# Double Binary Options

## Double Binary Options involve two assets and enable the speculator to back his directional views on each while in the process getting a far higher rate of return than by trading them separately.

The four types of Double Binary Options win if:

- Double Call: S
_{1}> K_{1}and S_{2}> K_{2} - CallPut: S
_{1}> K_{1}and S_{2}< K_{2} - PutCall: S
_{1}< K_{1}and S_{2}> K_{2} - Double Put: S
_{1}< K_{1}and S_{2}< K_{2}

where S_{1} & S_{2} are the underlying prices of asset 1 and asset 2, while K_{1} and K_{2} are their respective strike prices. Should either or both of the above conditions not be met at the expiry of the strategy then the strategy settles at zero.

The pricing formulae for the above strategies have been proposed by Kat and Heynon^{1}.

The remainder of this section concentrates on the Double Call with the reader being invited to apply the analysis to the three remaining bets.

## Double Binary Options Pricing

Double Binary Options require both elements of the bet to succeed in order to win, thus the Double Call requires both S_{1} and S_{2} to finish above their respective strikes of K_{1} and K_{2} as illustrated by Table 1

S |
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#### Table 4

In the table the three crosses represent the other three Double Binary Options (CallPut, PutCall and Double Put) winning scenarios and the aggregate probability of all four bets must come to one. In the same way, the two cells above S_{1} > K_{1} must equate to the probability of a vanilla call with underlying S_{1} and strike K_{1}.

### Double Call Price Profiles over Time

Figures 1a-c display how the price profiles change as time to expiry decreases from 25 days to 1 day with the following input:

K_{1 }= K_{2} =100.00

σ_{1} = σ_{2} = 5.0%

ρ = 0

Figure 1a illustrates the 25 days to expiry profile where with S_{1} = S_{2} = 102.00 the value of the Double Call is still less than 100 although both assets are two full points in-the-money. At either S_{1} or S_{2} = 98.00 the strategy has a fair value of 5.65.

**Figure 1a**

**Figure 1b**

As theta kicks Figure 1b shows that with 5 days to expiry at S_{1} = S_{2} = 102.00 the Double Call has ‘maxed out’ at 100 while at S_{1} and/or S_{2} = 98.00 the strategy is worthless.

The lower profile of Figure 1c describes a very narrow corridor where fair value does not equal either 0 or 100.

**Figure 1c**

At expiry the profile is discontinuous with two separate plateau, one at a value of 100 where S_{1} > K_{1} and S_{2} > K_{2} and other at zero at all other combinations of S_{1} and S_{2}.

This strategy can offer particularly high gearing for speculators who wish to combine forecasts on two separate assets within one instrument. A Double bet on two racehorses running in different races would be consistent with a Double Call with rho = 0, although a horserace aficionado would no doubt claim that factors such as both horses favouring the same track conditions (‘going’), same jockeys, weather etc. may well impact on rho.

Doubles in general require the input of three individually assessed variables, i.e. σ_{1}, σ_{2} and ρ, which could quite possibly cause considerable variation in fair value from one market-maker to another. Consequently it would be unsurprising should market-makers build in a wider bid/ask spread than would be their norm. This would not only reflect the market-maker’s lack of assuredness in evaluating the fair value of the strategy, but also the subsequent risk management of the position when one of those three variables, rho, is totally unhedgeable.

A cross section of the graph is shown in Fig 2 with S_{2} fixed as per legend. Where S_{2} = 100 the profile is limited to a maximum value of 50 since, even as S_{1} → ∞, the profile resembles that of a $100 binary call with underlying S_{2}=100, which is worth 50.

**Figure 2**

### Double Call and Theta

In Figures 3a-c the area where S_{1} < $100 and/or S_{2} < $100 has a negative theta as the value of the out-of-the-money Double Call falls over time. As the time to expiry falls to 1

**Figure 3a**

day, the theta becomes increasingly negative where S_{1} and S_{2} are just under K_{1} and K_{2} respectively. Alternatively the theta is sharply positive immediately above S_{1} = S_{2} = $100. Subsequently theta returns to zero as both S_{1} and S_{2 }are so far in-the-money that the Double Call is 100 and can go no higher.

