By Gerd Hardack, Dieter Karras, Ben Fine
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The major concept of pricing barrier options is that the value of the contract is determined by two parts: value of the vanilla option and the possibility of crossing or not crossing the barrier. Simply the value of a barrier option equals the value of a plain vanilla option multiplied by the possibility of crossing (knock-in) or not crossing (knock-out) the barrier. We consider the 16 single barrier options as follows, given by Reiner and Rubinstein under a BS framework. When the barrier level is lower than or equal to the strike price, the following values are given.
Many pricing platforms used in the industry followed the above framework in their implementation. However, if the duration of such a product is long, then the use of constant volatility might not generate satisfactory accuracy, and so more flexible pricing methods are needed. 2â•‡ Binomial method Like most other path-dependent options, barrier options can be priced via lattice trees, such as binomial, trinomial, or adaptive mesh models by solving the PDE using a generalized finite difference method.
Kemna, A. G. Z. & Vorst, A. C. F. (1990). ” Journal of Banking and Finance, 14:113–129. , & Taylor, D. R. (2003). “Discrete Closed Form Solutions for Barrier Options,” University of Witwatersrand Honours Project. Reiner, E. & Rubinstein, M. (1991). ” Risk, 4(8): 28–35. Roll, R. (1977). ” Journal of Financial Economics, 5: 251–258. Rubinstein, M. (2000). ” Journal of Derivatives, 8(2): 47–50. Turnbull, S. M. & Wakeman, L. M. (1991). ” Journal of Financial and Quantitative Finance, 26: 377–389. 1â•‡ Implied volatility Implied volatility of an option contract is the volatility of that implied (calculated) from the price of the option with an equation, like Black and Scholes (BS), which contains the market view of the contract holders about the future asset movement.