What are these options called, and where can I find their prices? For example, in a Black Scholes setting?

Note that this is not an appropriate valuation model to use for non-European contracts in most real-world markets. After reading your question for a second time, I agree with Quantuple's comment that you seem to be looking for the solution to an up-and-in binary call option. Consequently, we can skip the second indicator and your payoff is just.

You thus have the standard result that. A more interesting case is the down-and-out binary call. This is how I initially understood your question. This option knocks out, should the spot price breach the barrier before maturity.

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Otherwise it has a digital payoff of one. Using the method of images, see e. Buchen , the solution can be shown to be. By posting your answer, you agree to the privacy policy and terms of service. Sign up or log in to customize your list. Stack Exchange Inbox Reputation and Badges. Questions Tags Users Badges Unanswered. Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics.

Join them; it only takes a minute: Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top. Barrier digital options and pricing. What do you call options which behave like barrier options but for a digital option? I guess they are called barrier digital options. They are path-dependent and for this reason there isn't a closed form solution.

You can use Monte-Carlo or Binomial trees. Alternatively you can approximate a digital option as a N call spreads, with N big.

I disagree with your statement that these options don't have a closed-form solution in the Black-Scholes framework. These options are one of the two elementary building blocks of barrier options with a plain vanilla payoff which is widely known to have a closed-form solution.

The other component is a barrier option on an asset-or-nothing payoff. Do you have some references? If not, then perhaps prices can be obtained in a simpler continuous framework e. Sounds to me like an up-and-in binary or digital option. This wasn't fully clear to me from the question. I read this as a down-and-out call. But maybe Tony can clarify this. In any case - the framework that I referenced can be used for this.

Up-and-In Binary Call After reading your question for a second time, I agree with Quantuple's comment that you seem to be looking for the solution to an up-and-in binary call option. LocalVolatility 3, 2 7 How do you pick the bs vol to use though? That's a valid though very different question.

black scholes - Barrier digital options and pricing - Quantitative Finance Stack Exchange

Tony was asking about the valuation formula assuming that the underlying follows a GBM. Since this model only poorly reflects observable market prices and dynamics, I wouldn't recommend using it in the first place to value any non-European payoff. Fair, thought they do ask where they can find their pricing - yeah they say for example in a BS setting , but i think you should definitely note in the answer that this is not a valid way to value barrier options, indeed even local vol isn't correct.

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