#### Figure 3b

#### Figure 3c

The thetas, as illustrated by the contours in Figures 3, determine for those that invest in order to take in time decay which areas to buy and/or sell. For example, in Figure 3b the ‘hot spot’ is a semi-elliptical shape with theta (according to the legend) of between 10-15. This area shows both S_{1} and S_{2} to be in-the-money, so although the positive theta might give it away, it is necessary to buy the Double Call at this point. But this is a transitory trade. Should either S_{1} or S_{2} slip out-of-the-money then the theta turns sharply negative and the criterion for the trade is compromised; alternatively if either S_{1} or S_{2} travel deeper in-the-money then yet again the theta falls away from the ‘hot spot’ and the trade should be exited.

Using vanilla binary calls creates issues as the risk reversal is always a threat, with Doubles the risk is even greater.

### Double Call and Vega

In the following examples the vega is illustrated for asset1 with σ_{2} remaining constant at 5%.

#### Figure 4a

The vega for asset 1 with asset 2’s implied volatility constant at 5% shows positive vega below the strike and negative above. If a cross-section of the graph was taken at S_{2} = $100 then the profile would be very similar to that of the vanilla Binary Call vega.

As time passes the absolute value of the vega does not change a great deal (all three illustrations have the scale remaining between ±0.6) although the profiles become narrower with respect to S_{1}.

#### Figure 4b

#### Figure 4c

### Double Call Deltas

Double Call deltas are always positive. Each asset has its own delta and Figures 5a-c shows deltas for asset 1. At S_{1} = $100 the same steepling effect takes place as with an at-the-money Binary Call and yet again the cross-section at S_{2} = $100 resembles the delta of the vanilla Binary Call.

#### Figure 5a

As one can see from the changes in the scales of the graphs, the delta causes the same problems to the market-maker as time to expiry passed with delta-neutral hedging becoming increasingly difficult over the last day where at the beginning of the day the delta for the at-the-money is already ±1.5 and rising sharply.

#### Figure 5b

#### Figure 5c

### Double Call Gammas

The gamma for S_{1} is zero or positive while S_{1} < $100 and positive or negative for S_{1} > $100. At S1 = $100 gamma is zero.

#### Figure 6a

As ever gamma (along with theta and vega) has the usual risk reversal issues.

#### Figure 6b

#### Figure 6c

### Double Call and Rho

The following section holds S_{2} = 100.00 while S_{1} and rho are the variables. The first characteristic of Figure 7a-c is that the fair value of the Double Call never exceeds 50 and this is because the price of one of the assets has been constrained to the strike price of 100.00. If S_{1}→∞ the strategy would still only be worth a maximum of 50.

#### Figure 7a

At S_{1} = 100.00 and ρ = ―1.0 the Double Call is worthless since any upward move in S_{1} is associated with an equal downward move in S_{2}.

At S_{1} = 100.00 and ρ = 0.0 a cross-section of the profile would be similar to the binary call with S = K = 100.00.

Another feature of the interaction of time and rho is that as time elapses the effect of rho diminishes. This is reflected in the lower of the diagrams where the ‘overhead’ view of the price shows a narrower and more vertical price profile as rho ranges from ―1 to +1. As t→0 the profile for ―1 ≤ rho ≤ +1 becomes that of a vanilla binary call where S = K = 100.00.

What the profiles of Figures 7 illustrate is that rho itself has a ‘delta’ but since rho itself is unhedgeable and is a conceptual value provided by statistical analysis (much as historical volatility) such a delta has limited value. By hedging the pair of underlying in a delta-neutral manner is as close as one is going to get in terms of a simple hedge of rho although by using options, both conventional and/or binary, an avid ‘rho-neutral’ speculator may be able to ‘flatten out’ more efficiently, (maybe!).

#### Figure 7b

#### Figure 7c

### Eachway Double Call

Eachway versions of Doubles are yet another variation on a theme. Yet again should the settlement prices be 0, 40 and 100 with strikes K_{11}, K_{12} and K_{21}, K_{22} the Eachway Call would then be:

Eachway Double Call = 0.4 x Double Call(K_{11},K_{12}) + 0.6 x Double Call(K_{21},K_{22})

#### Figure 8a

As time to expiry erodes the curved contours of Figure 8a transform to the square contours of Figure 8c where there is the clear outline of the three separate plateau defining the losing bet, the ‘place’ with a settlement price of 40, and the outright win where S_{1} > K_{21} and S_{2} > K_{22}.

