• those banks which solely use purchased options will be free to use the simplified approach described in Section I below;
• those banks which also write options will be expected to use one of the intermediate approaches as set out in Section II or a comprehensive risk management model under the terms of Part B of this paper. The more significant its trading, the more the bank will be expected to use a sophisticated approach.

2. In the simplified approach, the positions for the options and the associated underlying, cash or forward, are not subject to the standardised methodology but rather are "carved-out" and subject to separately calculated capital charges that incorporate both general market risk and specific risk. The risk numbers thus generated are then added to the capital charges for the relevant category, i.e., interest rate related instruments, equities, foreign exchange and commodities as described in A.1-4. The delta-plus method uses the sensitivity parameters or "Greek letters" associated with options to measure their market risk and capital requirements. Under this method, the delta-equivalent position of each option becomes part of the standardised methodology set out in A.1-4 with the delta-equivalent amount subject to the applicable general market risk charges. Separate capital charges are then applied to the gamma and vega risks of the option positions. The scenario approach uses simulation techniques to calculate changes in the value of an options portfolio for changes in the level and volatility of its associated underlyings. Under this approach, the general market risk charge is determined by the scenario "grid" (i.e., the specified combination of underlying and volatility changes) that produces the largest loss. For the delta-plus method and the scenario approach the specific risk capital charges are determined separately by multiplying the delta-equivalent of each option by the specific risk weights set out in A.1 and A.2.

I. Simplified approach

3. Banks which handle a limited range of purchased options only will be free to use the simplified approach set out in Table 8 for particular trades. As an example of how the calculation would work, if a holder of 100 shares currently valued at $10 each holds an equivalent put option with a strike price of $11, the capital charge would be: $1,000 x 16% (i.e., 8% specific plus 8% general market risk) = $160, less the amount the option is in the money ($11 - $10) x 100 = $100, i.e., the capital charge would be $60. A similar methodology applies for options whose underlying is a foreign currency, an interest rate related instrument or a commodity.

Position	Treatment
Long cash and Long put or Short cash and Long call	The capital charge will be the market value of the underlying security multiplied by the sum of specific and general market risk charges for the underlying less the amount the option is in the money (if any) bounded at zero
Long call or Long put	The capital charge will be the lesser of: the market value of the underlying security multiplied by the sum of specific and general market risk charges37 for the underlying the market value of the option

II. Intermediate approaches

(a) Delta-plus method

4. Banks which write options will be allowed to include delta-weighted options positions within the standardised methodology set out in A.1-4. Such options should be reported as a position equal to the market value of the underlying multiplied by the delta. However, since delta does not sufficiently cover the risks associated with options positions, banks will also be required to measure gamma (which measures the rate of change of delta) and vega (which measures the sensitivity of the value of an option with respect to a change in volatility) sensitivities in order to calculate the total capital charge. These sensitivities will be calculated according to an approved exchange model or to the bank's proprietary options pricing model subject to oversight by the national authority.

5. Delta-weighted positions with debt securities or interest rates as the underlying will be slotted into the interest rate time-bands, as set out in A.1, under the following procedure. A two-legged approach should be used as for other derivatives, requiring one entry at the time the underlying contract takes effect and a second at the time the underlying contract matures. For instance, a bought call option on a June three-month interest-rate future will in April be considered, on the basis of its delta-equivalent value, to be a long position with a maturity of five months and a short position with a maturity of two months. The written option will be similarly slotted as a long position with a maturity of two months and a short position with a maturity of five months. Floating rate instruments with caps or floors will be treated as a combination of floating rate securities and a series of European-style options. For example, the holder of a three-year floating rate bond indexed to six month LIBOR with a cap of 15% will treat it as:

1. a debt security that reprices in six months; and
2. a series of five written call options on a FRA with a reference rate of 15%, each with a negative sign at the time the underlying FRA takes effect and a positive sign at the time the underlying FRA matures.

6. The capital charge for options with equities as the underlying will also be based on the delta-weighted positions which will be incorporated in the measure of market risk described in A.2. For purposes of this calculation each national market is to be treated as a separate underlying. The capital charge for options on foreign exchange and gold positions will be based on the method set out in A.3. For delta risk, the net delta-based equivalent of the foreign currency and gold options will be incorporated into the measurement of the exposure for the respective currency (or gold) position. The capital charge for options on commodities will be based on the simplified or the maturity ladder approach set out in A.4. The delta-weighted positions will be incorporated in one of the measures described in that section.

