Article 5 - Formula to calculate the supervisory delta of call and put options mapped to the interest rate risk category and supervisory volatility suitable for such formula

1.Institutions shall calculate the supervisory delta (δ) of call and put options, when mapped to the interest rate risk or the commodity risk categories, that is compatible with market conditions in which interest rates or commodity prices may be negative, as follows:

\delta = \mathrm{sign} \cdot N \left( \mathrm{type} \cdot \frac{\ln \left(\frac{P + \lambda}{K + \lambda}\right) + 0.5 \cdot \sigma^2 \cdot T}{\sigma \cdot \sqrt{T}} \right)

where:

\mathit{type} = \begin{cases} -1 & \text{where the transaction is a put option} \\ +1 & \text{where the transaction is a call option} \end{cases}

\mathrm{sign} = \begin{cases} -1 & \text{where the transaction is a sold call option or a bought put option} \\ +1 & \text{where the transaction is a sold put option or a bought call option} \end{cases}

N(x) = the cumulative distribution function for a standard normal random variable which reflects the probability that a normal random variable with mean zero and variance of one is less than or equal to ‘x’;

P = the spot or forward price of the underlying instrument of the option;

K = the strike price of the option;

T = the expiry date of the option, expressed in years using the relevant business day convention;

λ = the shift adequate to move both P and K into positive territory, determined in accordance with paragraph 2;

σ = the supervisory volatility of the option determined in accordance with paragraph 3.

2.For the purposes of paragraph 1, institutions shall calculate the shift (λ) for any call and put options as follows:

where:

3.For the purposes of paragraph 1, institutions shall determine the supervisory volatility of the option on the basis of the risk category of the transaction and the nature of the underlying instrument of the option in accordance with the following table: