Black 76 on index futures

# BLACK-76 OPTION MODEL

## Option On Index Futures

BLACK-76: OPTION ON INDEX FUTURES

The model is a simple application of Black-Scholes formula. For further information on Black 76 model, users are recommended to read the information in Wikipedia. The calculator presented here is application of the model for options on equity index futures, although the model can be practically applied for any option on a futures contract.

The fields in our calculators are sufficiently descriptive for user to understand. The 'Rate' field however, is not the continuous rate , though it will be converted as required by the model. Hence, we have two additional fields to decribe the rate input - 'Rate Basis' and 'Day Count'. Reader who have been reading our materials will be able to understand the significance of the fields.

From a theroretical point of view, user can price an option on index futures using the calculators. In the case of Malaysian KLCI index futures, it is refered to as OKLI by the exchange (Bursa Malaysia). The trading and settlement value of the option are stated in terms of points instead of monetary numbers, in which 1 index point is equivalent to MYR50. By the same token, since the calculator only takes the prices as index point, the resulting 'Value' are also in terms of index point.

We added the ability for user to input the market value of the option. User can then calculates the implied volatility of the option based on the market/ traded/ settlement value of the option and is way more practical than the ability to price the option.

Four 'greeks' are included in the calculator - delta, gamma, vega and vanna.

• Delta - The first derivative of the option value with respect to futures price. It is percentage change in value for a small change in price and often taken as a equivalent position in the futures market. For instance if delta is 50%, it is equivalent of 0.5 of the futures contract. So if you are holding 10 options contract it is equivalent to 5 futures contract
• Gamma - The second derivative of the option value with respect to futures price. It is the percentage change in delta for a small change in futures price or the percentage change in the equivalent futures contract position
• Vega - The first derivative of the option value with respect to volatility. The number represents the estimated change in the value of the option if volatility changes by 1.00%
• Vanna - The first derivative of the delata with respect to volatility. The number represents the estimated change in the delta of the option if volatility changes by 1.00%
• The greeks provided are calculated using forward differential method, i.e shifting the relevant value upwards by a small margin as opposed to calculating from a formula. For practical purposes, the difference is negligible.

We have also provided 3D charts of the option value and various greeks for user analysis. Input values are from the calculators are used except for 'Futures Price' and 'Volatility'. They are the variables used to derived the values for the charts. Click the 'Calculate' button to redraw the chart if you make any changes to the input values.

#### Option Value

1. Option value is positively correlated with volatility
2. Option value is higher when the option is in the money. For a call, the higher the price, the higher the value while for put a put a lower price will give a higher value.

#### Delta

1. Delta is nothing more than the cumulative probability
2. Delta is positive for a long call and negative for a long put.
3. Delta positively related to the value of the option. The higher the value, the higher is the delta. Alternativey, a deeper in the money option will have a higher absolute delta value

#### Gamma

1. Gamma is the higher when it at the the money i.e. strike is the same as the market underlying price.
2. Buyer of option is long gamma and seller of option is short gamma.

#### Vega

1. Vega is positively related to volatility.
2. Vega is at its highest when the option is at the money.
3. Buyer of option is long vega and seller of option is short vega, or often reffered to as long and short volatility respectively.

#### Vanna

1. Vanna is around zero when it is at the money and volatile around that point and pronounced at lower volatility.