![\"Shengyu](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/07\/img_SimTrade_Photo1_Shengyu_Zheng.jpg\" "\\"\\"")
<\/a><\/p>\nIn this article, Shengyu ZHENG<\/a> (ESSEC Business School, Grande Ecole<\/i> Program – Master in Management, 2020-2023) explains barrier options which are the most traded exotic options in derivatives markets.<\/p>\n Barrier options are path dependent. Their payoffs are not only a function of the price of the underlying asset relative to the option strike, but also depend on whether the price of the underlying asset reached a certain predefined barrier during the life of the option.<\/p>\n The two most common kinds of barrier options are knock-in (KI) and knock-out (KO) options.<\/p>\n KI barrier options are options that are activated only if the underlying asset attains a prespecified barrier level (the \u201cknock-in\u201d event). With the absence of this knock-in event, the payoff remains zero regardless of the trajectory of the price of the underlying asset.<\/p>\n KO barrier options are options that are deactivated only if the underlying asset attains a prespecified barrier level (the \u201cknock-out\u201d event). In the presence of this knock-out event, the payoff remains zero regardless of the trajectory of the price of the underlying asset.<\/p>\n The determination of the occurrence of a barrier event (KI or KO conditions) is essential to the ultimate payoff of the barrier option. In practice, the details of the KI or KO conditions are precisely defined in the contract (called \u201cConfirmations\u201d by the International Swaps and Derivatives Association (ISDA) for over-the counter (OTC) traded options).<\/p>\n The observation period denotes the period where a barrier event (KI or KO) can be observed, that is to say, when the price of the underlying asset is monitored. There are three styles of observation period: European style, partial-period American style, and full-period American style.<\/p>\n There are two typical types of monitoring methods in terms of the determination of a knock-in\/knock-out event: continuous monitoring and discrete monitoring. The monitoring method is one of the key factors in determining the premium of a barrier option.<\/p>\n For the most cases, the Barrier Reference Asset is the underlying asset itself. However, if specified in the contract, it can be another asset or index. It can also be other calculatable properties, such as the volatility of the asset. In this case, the methodology of calculating such properties should be clearly defined in the contract.<\/p>\n For knock-out options, there could be a rebate. A rebate is an extra feature and it corresponds to the amount that should be paid to the buyer of the knock-out option in case of the occurrence of a knock-out event.<\/p>\n Analogous to the call-put parity relation for plain vanilla options, there is an in-out parity relation for barrier options stating that a long position in a knock-in option plus a long position in a knock-out option with identical strikes, barriers, monitoring methods and maturity is equivalent to a long position in a comparable vanilla option. It could be stated as follows:<\/p>\n Where K denotes the strike price, T the maturity, and B the barrier level.<\/p>\n It is worth noting that this parity relation is valid only when the two KI and KO options are identical, and there is no rebate in case of a knock-out option.<\/p>\n There are four types of basic barrier options traded in the market: up-and-in option, up-and-out option, down-and-in option, and down-and-out option. \u201cUp\u201d and \u201cdown\u201d denotes the direction of surpassing the barrier price. \u201cIn\u201d and \u201cout\u201d depict the type of barrier condition, i.e. knock-in or knock-out. These four types of barrier features are available for both call and put options.<\/p>\n An up-and-in option is a knock-in option whose barrier condition is achieved if the underlying price arrives higher than the barrier level during the observation period.<\/p>\n Figure 1 illustrates the occurrence of an up-and-in barrier event for a barrier option with full-period American style and discrete monitoring (the closing time of each trading day).<\/p>\n Figure 1. Illustration of an up-and-in barrier option An up-and-out option is a knock-out option whose barrier condition is achieved if the underlying price arrives higher than the barrier level during the observation period.