Duke of York

Duke of YorkThe grand old Duke of York, He had ten thousand men.

He marched them up to the top of the hill

And he marched them down again.

And when they were up, they were up;

And when they were down, they were down.

But when they were only halfway up,

They were neither up nor down!

Duke of York GreeksBelow 1st StrikeBetween 1st & 2nd StrikesBetween 2nd & 3rd StrikesBetween 3rd & 4th StrikesBetween Centre StrikesBetween 5th & 6th StrikesBetween 6th & 7th StrikesBetween 7th & 8th StrikesAbove 8th Strike
Delta+ve+ve+ve+ve+ve -ve-ve-ve-ve-ve
Gamma+ve-ve +ve-ve +ve-ve +ve-ve+ve -ve+ve -ve+ve -ve+ve
Theta-ve+ve -ve+ve -ve+ve -ve+ve-ve +ve-ve +ve-ve +ve-ve
Vega+ve-ve +ve-ve +ve-ve +ve-ve+ve -ve+ve -ve+ve -ve+ve

The binary options Duke of York strategy is named owing to the price profile of the strategy which sees the value of the instrument rise to a peak then fall back again as the underlying asset price rises from below the lowest strike to above the highest strike.

Duke of York Expiry Value

The binary option’s Duke of York strategy is a volatility trade, as is the standard binary options eachway tunnel, but with the Duke of York there are a further four strikes providing five different settlements levels, not counting the ‘dead heats’.

The example offered in Figure 1 is of the British £ v US$ Duke of York strategy which settles at 0, 10, 30, 60 on the way up ‘the hill’ and 100 at ‘the top of the hill’, and of course 60, 30, 10 and 0 on the march back down.

Duke of York

Fig.1 – GBP v USD Binary Options Duke of York Price Profile at Expiry

Duke of York Over Time

Figure 2 illustrates the Duke of York price profiles over time with 10% implied volatility. Midway between the central strikes the strategy could be considered a theta play when there are 25-days and 8-days to expiry since time decay has a positive effect on the price over a wide range of the underlying. When there are 3-days to expiry at 1.60 the Duke of York has a fair value of 74.5 which would create a 3-day return of 34.23% should ‘cable’ not drift more than a cent either side, i.e. below 1.59, above 1.61 at expiry.

Duke of York

Fig.2 – GBP v USD Binary Options Duke of York Price Profiles w.r.t. Time to Expiry

Although the 0.01-day profile has a step-like profile resembling Figure 1 the other profiles are smooth with high points midway between the central strikes. These profiles could easily be mistaken for a short straddle position except for one salient difference, short straddles create unlimited downside risk whereas the Duke of York’s losses are capped.

When there are 3-days or less to expiry, with the underlying outside the lowest and highest strikes the Duke also provides a directional play with substantial deltas as measured by the gradients of those price profiles. So, at 1.64 the 3-day profile has a fair value of 13.54 which on the underlying falling 2¢ to 1.62 rises to 48.3. If the underlying continued to fall and settled between 1.59 and 1.61, i.e. a fall of between 3¢ and 5¢, the Duke of York would return 100/13.54―1=639%.

Duke of York and Implied Volatility

As with Figure 3 with theta, the Duke provides the opportunity to take a position on the future implied volatility. If with the underlying at 1.60 the punter is of the view that volatility will fall then they would need to buy the Duke. At 18% implied volatility the strategy is worth 42.92 so if the punter buys the Duke and volatility falls to, say 14%, then the option is worth 52.2 and the punter ‘trousers’ a profit of 9.28.

Duke of York

Fig.3 – GBP v USD Binary Options Duke of York Price Profiles w.r.t. Implied Volatility

Evaluating the Binary Options Duke of York

Duke of York       =       100 x

(Payout1 x Binary Call(K1) +  Payout2 x Binary Call(K2) +

Payout3 x Binary Call(K3) +  Payout4 x Binary Call(K4) ―

Payout5 x Binary Call(K5) +  Payout6 x Binary Call(K6) ―

Payout7 x Binary Call(K7) +  Payout8 x Binary Call(K8))

where K1 is the lowest strike and K8 is the highest plus:

K1 + K2 + K3 + K4 = 1 and K5 + K6 + K7 + K8 = 1

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