#### Figure 8b

#### Figure 8c

### ‘Double’ Applications

For the most part, the ‘Doubles’ will be used for purely speculative intent, i.e. gold is above $1775 (current price $1750) and Brent crude is above $110 (current price $110) in 25 days time. Assuming volatilities of 20% and 25% for the respective products, the price of the gold vanilla binary call would be 38.32 and the oil binary call 23.82. A $10/pt bet would create a maximum downside of $621.40 with a potential profit $1,378.60 providing a 122% profit.

#### Figure 9a

Assuming a gold/oil correlation coefficient of rho = 0 the same Double Call would be worth 9.13 meaning that a trader can buy $68.06/pt and have the same downside risk as the above two individual calls. The upside is $6,185 creating a 895% profit. The downside is that with the individual bets either can win independently creating an overall profit while with the Double Call both must win to avoid losing the outlay. Yet again the choice boils down to the trader‘s strength of opinion and risk profile as to which alternative they choose.

The Gold/Oil example lends itself to another use. Above rho is assumed as zero, with the implied volatilities of Gold and Oil being taken from the vanilla binary options market, which will in turn have likely to have derived from the conventional options market. If one took the view that Gold and Oil are positively correlated then one might buy the Double Call and sell the individual Gold and Oil Binary Calls.

At $1750 the Gold 1775 Binary Call is worth 38.32 while at $110 the Oil $115 Binary Call is worth 23.82, plus the Double Call is worth 9.13. A purchase of 100 Double Calls would require an investment of $9,126.98 which could be roughly retrieved by a sale of 15 Gold and Oil Binary Calls:

100 Double Calls @ 9.13 *less* 15 Gold Calls @ 38.32 *less* 15 Oil Calls @ 23.82 = -19.1

Assuming a tick size of $10/pt the P&L profile is illustrated in Figure 9b.

#### Figure 9b

If Gold and Oil had a rho = 1 then the related price action of Gold and Oil would run along a straight line from Gold = $1,500, Oil = $105 to Gold = $2,000, Oil = $115, i.e. a straight line running from the bottom left to the top right of the lower graph of Figure 9b. In effect that combination of prices covers the profitable areas of the P&L profile and as such the strategy would be a worthy one for trading rho.

Another interesting application for these Doubles is as ‘out-performance’ bets. For example, the FTSE100 is trading at 6,120 and Vodafone is £1.36. A trader fancies the market up but also thinks that Vodafone will underperform the market. The implied volatility of FTSE100 is 15% and Vodafone 20% and there are no ex-dividend dates in the offing. The trader decides to buy the 6,200/$1.40 CallPut with 30 days to expiry. This CallPut wins providing at expiry FTSE100 is above 6,200 and Vodafone is below £1.40.

The strategy is worth 26.25 on the assumption that there is a zero correlation coefficient, but in this case, rho would not equal zero. Why? For a start Vodafone accounts for 6%, say, of the FTSE100 index, but furthermore since there is likely to be a positive correlation with many of the individual constituents of the FTSE100 index, rho will be higher than 6%. Presuppose FTSE and Vodafone have a rho of 0.6 then the value of the bet now falls to 18.52 since the probability of Vodafone closing above £1.40 conditional on FTSE100 closing above 6,200 has now risen creating a greater likelihood of a losing bet. Should FTSE gain 6,200 and Vodafone remain under £1.40 the strategy returns a 340% profit.

### Summary

1. They are not path dependent and therefore lend themselves to comparison with individual binary calls and puts. Indeed, it is quite feasible for a trader with a ‘Double’ position to hedge each side with binary calls and puts.

2. Rho is a critical element in the evaluation of double bets and the measure of the correlation coefficient provides problems comparable with evaluating historic volatility, i.e. what time frame is appropriate and how often within that time frame should samples be taken.

3. Double Calls and Puts are strategies for the more assertive trader whom believes they can forecast the future price level of two assets and thus reap the more favourable odds.

4. ‘Doubles’ can also be used for backing the view that one asset that is a member of a wider group of assets may under- or out-perform that basket of assets.

^{1} Heynan, R.C., and H.M.Kat (1996):”Brick by Brick,” *Risk Magazine*

^{2} Espen Gaarder Haug, ‘The Complete Guide to Option Pricing Formulas’ ISBN 0-7863-1240-8