Delta equivalent amount of cash-or-nothing digital (binary) options

Should the delta equivalent amount of cash-or-nothing digital (binary) options be limited to the maximum possible payment at expiry?

According to Article 329 (1) Options and warrants on interest rates, debt instruments, equities, equity indices, financial futures, swaps and foreign currencies shall be treated as if they were positions equal in value to the amount of the underlying instrument to which the option refers, multiplied by its delta for the purposes of this Chapter. For plain vanilla options delta is bounded by 1 for call options and -1 for put options. This is not the case, however, for digital (binary) options. The characteristic of digital (binary) options is a limited payoff. Therefore the risk exposure for the writer of the option is limited (option writer cannot lose more than the payoff amount). Digital (binary) options have another characteristic: when the options are close to expiry and the price of the underlying is close to strike price of the option, the option’s delta becomes very large and is not bounded by -1 and 1 (see e.g. p. 122 of Paul Willmott on Quantitative Finance, 2nd edition, 2006). The abovementioned issue was the subject of question ID 2015_2462. EBA rejected this question and explained that the issue is already explained or addressed in Article 4(3) of Regulation (EU) No 528/2014. Article 4(3) of Regulation (EU) No 528/2014 (point b in particular) states that the own funds requirements for non-delta risks for non-continuous options (e.g. digital (binary) options) shall be determined as the maximum of 0 and the difference between market value of the underlying, which in case of a digital option is the maximum possible payment at expiry (payoff) and risk weighted delta equivalent amount (understood in the manner described in Article 3(1)(b) of Regulation (EU) No 528/2014). This could be interpreted in line with the first characteristic of digital (binary) options mentioned above (the limited payoff): delta and non-delta risks amount to the maximum possible payment at expiry, so the total own funds requirements for writing a digital (binary) option would be limited to its payoff. On the other hand, according to Article 3(1)(b) of Regulation (EU) No 528/2014, risk-weighted delta equivalent amount shall be calculated as market value of the underlying multiplied by delta and then multiplied by a relevant risk weight. Having in mind the definition of market value of the underlying for non-continuous options (Article 4(3)(b)(i) of Regulation (EU) No 528/2014), this would mean that for cash-or-nothing digital (binary) options the risk weighted delta equivalent amount equals maximum possible payment at expiry multiplied by delta and then multiplied by a relevant risk weight. Taking into account the second characteristic of digital (binary) options mentioned above (the unbounded delta) this could mean that when the options are close to expiry and the price of the underlying is close to strike price of the option, the non-delta risk would be zero but the total own funds requirements for writing a digital (binary) option would be greater than the maximum possible payment at expiry, which economically makes no sense and is irrational from a hedging perspective. The example below clarifies the abovementioned issue: Institution writes a digital up option on equity X. The maximum possible payment at expiry: 100 EUR At the end of the day the option delta reaches 1 000. Risk weighted delta equivalent = 100 EUR * 1 000 * (8% + 8%) = 16 000 EUR The own funds requirements for non-delta risk = max(0; 100 EUR – 16 000 EUR) = 0 EUR Total own funds requirements for writing a digital (binary) option = 16 000 EUR The solution for the abovementioned issue would be to limit the delta equivalent amount to the maximum possible payment at expiry, which is equivalent to bounding the digital (binary) option’s delta (calculated using an appropriate option pricing model) to <-1;1> depending on the type of the option (up or down). In the example above this would mean: Risk weighted delta equivalent = 100 EUR * min (1;1 000) * (8% + 8%) = 100 EUR * 1 * (8%+8%) = 16 EUR The own funds requirements for non-delta risk = max(0; 100 EUR – 16 EUR) = 84 EUR Total own funds requirements for writing a digital (binary) option = 16 EUR + 84 EUR = 100 EUR = maximum possible payment at expiry