This post is published by Hamish Raw of https://hamishraw.com/

Binary Spreads aka Binary Pairs enables the speculator to back the view that on a certain date in the future the price of asset 1 will be K higher or lower than the price of asset 2. The target difference K thus assumes the role of the strike price and is independent of the overall levels of the two asset prices. A strike price of zero then indicates that one asset must have a higher price (if call) than another asset; conversely, if the Binary Spread is a put then one asset must have a lower asset than another. Immediately one can glean that the Binary Spread has a great deal in common with a vanilla binary option in that a Put Binary Spread is 100 less the same strike Call Binary Spread. **The following analysis of the Binary Spread considers the Call Binary Spread.**

Most derivatives traders have conducted a spread trade at some point in their trading career; these spread trades generally are futures trades and the most common are the spreading of one short-term interest rate month against another month of the same contract, e.g. sell 1 Dec Eurodollar 98.30, buy 1 Mar Eurodollar 98.35. If the spread widens the seller gains by $25 per spread per tick profit. If the current spread is trading at -5 and a Binary Spread has a strike of -5 ticks then the price would be roughly 50. Should the spread widen from -5 this would generate a settlement price of 100, in effect roughly doubling the speculators money. Therefore in terms of a straight comparison between a futures spread and a Binary spread, the latter can provide significantly higher gearing along with a limited risk scenario.

The formula for the Binary Spread is from Thijs van den Berg^{1} founder of www.sitmo.com

## Pricing

Figures 1a-c illustrate the price profiles over time of two assets, S_{1} & S_{2}, with a strike price of 0.00. Each plane is viewed from the side and overhead. The Rho is set at zero meaning that the two assets are totally unrelated to each other in terms of price movements.

The following illustrations are in terms of asset 1 (S_{1}) and the Call Binary Spread is profiled. Should the same price profile be in terms of asset 2 (S_{2}) then the profile would be presenting the Put Binary Spread.

Figure 1a profiles the 50 & 25 day Binary Spread which is relatively shallow compared to Figs 1b-c. Following the plane at S_[1] = S_[2] the price is 50 since S_[1]-S_[2]=0 i.e. the strike. This relationship is maintained through Figs 1b-c and reflects the likelihood of asset 1 being higher or lower than asset 2 is 50:50.

Figure 1b illustrates the 10 & 5 day profiles where the probability of S_{1} > S_{2} is represented by a narrower and steeper price profile, which in turn is accentuated by Figure 1c.

In Figure 1c where S_{1 }= $98 and S_{2} = $102 the bet is worthless while at S_{1} = $102 and S_{2} = $98 the bet is valued at 100. At expiry the diagonal lines become just one line with the final settlement price of the Binary Spread dependent on the particular terms of the strategy, i.e. where the final settlement prices of S_{1} & S_{2} are exactly 0.00 apart the terms may well define the strategy as a ‘dead heat’ and settle at 50.

If one were to take a cross-section of the graphs from S_{1} = $98, S_{2} = $102 to S_{1} = $102, S_{2} = $98, i.e. from bottom left to top right of each graph, the outline of the price profiles would resemble that of a single asset binary call.

The following analysis of the greeks determine how time, volatility and rho affect the above price profiles.

## Binary Spreads Over Time

Figures 2a-c graphically display the theta of the above price profiles. The theta is always zero when the difference between the two assets is 0.00, the strike price. When the difference is below 0.00 the theta is negative or zero, while a difference greater than zero offers a positive or zero theta. The absolute value of the theta increases as the time to expiry decreases so that the scale of Figure 2a is just ±2.0 while the 1 day theta of Figure 2c is ±50.0. But as ever the theta of an option grossly exaggerates the actual decay of the premium as time to expiry approaches zero.

As with writing vanilla binary options the same considerations apply; writing an out-of-the money Binary Spread in order to take in time value can spectacularly backfire should the option turn in-the-money since now instead of taking premium in the premium has a positive theta and is now increasing in value over time.

## Binary Spreads and Volatility

The vega in this section relates to a change of implied volatility of asset S_{1 }from 5% with asset S_{2} remaining constant at 5%.

