Pricing Up-and-In Barrier Options with In-Out Parity

Summary

The document asks what information a binomial tree must retain when valuing an up-and-in barrier option. The response assumes a European-style contract and proposes using knock-in/knock-out parity: the value of the corresponding vanilla European option equals the sum of the up-and-in and up-and-out option values. Under that relationship, one can value the up-and-out contract with a binomial tree and obtain the up-and-in value by subtracting it from the vanilla value, which the response says can be calculated with Black–Scholes.

This provides a practical route to the requested price without directly tracking every node for the knock-in claim. The excerpt does not derive the parity or discuss its conditions, and its proposed combination depends on matching contract terms and assumptions. It addresses a European option; it does not establish how to handle early exercise or other barrier conventions. The initial question about the minimal tree information structure is therefore not answered directly.

Key ideas

• For matching European contracts, vanilla value can be decomposed into knock-in and knock-out values.
• The response suggests valuing the up-and-out claim on a binomial tree and using parity to infer the up-and-in value.
• The vanilla European value is proposed to come from the Black–Scholes formula.
• The method is presented under a European-option assumption, with no detailed derivation or treatment of other contract terms.

Tags

# Barrier Option from binomial tree


# Barrier Option from binomial tree












What is the smallest information structure that is required for using the binomial tree to calculate the price of a barrier (up-and-in) option? My gut feeling is any node below the node that reaches the barrier price will be irrelevant.

## Answer by user23564 (score 4, accepted)

https://quant.stackexchange.com/a/43830

I'm assuming you're talking about a European option. I did a similar problem for my homework recently, I used the in-out parity for pricing the up and in barrier option.

Basically European Option = Knock up and in Option + Knock up and out option

You can price the up and out easily using Binomial and use BS formula for pricing the European Option, then use the above parity to get the knock up and in.

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