7. In addition to the above capital charges arising from delta risk, there will be further capital charges for gamma and for vega risk. Banks using the delta-plus method will be required to calculate the gamma and vega for each option position (including hedge positions) separately. The capital charges should be calculated in the following way:

1. for each individual option a "gamma impact" should be calculated according to a Taylor series expansion as:
   Gamma impact = ½ x Gamma x VU²

   where VU = Variation of the underlying of the option.

2. VU will be calculated as follows:
   • for interest rate options if the underlying is a bond, the market value of the underlying should be multiplied by the risk weights set out in Table 1 of A.1. An equivalent calculation should be carried out where the underlying is an interest rate, again based on the assumed changes in the corresponding yield in Table 1 of A.1;
   • for options on equities and equity indices: the market value of the underlying should be multiplied by 8%;
   • for foreign exchange and gold options: the market value of the underlying should be multiplied by 8%;
   • for options on commodities: the market value of the underlying should be multiplied by 15%.

3. For the purpose of this calculation the following positions should be treated as the same underlying:
   • for interest rates, each time-band as set out in Table 1 of A.1;
   • for equities and stock indices, each national market;
   • for foreign currencies and gold, each currency pair and gold;
   • for commodities, each individual commodity as defined in paragraph 5 of A.4.

4. Each option on the same underlying will have a gamma impact that is either positive or negative. These individual gamma impacts will be summed, resulting in a net gamma impact for each underlying that is either positive or negative. Only those net gamma impacts that are negative will be included in the capital calculation.
5. The total gamma capital charge will be the sum of the absolute value of the net negative gamma impacts as calculated above.
6. For volatility risk, banks will be required to calculate the capital charges by multiplying the sum of the vegas for all options on the same underlying, as defined above, by a proportional shift in volatility of 25%.
7. The total capital charge for vega risk will be the sum of the absolute value of the individual capital charges that have been calculated for vega risk.

(b) Scenario approach

8. More sophisticated banks will also have the right to base the market risk capital charge for options portfolios and associated hedging positions on scenario matrix analysis. This will be accomplished by specifying a fixed range of changes in the option portfolio's risk factors and calculating changes in the value of the option portfolio at various points along this "grid". For the purpose of calculating the capital charge, the bank will revalue the option portfolio using matrices for simultaneous changes in the option's underlying rate or price and in the volatility of that rate or price. A different matrix will be set up for each individual underlying as defined in paragraph 7 above. As an alternative, at the discretion of each national authority, banks which are significant traders in options will for interest rate options be permitted to base the calculation on a minimum of six sets of time-bands. When using this method, not more than three of the time-bands as defined in A.1 should be combined into any one set.

9. The options and related hedging positions will be evaluated over a specified range above and below the current value of the underlying. The range for interest rates is consistent with the assumed changes in yield in Table 1 of A.1. Those banks using the alternative method for interest rate options set out in paragraph 8 above should use, for each set of time-bands, the highest of the assumed changes in yield applicable to the group to which the time-bands belong. The other ranges are 8% for equities43, 8% for foreign exchange and gold, and ± 15% for commodities. For all risk categories, at least seven observations (including the current observation) should be used to divide the range into equally spaced intervals.

10. The second dimension of the matrix entails a change in the volatility of the underlying rate or price. A single change in the volatility of the underlying rate or price equal to a shift in volatility of + 25% and - 25% is expected to be sufficient in most cases. As circumstances warrant, however, the supervisory authority may choose to require that a different change in volatility be used and/or that intermediate points on the grid be calculated.

11. After calculating the matrix each cell contains the net profit or loss of the option and the underlying hedge instrument. The capital charge for each underlying will then be calculated as the largest loss contained in the matrix.

12. The application of the scenario analysis by any specific bank will be subject to supervisory consent, particularly as regards the precise way that the analysis is constructed. Banks' use of scenario analysis as part of the standardised methodology will also be subject to validation by the national authority, and to those of the qualitative standards listed in Part B which are appropriate given the nature of the business.

13. In drawing up these intermediate approaches the Committee has sought to cover the major risks associated with options. In doing so, it is conscious that so far as specific risk is concerned, only the delta-related elements are captured; to capture other risks would necessitate a much more complex regime. On the other hand, in other areas the simplifying assumptions used have resulted in a relatively conservative treatment of certain options positions. For these reasons, the Committee intends to keep this area under close review.

14. Besides the options risks mentioned above, the Committee is conscious of the other risks also associated with options, e.g., rho (rate of change of the value of the option with respect to the interest rate) and theta (rate of change of the value of the option with respect to time). While not proposing a measurement system for those risks at present, it expects banks undertaking significant options business at the very least to monitor such risks closely. Additionally, banks will be permitted to incorporate rho into their capital calculations for interest rate risk, if they wish to do so.