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AN ANALYTICAL SOLUTION FOR PARISIAN UP-AND-IN CALLS

Part of: Applications Mathematical finance

Published online by Cambridge University Press:  27 January 2016

NHAT-TAN LE Open the ORCID record for NHAT-TAN LE [Opens in a new window] ,
XIAOPING LU Open the ORCID record for XIAOPING LU [Opens in a new window]  and
SONG-PING ZHU Open the ORCID record for SONG-PING ZHU [Opens in a new window]
NHAT-TAN LE*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email ntl600@uowmail.edu.au, xplu@uow.edu.au, spz@uow.edu.au
XIAOPING LU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email ntl600@uowmail.edu.au, xplu@uow.edu.au, spz@uow.edu.au
SONG-PING ZHU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email ntl600@uowmail.edu.au, xplu@uow.edu.au, spz@uow.edu.au
*
✉ntl600@uowmail.edu.au
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Abstract

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We derive an analytical solution for the value of Parisian up-and-in calls by using the “moving window” technique for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an “in” barrier, the option holder cannot do or decide on anything before the option is activated, and once the option is activated it is just a plain vanilla call, which could be of American style or European style.

Keywords

Parisian options“moving window” techniqueanalytical solutionscoupled integral equations

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 62P05: Applications to actuarial sciences and financial mathematics

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 3 , January 2016 , pp. 269 - 279
Copyright
© 2016 Australian Mathematical Society 

References

Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–654; doi:10.1086/260062.Google Scholar
Cox, J. C., Ross, S. A. and Rubinstein, M., “Option pricing: A simplified approach”, J. Financ. Econom. 7 (1979) 229–263; doi:10.1016/0304-405X(79)90015-1.Google Scholar
Gao, B., Huang, J. Z. and Subrahmanyam, M., “The valuation of American barrier options using the decomposition technique”, J. Econom. Dynam. Control 24 (2000) 1783–1827; doi:10.1016/S0165-1889(99)00093-7.Google Scholar
Garcia, D., “Convergence and biases of Monte Carlo estimates of American option prices using a parametric exercise rule”, J. Econom. Dynam. Control 27 (2003) 1855–1879; doi:10.1016/S0165-1889(02)00086-6.Google Scholar
Haber, R. J., Schönbucher, P. J. and Wilmott, P., “Pricing Parisian options”, J. Derivatives 6 (1999) 71–79; doi:10.3905/jod.1999.319120.CrossRefGoogle Scholar
Hahn, D. W. and Özisik, M. N., Heat conduction (Wiley, 2012) Chapter 6, 236–272; doi:10.1002/9781118411285.ch6.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
Kim, I. J., “The analytic valuation of American options”, Rev. Financ. Stud. 3 (1990) 547–572; doi:10.1093/rfs/3.4.547.CrossRefGoogle Scholar
Longstaff, F. A. and Schwartz, E. S., “Valuing American options by simulation: A simple least-squares approach”, Rev. Financ. Stud. 14 (2001) 113–147; doi:10.1093/rfs/14.1.113.Google Scholar
Muthuraman, K., “A moving boundary approach to American option pricing”, J. Econom. Dynam. Control 32 (2008) 3520–3537; doi:10.1016/j.jedc.2007.12.007.CrossRefGoogle Scholar
Zhu, S. P., “A new analytical approximation formula for the optimal exercise boundary of American put options”, Int. J. Theor. Appl. Finance 9 (2006) 1141–1177; doi:10.1142/S0219024906003962.Google Scholar
Zhu, S. P., “An exact and explicit solution for the valuation of American put options”, Quant. Finance 6 (2006) 229–242; doi:10.1080/14697680600699811.CrossRefGoogle Scholar
Zhu, S. P. and Chen, W. T., “Pricing Parisian and Parasian options analytically”, J. Econom. Dynam. Control 37 (2013) 875–896; doi:10.1016/j.jedc.2012.12.005.Google Scholar
Zhu, S. P. and He, Z. W., “Calculating the early exercise boundary of American put options with an approximation formula”, Int. J. Theor. Appl. Finance 10 (2007) 1203–1227; doi:10.1142/S0219024907004615.Google Scholar
Zhu, S. P., Le, N. T., Chen, W. T. and Lu, X. P., “Pricing Parisian down-and-in options”, Appl. Math. Lett. 43 (2015) 19–24; doi:10.1016/j.aml.2014.10.019.Google Scholar