<\/p>\n Figure 2. Illustration of an up-and-out option<\/p>\n A down-and-in option is a knock-in option whose barrier condition is achieved if the underlying price arrives lower than the barrier level during the observation period.<\/p>\n Figure 3. Illustration of a down-and-in option A down-and-out option is a knock-out option whose barrier condition is achieved if the underlying price arrives lower than the barrier level during the observation period.<\/p>\n Figure 4. Illustration of a down-and-out option You can find below an R file to price barrier options.<\/p>\n Being the most popular exotic options, barrier options on stocks or indices have been actively traded in the OTC market since the inception of the market. Unavailable in standard exchanges, they are less accessible than their vanilla counterparts. Barrier options are also commonly utilized in structured products.<\/p>\n As one of the most traded but the simplest exotic derivative products, barrier options open an avenue for different applications. They are also very often incorporated in structured products, such as reverse convertibles. Knock-in\/knock out conditions are also common features in other types of more complicated exotic derivative products.<\/p>\n It is, therefore, important to be equipped with knowledge of this product and to understand the pricing logics if one aspires to work in financial markets.<\/p>\n ▶ All posts about options<\/a><\/p>\n ▶ Shengyu ZHENG Pricing barrier options with analytical formulas<\/a><\/p>\n ▶ Shengyu ZHENG Pricing barrier options with simulations and sensitivity analysis with Greeks<\/a><\/p>\n Broadie, M., Glasserman P., Kou S. (1997) A Continuity Correction for Discrete Barrier Option. Mathematical Finance<\/i>, 7:325-349.<\/p>\n Merton, R. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science<\/i>, 4:141-183.<\/p>\n Paixao, T. (2012) A Guide to Structured Products \u2013 Reverse Convertible on S&P500<\/p>\n Reiner, E.S., Rubinstein, M. (1991) Breaking down the barriers. Risk Magazine<\/i>, 4(8), 28\u201335.<\/p>\n Rich, D. R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual<\/i>, 7:267-311.<\/p>\n Wang, B., Wang, L. (2011) Pricing Barrier Options using Monte Carlo Methods, Working paper.<\/p>\n Haug, E. (1997) The Complete Guide to Option Pricing. London\/New York: McGraw-Hill.<\/p>\n Hull, J. (2006) Options, Futures, and Other Derivatives. Upper Saddle River, N.J: Pearson\/Prentice Hall.<\/p>\n The article was written in July 2022 by Shengyu ZHENG<\/a> (ESSEC Business School, Grande Ecole<\/i> Program – Master in Management, 2020-2023).<\/p>\n","protected":false},"excerpt":{"rendered":"Description<\/h2>\n
Knock-in (KI) barrier options<\/h3>\n
Knock-out (KO) barrier options<\/h3>\n
Observation<\/h2>\n
Observation period<\/h3>\n
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Monitoring method<\/h3>\n
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Barrier Reference Asset<\/h3>\n
Rebate<\/h2>\n
In-out parity relation for barrier options<\/h2>\n
![\"Knock-in](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/07\/img_SimTrade_Knock_in_knock_out_barrier_option_parity_relation.png\")
<\/p>\nBasic barrier options<\/h2>\n
Up-and-in option<\/h3>\n
\n![\"Example](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/08\/img_SimTrade_pic2.1_example_up_and_in_barrier_option.png\")
<\/p>\nUp-and-out option<\/h3>\n
![\"Example](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/08\/img_SimTrade_pic2.2_example_up_and_out_barrier_option.png\")
<\/p>\nDown-and-in option<\/h3>\n
\n![\"Example](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/08\/img_SimTrade_pic2.3_example_down_and_in_barrier_option.png\")
<\/p>\nDown-and-out option<\/h3>\n
\n![\"Example](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/08\/img_SimTrade_pic2.4_example_down_and_out_barrier_option.png\")
<\/p>\nDownload R file to price barrier options<\/h2>\n
![\"Download](\"https:\/\/www.simtrade.fr\/blog_simtrade\/wp-content\/uploads\/2022\/07\/img_SimTrade_Btn_Download_R_file_US.png\" "\\"Download")
<\/a><\/p>\nTrading of barrier options<\/h2>\n
Why should I be interested in this post?<\/h2>\n
Related posts on the SimTrade blog<\/h2>\n
References<\/h2>\n
Academic research articles<\/h3>\n
Books<\/h3>\n
About the author<\/h2>\n