The Barrier Binary Options
- 1 Huafa Industrial Share Co., LTD., Zhuhai, China
- 2 Sun Yat-Sen Business School, Guangzhou, China
Abstract
We extend the binary options into barrier binary options and discuss the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we show that the price function of a knock-in American binary option can be expressed in terms of the price functions of simple barrier options and American options. For the knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. By the properties of Brownian motion and convergence theorems, we show how to calculate the expectation of the local time. In the financial analysis, we briefly compare the values of the American and European barrier binary options.
- Derman, E. and Kani, I. (1996) The Ins and Outs of Barrier Options: Part 1. Derivatives Quarterly, 3, 55-67.
- Derman, E. and Kani, I. (1997) The Ins and Outs of Barrier Options: Part 2. Derivatives Quarterly, 3, 73-80.
- Haug, E.G. (2007) The Complete Guide to Option Pricing Formulas. McGraw-Hill Companies, New York.
- Merton, R.C. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4, 141-183. https://doi.org/10.2307/3003143
- Rich, D.R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual, 7, 267-311.
- Rubinstein, M. and Reiner, E. (1991) Unscrambling the Binary Code. Risk Magazine, 4, 20.
- Kunitomo, N. and Ikeda, M. (1992) Pricing Options with Curved Boundaries. Mathematical Finance, 2, 275-298. https://doi.org/10.1111/j.1467-9965.1992.tb00033.x
- Geman, H. and Yor, M. (1996) Pricing and Hedging Double-Barrier Options: A Probabilistic Approach. Mathematical Finance, 6, 365-378. https://doi.org/10.1111/j.1467-9965.1996.tb00122.x
- Haug, E.G. (2001) Closed Form Valuation of American Barrier Options. International Journal of Theoretical and Applied Finance, 4, 355-359. https://doi.org/10.1142/S0219024901001012
- Dai, M. and Kwok, Y.K. (2004) Knock-in American Options. Journal of Futures Markets, 24, 179-192. https://doi.org/10.1002/fut.10101
- Jun, D. and Ku, H. (2012) Digital Barrier Option Contract with Exponential Random Time. IMA Journal of Applied Mathematics, 78, 1147-1155. https://doi.org/10.1093/imamat/hxs013
- Jun, D. and Ku, H. (2013) Valuation of American Partial Barrier Options. Review of Derivatives Research, 16, 167-191. https://doi.org/10.1007/s11147-012-9081-1
- Hui, C.H. (1996) One-Touch Double Barrier Binary Option Values. Applied Financial Economics, 6, 343-346. https://doi.org/10.1080/096031096334141
- Gao, B., Huang, J.Z. and Subrahmanyam, M. (2000) The Valuation of American Barrier Options Using the Decomposition Technique. Journal of Economic Dynamics and Control, 24, 1783-1827. https://doi.org/10.1016/S0165-1889(99)00093-7
- Aitsahlia, F., Imhof, L. and Lai, T.L. (2004) Pricing and Hedging of American Knock-in Options. The Journal of Derivatives, 11, 44-50. https://doi.org/10.3905/jod.2004.391034