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.
Highlights
Barrier options on stocks have been traded in the OTC (Over-The-Counter) market for more than four decades
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 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
Summary
Barrier options on stocks have been traded in the OTC (Over-The-Counter) market for more than four decades. Any up-and-in call with strike above the barrier is equal to a standard call option since all stock movements leading to pay-offs are knock-in naturally. Rubinstein and Reiner [6] provided closed form formulas for a wide variety of single barrier options. Haug [9] has presented analytic valuation formulas for American up-and-input and down-and-in call options in terms of standard American options It was extended by Dai and Kwok [10] to more types of American knock-in options in terms of integral representations. There are many different types of barrier binary options It depends on: 1) in or out; 2) up or down; 3) call or put; 4) cash-or-nothing or asset-or-nothing.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
