# Down and In Binary Call Options

## Down and In Binary Call Options could well be one of the most useful speculative instruments as well as one of the most under-utilised of instruments.

Down and in binary call options are perfect for the chartist who wants to take a position based on a support level being touched but holding as in the condition where the strike (K) is the same as the barrier (B).

## Down and In Binary Call Options Structure

In the first illustration (Fig.1) for the S&P500 the example is of down-and-in binary call options with barrier at 1,100 and strike at 1,150. The knock-in profile (black) shows the S&Ps must fall to touch the barrier at 1,100 at which point the strategy transforms into a vanilla binary call option (red) with strike at 1,150. If the S&Ps do not touch the barrier by expiry the strategy settles at zero.

The knock-in and the binary call profiles intersect at the barrier. If the barrier is pitched at a major level of support it provides a compelling instrument for chartists to back their view that the level will hold and the underlying bounce off it. In this instance the chartist is invited to speculate on whether the index will bounce 50 points to expire above the strike at 1,100.

### Down and In Binary Call Options (K>B)

Figure 2 shows the down-and-in with 2, 10 and 50 days to expiry and 25% implied volatility. With just two days to expiry the knock-in is worthless at every underlying price because at the barrier the binary call option is already worthless. As the time to expiry increases to 10 days and 50 days at the S&P 500 level of 1,100 the binary call options increase in value as the probability of the S&P’s being higher than 1,150 at expiry increases. This requires the knock-in to increase in value also since at the barrier the binary call option and the knock-in must be equal in value.

The attraction of this strategy to the chartist who believes that the S&P500 1,100 level is a critical support level is the return on offer. With 10 days to expiry at the S&P level of 1125, i.e. just 25 points above support, the knock-in is worth 5.19 so if the S&Ps did fall to 1,100 and rally back above 1,150 at expiry then this strategy generates a profit of 94.81, a percentage return over 10 days of 1,827%.

Figure 3 illustrates the same down-and-in binary call option but over a range of implied volatilities, the time to expiry fixed at 5 days. An out-of-the-money binary call nearly always^{1} has positive vega meaning that an increase in volatility increases the value of the binary call which in turn demands an increase in value of the knock-in; the barrier also has a greater chance of being touched with higher volatility while the binary call has a better chance of success with higher volatility in the underlying. This can be contrasted with the knock-out where a higher volatility increases the chance of the binary call being knocked-out and settling at zero.

### Down and In Binary Call Options (K=B) aka ‘Support’

Figures 4 & 5 show the barrier and strike pitched at the same level of 1,100. If there was ever a strategy designed for a specific speculator then this bet has to be it. If the barrier (and strike) is pitched on a level of support the bet could simply be expounded as:”Will the support level hold (if hit)?”. So the down-and-in binary call (K=B) is firstly a speculative decision on whether the S&P’s are falling to the support/barrier/strike at 1,100, with a conditional punt that should it do so, then the support level will hold and the S&P’s be above 1,100 at the expiry of the down-and-in binary call.

The advantages to the chartist are:

- Back the support level with a minimum risk strategy. Often support levels are pivotal points from which the underlying may move from aggressively in either direction. A chartist who is buying futures at the underlying could be wearing a significant loss in a very short period of time if the support level is targeted and comes under sustained and heavy selling.
- Alternatively, should the support level hold and bounce aggressively from it then buying at the barrier may be almost impossible. This strategy alleviates the need to climb on board a rapidly rising market as the buyer of the knock-in has automatically received a long position in the market via his converted call.
- The market does not need to be monitored by the naked buyer of the strategy.

The above points outline the disadvantages of this bet to the market-maker:

- On triggering the barrier a market-maker short this strategy needs to buy double the short underlying position being held already as a hedge in the falling market. If the market bounces aggressively then buying back the hedge and buying more for the naked short call position could be extremely difficult.
- But the silver lining to this particular cloud is if the underlying gaps down through the barrier on the open one morning. The position is now a short call which is getting cheaper; furthermore the market-maker is still short the hedge from above the barrier. So not all doom and gloom for the market-maker.

All-in-all, if a big ‘hedgie’ starts buying the above knock-in then it might be a reasonable bet that the hedge fund will be doing its utmost to take the S&P’s down to the support with a large buy order at the support level.

### Down and In Binary Call Options (K<B)

The following two illustrations round off the different down and in binary call options with examples of the barrier being set at a higher level than the strike.

Here the knock-ins intersect at the S&P level which is equates to B + (B – K) and has the properties of a binary put with strike B + (B – K).

The gearing of the down-and-in binary call option where B>K does not present the opportunities that the same option with K>B does. This strategy could and would be useful to the chartist (again) who believes that there is a strong level of support between the strike and the barrier suggesting that the strike has a level of protection above it. But even then that would probably not suffice for the trader looking for gearing since the entry level of this strategy is high.

### The Knock-In

The knock-in component of the down-and-in call resembles a one-touch put with the pay-off adjusted so that instead of a winning price of 100, the one-touch price at the barrier equates to the call premium. It is not so. The knock-in is graphically displayed in Figure 8 alongside the one-touch put with barrier/strike at 1,100 and there is a clear disparity.

The knock-in evaluation is generally based on a ‘heat reflection’ theory(!), which mathematically, is totally beyond me, but should any punter/financial engineer wish to contribute please comment below.

In the Knock-Out section where K<B then the profile is simple: since the strike is below the barrier then should the underlying not touch the barrier it will settle at 100, else 0. This is nothing less than 100 less than the One-Touch Put yet this option would be priced as the binary call less the knock-in, so there is a clear discrepancy. More under down-and-out binary call options.

^{1} As implied volatility increases the vega falls to zero and should the implied volatility increase beyond this point the vega will turn negative as the strike constrains the option value to 50 below the strike. Increasing implied volatility subsequently has a disproportionate effect on the value of the binary call as the probability of the underlying falling further outweighs the impact of the underlying travelling over the strike.