# Down-and-Out One-Touch Call

## This knock-out instrument provides a cheap entry into a one-touch call as the barrier provides an automatic ‘stop-loss’. The knock-out barrier lowers the probability of the strike being touched as opposed to a straight one-touch call.

The down-and-out one-touch call has three possible outcomes:

i.            The underlying falls to trade at or below the barrier prior to the underlying rising to trade at or above the strike. In this case the down-and-out one-touch call is immediately knocked-out and settles at 0.

ii.            The underlying rises to trade at or above the strike prior to the underlying falling to trade at or below the barrier. In this case the strategy immediately wins and is settled at 100.

iii.            The underlying touches neither the strike nor the barrier before expiry. In this case the down-and-out one-touch call is a loser and settles at 0.

This can be summarised as either the strike is touched and the bet settles at 100, or it doesn’t touch the strike and settles at 0.

### Down-and-Out One-Touch Call Pricing

The relationship that holds for a binary call option with respect to the down-and-in call option and the down-and-out call option does not hold for the one-touch call option.

This is because prior to the barrier being hit the down-and-in one-touch call and the down-and-out one-touch call behave differently at the strike. The ‘Out’ automatically wins once the underlying has traded at or above the strike, whereas with the ‘In’ the strike does not in effect exist prior to the underlying hitting the barrier.

The down-and-out one-touch call therefore has two levels which, when hit, immediately terminate the bet. If it hits the lower level the bet loses, if the upper then the bet wins. In many respects this bet is akin to the double no-touch and like the double no-touch the Fourier method is used to price the bet.

Fig.1 shows how time to expiry affects the pricing of down and out one-touch calls.

Fig.1 – S&P500 Down and Out One-Touch Call Fair Value w.r.t. Time to Expiry

As time decays the barrier has increasingly less relevance to the value of the strategy. With increased time to expiry the bet’s price profile ultimately becomes a straight line between the barrier price of zero and the strike price of 100, instanced by the 25 day profile. As expiry looms closer, since the barrier has not been triggered the price profile progressively takes on the form of a straight 1500 one-touch call.

In Fig.2 increasing the implied volatility ultimately has the same effect of producing a straight line profile. Increasing the time to expiry and/or implied volatility does not push the curve in a convex manner passed the straight line profile.

Fig.2 – S&P500 Down and Out One-Touch Call Fair Value w.r.t. Implied Volatility

This strategy provides a cheap alternative to a one-touch call as the barrier increases the probability of the bet being worthless. Should the barrier be below where the strategy buyer believes there is a strong level of support then the barrier may be considered irrelevant by the buyer and therefore this cheaper version of the one-touch call even more attractive.

Alternatively, the barrier can be considered as a one-touch call with a built-in ‘stop’.

### Formula

The evaluation is based on the iterative Fourier Series which means that as time to expiry approaches zero the Gibbs Phenomenon1 takes effect which means that the value can overshoot the maximum price of 100 as the series converges. Increasing the terms does not alter the amount of the overshoot but concertinas the overshoots closer to the strike. The paper by Ebenfeld, Mayr and Topper (under Further Reading) offers a method to evaluate the optimum number of iterations, which although in reference to the Onion, applies equally well to the down and out one-touch call.

where:

and:

### References:

1. Carslaw, H.W. (1930). Introduction to the Theory of Fourier’s Series and Integrals. Dover 3rd edition.