# Double No Touch Options

## Double no touch options consist of two strikes/barriers, one below the current underlying price and the other above the current underlying price.

Double no touch options immediately loses if at any time prior to expiry the underlying touches either strike.

The strategy wins and settles at 100 if at expiry neither strike has been traded at or through.

In terms of taking a view on volatility, double no touch options are probably the most efficient of all instruments including conventional straddles and strangles. If the trader wants to sell volatility, which in the conventional sense would require selling at-the-money straddles or strangles, the trader would need to buy, not sell, double no touch options. This is very much a sudden death approach to trading, but at the same time a very lucrative approach to shorting volatility assuming strikes are not touched.

Double No-Touch GreeksBelow Lower StrikeBetween StrikesAbove Upper Strike
DeltaN/A+ve -veN/A
GammaN/A-veN/A
ThetaN/A+veN/A
VegaN/A-veN/A

### Double No Touch Options Over Time

The below graph shows the routes to expiry for 215/235 coffee double no touch options. With 25 days to expiry the profile is very shallow and never exceeds a price of 23.04. In other words, the odds of the underlying remaining within the 215 to 235 corridor without ever touching either strike is 23.04%. Over time the profiles rise and fill in the rectangle bounded 0 and 100 and the strike prices.

Fig.1 – Coffee Double No-Touch Options Fair Value w.r.t. Time to Expiry

#### DNT Over Time: Practical Implications

Understanding the profile of double no-touch options is of immense importance to those buyers of time decay because the most profitable no-touch trades on a daily basis would be positions which are of varying degrees in-the-money.

For example, with 100 days to go it doesn’t really matter where the underlying is in relation to the strikes as the buyer will not make a profit over one day based on time appreciation (the double no-touch theta is always positive or zero).

With 25 days and 8 days to go you’d want to be buying double no-touch options if the price of coffee is at or very near 225. With 8 days to go buy this double no-touch if the underlying was off centre where the greatest time appreciation exists. An analysis of time decay/appreciation and the most profitable underlying price levels at which to purchase double no-touch options is in the section double no-touch theta.

The price profiles of double no-touch binary options are greatly impacted by implied volatility, i.e. double no-touch vega is high. It follows that the greater the volatility of the underlying the greater the chance of one of the strikes being hit, hence the lower the price of the double no-touch.

### Double No Touch Options Over Implied Volatility

Figure 2 offers price profiles of double no-touch options over a range of implied volatilities. In this 20-day to expiry example, at the coffee price of 224 and at 36% implied volatility the double no-touch is worth 42.21 implying there’s a chance of the underlying avoiding hitting one of the strikes of 42.21%. As volatility falls to 20% the double no-touch gains value. At 4% the double no-touch has fair value of 100 over the range of coffee prices from 215 to 233.

Clearly when volatility is high the greatest gains from selling volatility, i.e. buying double no touch options, is when the price of coffee is midway between the strikes. As implied volatility falls the greatest gains have moved away from the midpoint of the strikes and have moved towards the strikes. Trading vega is covered in more detail at Double No Touch Vega.

Fig.2 – Coffee Double No-Touch Options Fair Value w.r.t. Implied Volatility

The double no-touch options price profiles rise from the lower strike until the mid-point of the strikes at which point the profiles fall back to zero at the upper strike. The gradient of the profile is always positive until the midpoint area where it becomes zero and then negative as the profile falls back to zero, i.e. the delta is positive below the midpoint and negative above it. More at double no-touch delta.

Double no-touch gamma is continuous and is zero at the strikes. Elsewhere between the strikes the gamma is always zero or negative.

#### DNT And Vol: Practical Implications

Double no-touch options are a challenge conceptually but as with financial instruments in general the more complex they are the better chance of making money from them. To that end the conventional premium seller should take a good look at buying double no-touch options as they provide many opportunities for taking advantage of time appreciation.

