# Up-and-Out One-Touch Put

## The up-and-out one-touch put offers a cheap alternative to the one-touch put since the probability of the bet winning is reduced by the possibility of the barrier being hit.

The buyers of the up and out one-touch put can choose a strategy that has the barrier so high that there is no probability of it being hit, but at this point the strategy has therefore reduced to a one-touch put. For the up-and-out one-touch put to have any point there must be a possibility of the barrier being hit. The buyer of the up and out one touch put must therefore believe there are extenuating circumstances which have created a situation where the market has got the probability of the barrier being hit wrong; that the market has assumed too high a probability. The higher the market view of the probability of the barrier being hit, the lower the price of the up and out one-touch put.

Fig. 1 represent the up-and-out one-touch put for the Indian Nifty50 stock market index. Fig.1 displays the price profiles of the 5500 strike, 6500 barrier option over a range of implied volatility. When the implied volatility is 10% (black profile) the barrier has little impact on the value. The blue 50% profile is a straight line indicating that the barrier has now created a constraint on the value.

Fig.2 illustrates the same up-and-out one-touch put against times to expiry. With 1000 index points between the strike and the barrier the three days-or-less price profiles are already zero prior to the underlying reaching the level of the barrier, with the eight day profile worth 0.01 at the underlying of 6475. Therefore in this instance, with the barrier set at a distance from the strike, the barrier has little influence on the knock-out’s value.

To make this strategy worthwhile the up-and-out one-touch put must be worth less than the one-touch put with same expiry and strike price. Figs. 1 & 2 illustrate that as implied volatility and time to expiry rise the up-and-out one-touch put decreases in value.

Fig.3 illustrates the up-and-out one-touch put compared to the one-touch put, time to expiry and implied volatility both quite high. Clearly there is a considerable advantage price-wise in buying the knock-out compared to the straight one-touch, but as has been mentioned, the buyer needs a specific belief that there is a extraneous factor that will stop the underlying reaching the barrier.

At the underlying of 6500 the one-touch put is worth 21.9. Also at 6500 the 6500 strike one-touch call is worth 100, so if the difference between the up-and-out one-touch put and the one-touch put is rebased to 100 (by multiplying 21.9 by multiplying by 4.57) and this is done for every level of underlying then the graph ‘Implied Probability of Barrier Being Hit’ is achieved. Clearly this profile provides the signal to buy the up-and-out one-touch put if the buyer considers that the probability as provided by the black profile (and as indicated by the left-hand axis) is too high, implying that the knock-out is too cheap.

### Formula

where:

and:

### Further Reading:

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

Ebenfeld, S., Mayr,M.R.& Topper,J. (2002). An Analysis of Onion Options and Double-no-Touch Digitals.

Luo, L.S.J. (2001). Various types of double-barrier options. *Journal of Computational Finance, *pages 125-138.