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Is there a relation between the price of a one touch option and the price of a binary option?

By one touch option, I mean an option that pays off a fixed amount if the price of the underlying is above a certain threshold at any point in time before or at maturity.

By binary option, I mean an option that pays off a fixed amount if the price of the underlying is above a certain threshold at maturity.

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The One Touch option is approximately twice the price of the equivalent Binary Option. This result can be shown to be exact if the underlying process is a pure Wiener process using the Reflection Principal. For lognormal processes this result is a good approximation but not exact.

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One touch option is an American barrier digital

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