# Up and In Binary Put Options definition and price profile

Up and In binary put options could well be one of the most useful speculative instruments as well as one of the most under-utilised of instruments. This strategy is perfect for the chartist who wants to take a position based on a resistance level being touched but holding as in the condition where the strike (K) is the same as the barrier (B).

In the first illustration (Fig.1) for the Indian NSE Nifty50 Index the example is of a up and in binary put options with barrier at 5,600 and strike at 5,500. The knock-in profile (black) shows the Nifty50 must rise to touch the barrier at 5,600 at which point the strategy transforms into a vanilla binary put option (red) with strike at 5,500. If the Nifty50 does not touch the barrier by expiry the strategy settles at zero.

The knock-in and the binary put profiles intersect at the barrier. If the barrier is pitched at a major level of resistance it provides a compelling instrument for chartists to back their view that the level will hold and the Nifty50 bounce off it. In this instance the chartist is invited to speculate on whether the index will bounce 50 points lower to expire below the strike at 5,500.

## Up and In Binary Put Options (K<B)

Figure 2 shows up and in binary put options with 2, 10 and 50 days to expiry and 25% implied volatility. With just two days to expiry the knock-in is worth less than 1.0 below the index level of 5460, 140 ticks below the barrier. This is explained by the remote probability of the Nifty50 reaching the barrier within two days, but should it do so the binary put at the barrier is only worth 16.74. As the time to expiry increases to 10 days and 50 days at the Nifty50 level of 5,600 the binary put options increase in value as the probability of the Nifty50’s being lower than 5,500 at expiry increases. This requires the knock-in to increase in value also since at the barrier the binary put option and the knock-in must be equal in value.

The attraction of this strategy to the chartist who believes that the Nifty50 5,600 level is a critical resistance level is the return on offer. With 10 days to expiry at the Nifty50 level of 5450, i.e. 150 points below resistance, the knock-in is worth 13.83 so if the Nifty50 did rise to 5,600 and collapse back below 5,500 at expiry then this strategy generates a profit of 86.17, a percentage return over 10 days of 623%.

Figure 3 illustrates the same up and in binary put option but over a range of implied volatilities, the time to expiry fixed at 5 days. An out-of-the-money binary put nearly always1 has positive vega meaning that an increase in volatility increases the value of the binary put 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 put 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 up and out binary put being knocked-out and settling at zero.

## Up and In Binary Put Options (K=B)

Figures 4 & 5 show the barrier and strike pitched at the same level of 5,600. 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 resistance the bet could simply be expounded as:”Will the resistance level hold (if hit)?”. So the down and in binary put (K=B) is firstly a speculative decision on whether the Nifty50’s are rising to the resistance/barrier/strike at 5,600, with a conditional punt that should it do so, then the resistance level will hold and the Nifty50 will be below 5,500 at the expiry of the up and in binary put.

The advantages to the chartist are:

1. Back the resistance level with a minimum risk strategy. Often resistance levels are pivotal points from which the underlying may move from aggressively in either direction. A chartist who is selling futures at the barrier could be wearing a significant loss in a very short period of time if the resistance level is targeted and comes under sustained and heavy buying.
2. Alternatively, should the resistance level hold and bounce aggressively from it then selling at the barrier may be almost impossible. This strategy alleviates the need to climb on board a rapidly falling market as the buyer of the knock-in has automatically received a short position in the market via his converted put.
3. 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:

1. On triggering the barrier a market-maker short this strategy needs to sell double the long underlying position being held already as a hedge in the rising market. If the market bounces aggressively off the resistance then selling back the hedge and selling more for the naked short binary put position could be extremely difficult.
2. But the silver lining to this particular cloud is if the Nifty50 gaps up through the barrier on the open one morning. The position is now a short put which is getting cheaper; furthermore the market-maker is still long the hedge from below 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 Nifty50 up to the resistance level with a large sell order at the resistance level.

## Up and In Binary Put Options (K>B)

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

Here the knock-ins intersect at the Nifty50 level which equates to B ― (K ― B) and has the properties of a binary call with strike B ― (K ― B).

The gearing of the up and in binary put 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 resistance between the strike and the barrier suggesting that the strike has a level of protection below 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 up and in put resembles a one-touch call with the pay-off adjusted so that instead of a winning price of 100, the one-touch price at the barrier equates to the binary put premium. It is not so. The knock-in is graphically displayed in Figure 8 alongside the one-touch call with barrier/strike at 5,600 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 Up and Out Binary Put section where B<K then the profile is simple: since the strike is above 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 Call yet this option would be priced as the binary put less the knock-in, so there is a clear discrepancy. More under up and out binary put 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 above the strike. Increasing implied volatility subsequently has a disproportionate effect on the value of the binary put as the probability of the underlying rising further outweighs the impact of the underlying travelling over the strike.

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