# One Touch Calls

One touch calls are plain barrier options where the underlying has to only trade at or above the strike price for the strategy to immediately settle at 100. So, in the example of Figure 1 the S&P 500 Index has to move to 1400 or higher for one-touch call options to immediately be triggered and settle at 100 with immediate liquidation of the position.

One touch calls are American options as opposed to the European version of binary call options. In options terminology a European option can only be exercised at expiry while the American option can be exercised at any time up to and including expiry. Since there is absolutely no point in waiting until expiry to exercise one touch call options they are generally settled immediately.

One-Touch Call Option GreeksOut-of-the-MoneyIn-the-Money
Delta+veN/A
Gamma+ve N/A
Theta-veN/A
Vega+veN/A

## One Touch Calls v Binary Calls

Unlike the European binary call option the one touch call options have no ambiguous area where the underlying settles exactly on the strike. In one touch terms this immediately defines the winning bet so that there is no ‘dead heat’ possibility, the option has just two settlement prices, 0 or 100. The expiry price profile is the same as Figure 1 of the binary call option page but without the ‘dead heat’ settlement at 50.

This characteristic means that with the same strike, implied volatility and time to expiry one touch call options will be twice the price of the European binary call option. Why? At the strike one touch call options will be worth 100, while the binary call option will be worth 50. At any time prior to expiry there would be the possibility of buying the binary call option twice and selling the one touch once, and vice versa, and locking in a profit.

Unfortunately financial markets are never quite so easy to take money from since at the strike the binary options one touch call disappears having settled at 100, leaving a binary call option to ‘leg out of’. Furthermore, there is additional risk if the position is short two binary call options and long one one touch call, and the underlying price barrels through the strike with no chance of the short binary call options being closed out at 50, e.g. on ‘the open’ the underlying price might gap up through the strike and hence any ‘arbitrage profits’ turn into losses.

## One Touch Calls w.r.t. Time

Fig.1 – S&P 500 One Touch Call Option Fair Value Profiles w.r.t. Time to Expiry

Figure 1 shows one touch calls over time for the S&P 500 Index. Although the 25-day profile has a fairly mild gradient reflecting a low delta, the profile with just 0.2 days to expiry reveals extremely high gearing; in fact this option (and one touch put options) have probably the highest gearing of any financial instrument. And of course a delta of such severity would have a correspondingly aggressive gamma.

## One Touch Calls w.r.t. Implied Volatility

Figure 2 details the effect of different implied volatilities on one touch call options.

Fig.2 – S&P 500 One Touch Call Fair Value Profiles w.r.t. Implied Volatility

Yet again the profiles are fairly standard with the lower implied volatilities representing a lower probability of the strike being reached.

With both illustrations it is evident that the risk reversal cannot take place where theta and vega change sign. For this reason the one touch calls are a far more efficient method of trading time decay and implied volatility although selling theta and volatility means immediate loss should the strike be hit. Therefore the seller of one touch call options who wishes to take in time decay or wishes to short implied volatility needs to be fairly confident that the underlying price is also a bit ‘toppy’.

From a practical standpoint, if the CME offered these options there is likely to be considerable demand from the over-the-counter brigade who are pricing up and trading S&P 500 barrier options for their clients. These barrier options may consist of knock-ins and knock-outs which are very popular in the OTC market so liquid one touch call options (and put options) would be a very simple method of hedging away the position prior to the barrier being triggered.

### Formula

$\textup{One-Touch&space;Call}=\left&space;(&space;\frac{K}{S}&space;\right&space;)^{\frac{2r}{\sigma&space;^{2}}}N\left&space;(&space;d_{3}&space;\right&space;)+\frac{S}{K}N\left&space;(&space;d_{1}&space;\right&space;)$

where:

$d_{1}=\frac{ln\left&space;(&space;\frac{S}{K}&space;\right&space;)+\left&space;(&space;r-D+\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

$d_{3}=\frac{ln\left&space;(&space;\frac{S}{K}&space;\right&space;)-\left&space;(&space;r+\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

and:

$\textup{S}=\textup{price&space;of&space;the&space;underlying}$

$\textup{K}=\textup{strike&space;price}$

$\textup{r}=\textup{risk&space;free&space;rate&space;of&space;interest}$

$\textup{D}=\textup{continuous&space;dividend&space;yield&space;of&space;underlying}$

$\textup{t}=\textup{time&space;in&space;years&space;to&space;expiry}$

$\sigma&space;=\textup{annualised&space;standard&space;deviation&space;of&space;asset&space;returns}$

share