Approximating a One-Touch Option with a European Digital

Summary

The question asks how to gain exposure to an underlying asset touching a target level at any time before expiry, rather than finishing beyond that level. It considers whether an American vanilla option or spread could approximate a payoff that increases when a price threshold is reached, while noting that delta hedging would require repeated position adjustments.

The response offers a valuation relationship: an American one-touch call is approximately worth half of a corresponding European digital call, and exactly half under zero drift, such as when dividend yield equals the interest rate. Its rationale is that after the barrier is first touched, the European digital has a one-half chance of finishing in the money under that condition. This is a concise theoretical pointer, not a complete replication strategy or hedge plan; the discussion does not quantify errors when drift is nonzero or address market frictions and contract details.

Key ideas

• A one-touch option pays based on reaching a barrier before expiry.
• Under zero drift, the one-touch call is stated to be worth half the equivalent European digital call.
• The approximation relies on a one-half chance of finishing in the money after the barrier is hit.
• The reply does not provide a detailed hedge or discuss nonzero-drift errors.

Tags

# How to approximate the payoff for touching instead of expiring in the money?


# How to approximate the payoff for touching instead of expiring in the money?












Is there a way to convert an American option (vanilla or spread?) to approximate touching a price instead of expiring at a given price?

For example, suppose you want to buy a call option such that the value doubles if the price goes up 10% at any time before expiration.

I can see using delta to purchase more contracts, but the delta is always changing so it would require constant buying and selling to update the delta.

I could see using BS (or other) model to calculate some position to open, but that too would require changing the position over time.

Also, can you recommend a resource (book or course) that teaches answers to this sort of question?

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/81644

There’s a pretty well known theorem that an American one-touch call option is worth approximately 50% of the equivalent European digital call. In fact , exactly 50% if the underlying has ‘zero drift’ (for example, dividend yield equals interest rate). This is because at the moment that the barrier is first hit, the European version has a 50% chance of ending in the money. Does this address your issue ?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.