Down-and-Out Call Value Compared with a Vanilla Call

Summary

The question challenges the claim that a down-and-out call with a barrier below its strike can have the same value as a vanilla call. A knock-out call can expire worthless if the underlying reaches its barrier, so the question asks why it would ever cost as much as an otherwise comparable vanilla option. The reply directly corrects the premise: the down-and-out call is worth less than the vanilla call.

As a limiting example, the answer says that when the barrier equals the strike, the down-and-out call is worth zero. This illustrates how the barrier can eliminate the option's payoff and why its presence reduces value. The exchange is brief and gives no pricing formula, assumptions, numerical comparison, or discussion of barrier monitoring and rebates. Its contribution is the basic valuation distinction, not a general method for pricing barrier options.

Key ideas

• A down-and-out call can lose its value if the underlying reaches the barrier.
• The reply corrects the claim that a down-and-out call below the strike matches a vanilla call in price.
• The down-and-out call is worth less than the vanilla call under the stated comparison.
• The answer gives zero value when the barrier is set equal to the strike as an illustrative limiting case.

Tags

# Down-Out Call and Vanilla call price


# Down-Out Call and Vanilla call price












We all know from text books and practice that a knock out call is usually cheaper than a vanilla call option. Economically speaking, this comes from the fact that there is a probability bigger than zero of future spot price movements hitting the barrier. Consequently, the knock out option does not participate in every possible future spot price scenarios and that is why we do not have to pay the same premium as for a vanilla call.

If we now have a look at a Down-and-out Call with Barrier below strike, we see that the price matches a plain vanilla call option.

So, why should I buy a down and out call? Consider we hit the barrier below strike, then the option expires worthless. Consider the same case with a plain vanilla call. The option does not expire worthless and even has the possibility to participate on future spot upward movements. So why do we have to pay the same amount of premium for a down and out call as for a plain vanilla call?

Thanks,

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/38008

> If we now have a look at a Down-and-out Call with Barrier below strike, we see that the price matches a plain vanilla call option.

This statement is false. The DAO Call is worth less than the plain vanilla Call. For instance if the Barrier is equal to the strike then the DAO Call is worth zero.

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