The vega is positive (or zero) with S_{1} – S_{2} < K (0.00) and negative where S_{1} – S_{2} > K (0.00) so yet again a cross-section of the graph running from S_{1} = $98, S_{2} = $102 to S_{1} = $102, S_{2} = $98 would resemble the profile of a binary call vega on a single asset.

The scale of the vega does not change remaining at ±0.04 for 25 days to 1 day. As time decay reduces the premium when out-of-the-money and increases it towards 100 when in the money, there is a constraint placed on the vega which does not exist for conventional options. The effect that does take place is a concertinaing of the vega profiles over time, as progressively illustrated by Figures 3a-c. with vega always zero where S_{1} = S_{2}.

## Binary Spreads & Deltas

This section plots the delta of S_{1} with the delta of S_{2} being the (negative) put delta.

As with binary deltas in general increasing the time to expiry lowers the absolute values of call and put deltas and the Binary Spread is no different. The at-the-money delta with 25 days to expiry is 0.2200 where S_{1} = S_{2} = $98 falling to 0.2113 where S_{1} = S_{2} = $102. With just 1 day to expiry these two deltas have increased to 1.0999 and 1.0567 respectively. The delta is the familiar Dirac delta which requires the sum of the cross-section below the profile to aggregate to 1. Therefore as time approaches zero and the cross-section profile approaches zero, the delta takes on the steepling effect as the delta tends to infinity.

Figures 4a-c offer delta profiles as time falls from 25 to 1 day, while Figure 4d provides the delta of the Put Binary Spread which always is the negative of the call delta.

Hedging the Delta exposure

The below example in Table 1 provides an example of a short 100 Binary Spreads with buying 48 future spreads against it. The position has a strike of 0.00 with both futures, S_{1} & S_{2}, price 100.00. The Binary Spread is initially worth 50, one point being worth $10 with Contract Value = 100 x $10 = $1,000.

At S_{1} = $99.40, S_{2} = 99.60 the Binary Spread has fallen to 40.40 as the spread is now 99.40 – 99.60 = ―0.20, i.e. the Binary Spread is now out-of-the-money. The future spread makes a $20,000 loss against the $9,600 profit of the short 100 Binary Spreads.

At S_{1} = 100.60, S_{2} = 99.60 the Binary Spread has risen to 88.63 as the spread has now risen to 100.60 – 99.60 = +1.00, i.e. the Binary Spread is now well in-of-the-money. The future spread makes a $100,000 profit against the $38,630 loss of the short 100 Binary Spreads.

Futures Price | S1=99.40S2=99.60 | S1=100S2=100 | S1=100.60S2=99.60 |

Binary Spread | 40.40 | 50.00 | 88.63 |

Delta | 0.4709 | 0.4820 | 0.2312 |

Call Profit | 100 x (50.00-40.40) x $10 = +$9,600 | N/A | 100 x (50.00-88.63) x $10 = -$38,630 |

S1 Futures P&L | +48.2 x ((99.40-100.00) x $10) / 0.01 = -$28,920 | N/A | +48.2 x ((100.60 – 100.00) x $10) / 0.01 = +$28,920 |

S2 Futures P&L | -48.2 x ((99.60-100.00) x $10) / 0.01 = +$19,280 | N/A | -48.2 x ((99.60 – 100.00) x $10) / 0.01 = -$19,280 |

P&L | +$9,600 – $28,920 + $19,280 = -$40 | N/A | -$38,630 + $28,920 + $19,280 = $9,570 |

48.2 spreads might be an issue so if the futures spreads were rounded to 48 the bottom line P&L becomes $0 and $9,370.

## Binary Spreads & Gammas

The above example of Table 1 presents the advantage of going short at-the-money binary calls and purchasing futures against it.

