One-touch barrier binary option values

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We can buy a put option struck at 95 to hedge it but it is more inexpensive to buy an up-an-out put with a strike price 95 and barrier Once the stock price rises to when we can sell it and this put disappears simultaneously. The relationship between knock-in option, knock-out option and knock-less option standard option of the same type call or put with the same expiration date, strike and barrier level can be expressed as. This relationship only holds for the European barrier options. It has not been obtained for the American version when we get the American values from the sections above.

We plot the value of the American barrier binary options using the free-boundary structure in the above sections. Note that the value of in Equations 4. The American value curves in Figure 3 and Figure 4 are simulated from 15 by inserting different American binary option values. Figure 3 shows that the value of the American down-in cash-or-nothing call options asset-or-nothing call option follows a similar curve increases with stock price before the in barrier and then decreases due to the uncertainty of knock-in.

Figure 4 shows the value of the American up-in cash-or-nothing put option asset-or-nothing put is similar. As we can see before the barrier, the option value is increasing and gets its peak at the barrier. Then the value goes down as the stock price continues to go up after the barrier level.

Generally, the price of the American version options is larger than the European version. Figures show the values for the knock-out binary options. Figure 5 illustrates that the value of the up-out cash-or-nothing put option is a decreasing function of the stock price below the barrier. However, in Figure 6 the up-out asset-or-nothing put first goes up and then down to the barrier. We can see the value of the down-out cash-or-nothing call option in Figure 7 is strictly increasing as the asset price above the barrier.

The asset-or-nothing call value in Figure 8 is also in the similar situation but with different amount of payoff size. All of the out figures show that the smooth-fit condition is not satisfied at the stopping boundary K. Figure 3. A computer comparison for the values of the European and the American down-in cash-or-nothing call options with parameters and.

Figure 4. A computer comparison for the values of the European and the American up-in cash-or-nothing put options with parameters and. Figure 5.

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A computer comparison for the values of the European and the American up-out cash-or-nothing put options with parameters and. Figure 6. A computer comparison for the values of the European and the American up-out asset-or-nothing put options with parameters and. Figure 7. A computer comparison for the values of the European and the American down-out cash-or-nothing call options with parameters and. The results of this paper also hold for an underlying asset with dividend structure.

With minor modifications, the formulas developed here can be applied to handle those problems.

Figure 8. A computer comparison for the values of the European and the American down-out asset-or-nothing call options with parameters and. Derivatives Quarterly, 3, Risk Magazine, 4, Mathematical Finance, 2, Mathematical Finance, 6, International Journal of Theoretical and Applied Finance, 4, Journal of Futures Markets, 24, Review of Derivatives Research, 16, Applied Financial Economics, 6, Journal of Economic Dynamics and Control, 24, The Journal of Derivatives, 11, Springer, Berlin.

Journal of Theoretical Probability, 18, DOI: 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. Share and Cite:. Gao, M. Journal of Mathematical Finance , 10 , Preliminaries American feature entitles the option buyer the right to exercise early. Conflicts of Interest The authors declare no conflicts of interest. References [ 1 ] Derman, E. Please enable JavaScript to view the comments powered by Disqus.

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