### Fourier Series Method v Tunnel

If the fair value of tunnels can be calculated by subtracting the higher strike binary call from the lower strike binary call option, can double no touch options, the no-touch equivalent of the tunnel, be calculated in the same manner? This would involve aggregating the one-touch put and one-touch call and subtracting the result from 100. Conditional probability says “No”. The formula for double no-touch options is provided below and is based on the Fourier series.

#### Double No Touch Option v One touch Call & One Touch Put

Figure 3 offers the double no-touch options pricing method using the appropriate Fourier series, plus the inappropriate tunnel method of subtracting the aggregate price of the one-touch call and the one-touch put from 100.

Fig.3 – Double No-Touch Options Fair Value Evaluations

What is immediately apparent is that the tunnel method creates double no-touch options fair value that can be negative, thereby immediately eliminating it as a viable evaluation. This is basically because at the lower strike the upper strike one-touch call may have some value, while at the upper strike the lower strike one-touch put may also have value. Therefore at either strike, where the actual value of double no-touch binary options should be 0, they are in fact negative.

#### Formula¹

$\textup{Double&space;No-Touch&space;Options}=2R\pi&space;\left&space;(&space;\frac{S}{K_{1}}&space;\right&space;)^{\alpha&space;}\times$

$\sum_{i=1}^{N}n\left&space;[&space;\frac{1-\left&space;(&space;-1&space;\right&space;)^{n}e^{-\alpha&space;L}}{n^{2}\pi&space;^{2}+\alpha^{2}L^{2}}&space;\right&space;].e^{\left&space;[&space;-0.5\sigma&space;^{2}\left&space;(&space;\frac{\pi&space;n}{L}&space;\right&space;)^{2}-\beta&space;\right&space;]t}.sin\left&space;[&space;\frac{n\pi&space;}{L}\times&space;ln\left&space;(&space;\frac{S}{K_{1}}&space;\right&space;)&space;\right&space;]$

where:

$\textup{L}=ln\left&space;(&space;\frac{K_{2}}{K_{1}}&space;\right&space;)$

$\alpha&space;=\frac{1}{2}-\left&space;(&space;\frac{r-D}{\sigma&space;^{2}}&space;\right&space;)$

$\beta&space;=r+\frac{\sigma&space;^{2}\alpha&space;^{2}}{2}$

and:

$\textup{R}=1\left&space;(&space;\textup{or&space;100&space;in&space;the&space;context&space;of&space;this&space;site's&space;winning&space;bet}&space;\right&space;)$

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

$K_{1}=\textup{Lower&space;Strike/Exercise&space;Price}$

$K_{2}=\textup{Higher&space;Strike/Exercise&space;Price}$

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

$\textup{D}=\textup{Continuous&space;Dividend&space;Yield&space;of&space;the&space;Underlying}$

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

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

$\textup{n}=\textup{number&space;of&space;iterations}$

#### Optimal Value of n ²

With most iterative processes the greater the value n above the more accurate the calculation. The following equation provides the optimal value of n once it has been decided within what level of accuracy ε the calculation is required to provide.

For instance, if the bet is to be accurate to within 0.005 of a point then enter ε = 0.00005 into the above equation. The error term needs to reflect that the binary is the probability multiplied by 100 hence the division by 100. The term |α|  is the absolute value of α.

#### Gibbs Phenomenon ³

When using the above Fourier method of evaluating double no-touch options one may encounter some strange values when time to expiry is very short. Fig.4 illustrates the fair values of double no-touch options when time to expiry is just 0.0000001 days, i.e. less than one tenth of a second! OK, extreme, but it is to make the point.

Fig.4 – Double No-Touch Options and the Gibbs Phenomenon

When the iterations are restricted to n=10 and n=100 the option wildly overshoots the maximum value of 100. This feature is known as the Gibbs Phenomenon³.

#### References

¹Cho H. Hui (1996). One-touch double barrier binary option values

²Stefan Ebenfeld et al (2002). An Analysis of Onion options and Double No-Touch Digitals

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