Futures Price | S1=99.40S2=99.60 | S1=100S2=100 | S1=100.60S2=99.60 |

Binary Spread | 40.40 | 50.00 | 88.63 |

Delta | 0.4709 | 0.4820 | 0.2312 |

Gamma | 0.1328 | -0.0003 | -0.3388 |

At inception the trade was marginally long gamma (Gamma = -100 x -0.0003 = 0.03) while at S_{1} = 99.40, S_{2} = 99.60 the position is short gamma ( Gamma = -100 x 0.1328 = -13.28) which explains the $40 loss. On the other hand as the spread ‘went out’ to 1.00 the Binary Spread became increasingly long gamma ( Gamma = -100 x -0.3388 = 33.88) so that at S_{1} = 100.60, S_{2} = 99.60 the net delta has become 48.2 – 23.12 = 25.08 future spreads, explaining the increased profit.

Figures 5a-c illustrate the Binary Spread gamma with the usual variables.

## Binary Spreads and Rho

Figures 6a-c illustrate how, with S_{2} held constant at 100.00, rho and time to expiry act on the price. When the Binary Spread is at-the-money the price is always 50, irrespective of rho or time to expiry. As time to expiry decreases the influence of rho also decreases, so that with minimal time to expiry (Figure 6c) rho virtually has zero effect.

i. Rho = 1

When rho = 1, irrespective of time to expiry, when S_{1 }> S_{2} (100.00) then the price is 100. With rho = 1 the price of S_{2} will move exactly in tandem with S_{1} since all points on the distribution lie on a straight line. Therefore if S_{1} rises then S_{2} will rise in proportion: if S_{1} falls then S_{2} falls in proportion. This means that S_{1} will always be higher than S_{2} therefore the price will always be 100.

Likewise should S_{1 }< S_{2} this relationship will always exist and the Binary Spread will always be worth 0.

ii. Rho = 0

Figure 6 illustrates the Binary Spread (rho = 0) with the Binary Call so although one might consider that the two profiles should match, the Binary Call is worth less when out-of-the-money and more when in-the-money. The Binary Spread profile would actually be consistent with an equivalent Binary Call but with a higher implied volatility. This subsequently provides the interpretation for the disparity between the two profiles since the Binary Call value is based on just one volatility while the Binary Spread value requires the input of the volatilities of each underlying asset.^{2}

iii. Rho = ―1

The profile of the Binary Spread when rho = ―1 is a stepped increase in price. When S_{2} is initially held at 100.00 while S_{1} varies at each level of S_{1} there becomes a series of prices of S_{2} generating different win/lose scenarios.

Example 1: S_{1} = S_{2} = 100.00, K = 0.00

If S_{1} rises by 0.01 then S_{2} will fall by a multiple of 0.01, in this case the multiple is assumed at 1.0, and consequently S_{1} = 100.01, S_{2} = 99.99 and S_{1} > S_{2} therefore the strategy wins. In this instance, as both S_{1} and S_{2} have starting points at 100.00 the price movement is irrelevant as any upward move in S_{1} will result in decrease in price of S_{2}.

Conversely, if S_{1} falls then S_{2} rises meaning that whatever the amount of price movement S_{1} < S_{2} and the strategy always loses.

Example 2: S_{1} = 100.20, S_{2} = 100.00, K = 0.00

In this instance the amount of the price movement becomes relevant. If the price movement is 0.01 then S_{1} = 100.21, S_{2} = 99.99 so S_{1} > S_{2} and the strategy wins. Alternatively, S_{1} = 100.19, S_{2} = 100.01 and still S_{1} > S_{2}. But if the price movement is 0.1 then the two combinations of prices are S_{1} = 100.30 : S_{2} = 99.90 and S_{1} = 100.10: S_{2} = 100.10, i.e. the strategy is now at-the-money.

If the price movement is greater than 0.10 then it is possible for this strategy to lose should S_{1} fall.

In effect when rho = ―1 a range of new variables comes into play and this is exacerbated when the value of the price movement is large enough to distort the value of the strategy, i.e. should S_{1} =102.00, S_{1} = 100.00 and K = 0.00 then the strategy should be a certain winner. Yet if the assumption was that the price movement was in excess of 1.00 then the strategy becomes a loser. The conclusion may well be to avoid trading a strategy with rho = ―1!

