; Kolkiewicz, A. W.; Tan, K. S., Valuation of the reset options embedded in some equity-linked insurance products, N. Am. Actuar. J., 5, 3, 1-18, (2001) · Zbl 1083.91511 · doi:10.1080/10920277.2001.10595994
[8] Carr, P.; Jarrow, R.; Myneni, R., Alternative characterizations of the American put option, Math. Finance, 2, 87-106, (1992) · Zbl 0900.90004 · doi:10.1111/j.1467-9965.1992.tb00040.x
[9] Chesney, M.; Gibson, R., State space symmetry and two-factor option pricing models, Adv. Futures Options Res., 8, 85-112, (1993)
[10] Chiarella, C.; Kucera, A.; Ziogas, A.
[11] Dai, M.; Kwok, Y. K.; Wu, L., Optimal shouting policies of options with strike reset right, Math. Finance, 14, 3, 383-401, (2004) · Zbl 1134.91407 · doi:10.1111/j.0960-1627.2004.00196.x
[12] Dewynne, J. N.; Howison, S. D.; Rupf, I.; Wilmott, P., Some mathematical results in the pricing of American options, European J. Appl. Math., 4, 381-398, (1993) · Zbl 0797.60051 · doi:10.1017/S0956792500001194
[13] Duffie, D., Dynamic asset pricing theory, (1992), Princeton University Press: Princeton University Press, Princeton, NJ
[14] Evans, J. D.; Kuske, R.; Keller, J. B., American options on assets with dividends near expiry, Math. Finance, 12, 3, 219-237, (2002) · Zbl 1031.91047 · doi:10.1111/1467-9965.02008
[15] Friedman, A., Analyticity of the free boundary for the Stefan problem, Arch. Ration. Mech. Anal., 61, 97-125, (1976) · Zbl 0329.35034 · doi:10.1007/BF00249700
[16] Geske, R.; Johnson, H., The American put option valued analytically, J. Finance, 39, 1511-1524, (1984) · doi:10.1111/j.1540-6261.1984.tb04921.x
[17] Goard, J., Exact solutions for a strike reset put option and a shout call option, Math. Comput. Modelling, 55, 1787-1797, (2012) · Zbl 1255.91399 · doi:10.1016/j.mcm.2011.11.033
[18] Huang, J.-Z.; Subrahmanyan, M. G.; Yu, G. G., Pricing and hedging American options: A recursive integration method, Rev. Financ. Stud., 9, 3, 277-300, (1996) · doi:10.1093/rfs/9.1.277
[19] Jacka, S. D., Optimal stopping and the American put, Math. Finance, 1, 1-14, (1991) · Zbl 0900.90109 · doi:10.1111/j.1467-9965.1991.tb00007.x
[20] Ju, N., Pricing an American option by approximating its early exercise boundary as a multipiece exponential function, Rev. Financ. Stud., 11, 627-646, (1998) · doi:10.1093/rfs/11.3.627
[21] Karatzas, I., On the pricing of American options, Appl. Math. Optim., 17, 37-60, (1988) · Zbl 0699.90010 · doi:10.1007/BF01448358
[22] Kim, I. J., The analytic valuation of American options, Rev. Financ. Stud., 3, 547-552, (1990) · doi:10.1093/rfs/3.4.547
[23] Knessl, C., A note on a moving boundary problem arising in the American put option, Stud. Appl. Math., 107, 157-183, (2001) · Zbl 1152.91522 · doi:10.1111/1467-9590.00183
[24] Kolodner, I. I., Free boundary problems for the heat conduction equation with applications to problems of change of phase, Comm. Pure Appl. Math., 9, 1-31, (1956) · Zbl 0070.43803 · doi:10.1002/cpa.3160090102
[25] Kuske, R.; Keller, J. B., Optimal exercise boundary for an American put option, Appl. Math. Finance, 5, 107-116, (1998) · Zbl 1009.91025 · doi:10.1080/135048698334673
[26] Kwok, Y. K., Mathematical models of financial derivatives, (1998), Springer: Springer, Singapore · Zbl 0931.91018
[28] Mcdonald, R.; Schroder, M., A parity result for American options, J. Comput. Finance, 1, 5-13, (1998) · doi:10.21314/JCF.1998.010
[29] Mckean, H. P. Jr., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Ind. Manage. Rev., 6, 32-39, (1965)
[30] Merton, R. C., The theory of rational option pricing, Bell J. Econ., 4, 141-183, (1973) · Zbl 1257.91043 · doi:10.2307/3003143
[31] Merton, R. C., On the problem of corporate debt: The risk structure of interest rates, J. Finance, 29, 449-470, (1974)
[32] Samuelson, P. A., Rational theory of warrant pricing, Ind. Manage. Rev., 6, 13-31, (1965)
[33] Tao, L. N., The analyticity of solutions of the Stefan problem, Arch. Ration. Mech. Anal., 72, 285-301, (1980) · Zbl 0416.35017 · doi:10.1007/BF00281593
[34] Tao, L. N., The Cauchy-Stefan problem, Acta Mech., 45, 49-64, (1982) · Zbl 0506.73106 · doi:10.1007/BF01295570
[35] Thomas, B., Something to shout about, Risk, 6, 56-58, (1993)
[36] Wilmott, P., Paul Wilmott on quantitative finance, (2000), Wiley: Wiley, Chichester · Zbl 1127.91002
[37] Windcliff, H.; Forsyth, P. A.; Vetzal, K. R., Shout options: A framework for pricing contracts which can be modified by the investor, J. Comput. Appl. Math., 134, 213-241, (2001) · Zbl 1017.91060 · doi:10.1016/S0377-0427(00)00551-3
[38] Zhang, J. E.; Li, T., Pricing and hedging American options analytically: A perturbation method, Math. Finance, 20, 1, 59-87, (2010) · Zbl 1182.91181 · doi:10.1111/j.1467-9965.2009.00389.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.