Abstract
A barrier option is one of the most popular exotic options which is designedto give a protection against unexpected wild fluctuation of stock prices.Protection is given to both the writer and holder of such an option.Kunitomo and Ikeda (1992) analytically obtained a pricing formula forexponential double barrier knockout options. Since the logarithm of theirproposed barriers for the stock price process S(t), whichisassumed to be geometric Brownian motion, are nothing but straight lineboundaries, the protection provided by them is not uniform over time. Toremedy this problem, we propose square root curved boundaries±b√tfor the underlying Brownian motion process W(t). Since thestandarddeviation of Brownian motion is proportional to √t, theseboundaries(after transformation) can be made to provide more uniform protectionthroughout the life time of the option. We will apply asymptoticexpansions of certain conditional probabilities obtained by Morimoto (1999)to approximate pricing formulae for exponential square root double barrierknockout European call options. These formulae allow us to computenumerical values in a very short time (t < 10−6sec), whereas it takesmuch longer to perform Monte Carlo simulations to determine optionpremiums.
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References
Anderson, T. W. (1960) A modification of the sequential probability ratio test to reduce the sample size, Ann. Math. Statist. 31, 165–197.
Broadie, M., Glasserman, P., and Kou, S. (1997) A continuity correction for discrete barrier options, Math. Finance 7, 325–349.
DeLong, D. M. (1981) Crossing probabilities for a square root boundary by a bessel process, Commun. Statist.-Theor. Math. A10, 2197–2213.
Geman, H. and Yor, M. (1996) Pricing and hedging double-barrier options: A probabilistic approach, Math. Finance 6, 365–378.
Goldman, M. B., Sosin, H. B., and Gatto, M. A. (1979) Path dependent options: Buy at the low, sell at the high, J. Finance 34, 1111–1127.
Harrison, J. M. and Kreps, D. M. (1979) Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory 20, 381–408.
Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in the theory of continuous trading, Stoch. Process. Appl. 11, 215–260.
Kunitomo, N. and Ikeda, M. (1992) Pricing options with curved boundaries, Math. Finance 2, 275–298.
Morimoto, M. (1999) On Pricing Barrier Options Related to Brownian Motion with the Class of Square Root Curved Boundaries, Ph.D. Thesis, Boston University.
Merton, R. C. (1973) Theory of rational option pricing, Bell. J. Econ. Management Sci. 4, 141–183.
Øksendal, B. (1998) Stochastic Differential Equations, An Introduction with Applications, 4th edn, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo.
Siegmund, D. (1985) Sequential Analysis, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo.
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Morimoto, M., Takahashi, H. On Pricing Exponential Square Root Barrier Knockout European Options. Asia-Pacific Financial Markets 9, 1–21 (2002). https://doi.org/10.1023/A:1021107418014
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DOI: https://doi.org/10.1023/A:1021107418014