## Eachway Spreads

As with any European binary option, i.e. an option that has no barriers or ‘one-touch’ features, it is possible to introduce further strikes and create staggered settlement prices. Figures 9.2.8 illustrate profiles of an Eachway Spread with strike prices of K_{1} = ―1.5 and K_{2} = 1.00 with settlement prices of 0, 40 and 100.

Figure 8ba has five days to expiry and takes on the profile of a plain Binary Spread. With just 1 day to expiry the two separate strikes are now exerting their influence separately so that there is clear definition of the three settlement prices. While with 0.01 days to expiry the shape of the Eachway Spread is a great deal more concise.

The greeks of the Eachway Spread have lower absolute values as one would imagine based on the Eachway Call and subsequently is a less risky trade than the Binary Spread for all concerned.

### Applications

Binary Spreads and Eachway Spreads have applications wherever spread trading is currently being undertaken. Clearly there is no need for the two assets to have any correlation but it is generally common for spread traders to operate with assets that do have positive correlation.

i. Interest Rate Futures

Probably the highest volume of spread trades takes place between interest rate futures with traders operating down a specific currency curve or between different currency curves.

Short-term interest rate futures, e.g. Eurodollars, Euribor, Short Sterling, Oz 90-Day Bills, EuroSwiss are home to spread traders trading one month against another. Spreads, butterflies and condors are all spreading strategies within this market so subsequently it would be perfectly feasible for there to be Binary Spreads in these markets.

It is not unusual for traders to trade same currency 2, 5 and 10 year bonds against each other, while a more risky trade might be spreading the Bund against the US 10 year Note. In the 1990s when Italy was joining the Euro convergence took place and funds actively traded Italian BTP calls against Bund calls, so this would be a clear example where a Binary Spread could be applied.

ii. Stocks and Shares

Trading that one company outperforms another is a traditional form of spread trading. The shares are generally in competing companies and within a specific sector so the correlation is positive.

iii. Commodities

Oil is an obvious potential candidate for Binary Spreads with trading of the ‘crack’ spread already prevalent. This involves the trading of the price of oil between two levels of the production process.

iv. Shares v Commodities/Shares v Indices

Trading gold company shares against bullion has been a long-standing play, as to has trading a share in an index against the index itself. Both are forms of hedging based on the fact that the pair of assets will have positive correlation.

v. Reverse Yield Gap

This spread has a negative correlation and is based on the yield of an index of shares against an interest rate.

All the above examples would be potential candidates for Binary Spreads and Eachway Spreads. The advantages as ever revolve around the limited risk scenario of binary options plus the fact that a Binary Spread would need just one margin call, as opposed to the two margin calls of the Bund v US 10 year Note spread.

**Summary**

There are now Binary Spread platforms trading on the internet which probably underlines their relative simplicity, practicality and commerciality. These platforms, as with the current crop of internet-based binary options platforms, offer very short-term strategies. It is hoped that this section has broadened the readers outlook as to the potential uses, trading opportunities, pricing and risk management techniques so that when longer term Binary Spreads become available the reader will be able to form more knowledgeable assessments as to the fair value of a Binary and Eachway Spread strategy.

Section 9.2.9 on applications demonstrated current spread markets where traders are offsetting one instrument with another in the hope of gaining a profitable edge. In effect these traders are doing nothing but trading the correlation coefficient. The Binary Spread offers the binary options trader to make a critical judgement on the price of the strategy in the marketplace by forming solid and researched views on the volatility of the individual assets. This implied volatility assessment is likely to be influenced heavily by the implied volatility of the individual conventional options of each asset.

Binary Spreads and Eachway Spreads offer a new dimension to the market; in effect they are not ‘reinventing the wheel’ as spread trading is already in existence. What these binary strategies offer is the ability to get involved in these markets with limited downside risk plus only one initial margin call or premium charge. Furthermore, since spreading the underlying requires brokers for two individual ‘legs’ the cost of trading Binary Spreads and eachway Spreads is likely to be relatively cheap also.