## NSCCL SPAN

The objective of SPAN is to identify overall risk in a portfolio of futures and options contracts for each member. The system treats futures and options contracts uniformly, while at the same time recognizing the unique exposures associated with options portfolios like extremely deep out-of-the-money short positions, inter-month risk and inter-commodity risk.

Because SPAN is used to determine performance bond requirements (margin requirements), its overriding objective is to determine the largest loss that a portfolio might reasonably be expected to suffer from one day to the next day.

In standard pricing models, three factors most directly affect the value of an option at a given point in time:

- Underlying market price
- Volatility (variability) of underlying instrument
- Time to expiration

As these factors change, so too will the value of futures and options maintained within a portfolio. SPAN constructs scenarios of probable changes in underlying prices and volatilities in order to identify the largest loss a portfolio might suffer from one day to the next. It then sets the margin requirement at a level sufficient to cover this one-day loss.

**Mechanics of SPAN**

The complex calculations (e.g. the pricing of options) in SPAN are executed by the Clearing Corporation. The results of these calculations are called Risk arrays. Risk arrays, and other necessary data inputs for margin calculation are then provided to members on a daily basis in a file called the SPAN Risk Parameter file.

Members can apply the data contained in the Risk parameter files to their specific portfolios of futures and options contracts to determine their SPAN margin requirements.

Hence members need not execute complex option pricing calculations which are performed by NSCCL. SPAN has the ability to estimate risk for combined futures and options portfolios and re-value the same under various scenarios of changing market conditions.

**Risk Arrays**

The SPAN risk array represents how a specific derivative instrument (for example, an option on Nifty 50 index at a specific ' price) will gain or lose value from the current point in time to a specific point in time in the near future (typically it calculates risk over a one day period called the 'look ahead time'), for a specific set of market conditions which may occur over this time duration.

The specific sets of market conditions evaluated are called the risk scenarios, and these are defined in terms of:

- how much the price of the underlying instrument is expected to change over one trading day and,
- how much the volatility of that underlying price is expected to change over one trading day.

The results of the calculation for each risk scenario - i.e. the amount by which the futures and options contracts will gain or lose value over the look-ahead time under that risk scenario - is called the risk array value for that scenario. The set of risk array values for each futures and options contract under the full set of risk scenarios, constitutes the Risk Array for that contract.

In the Risk Array losses are represented as positive values and gains as negative values. Risk array values are typically represented in the currency (Indian Rupees) in which the futures or options contract is denominated.

SPAN further uses a standardized definition of the risk scenarios defined in terms of

- The underlying 'price scan range' or probable price change over a one day period,
- and the underlying price 'volatility scan range' or probable volatility change of the underlying over a one day period.

These two values are often simply referred to as the 'price scan range' and the 'volatility scan range'. There are sixteen risk scenarios in the standard definition. These scenarios are listed as under:

- Underlying unchanged; volatility up
- Underlying unchanged; volatility down
- Underlying up by 1/3 of price scanning range; volatility up
- Underlying up by 1/3 of price scanning range; volatility down
- Underlying down by 1/3 of price scanning range; volatility up
- Underlying down by 1/3 of price scanning range; volatility down
- Underlying up by 2/3 of price scanning range; volatility up
- Underlying up by 2/3 of price scanning range; volatility down
- Underlying down by 2/3 of price scanning range; volatility up
- Underlying down by 2/3 of price scanning range; volatility down
- Underlying up by 3/3 of price scanning range; volatility up
- Underlying up by 3/3 of price scanning range; volatility down
- Underlying down by 3/3 of price scanning range; volatility up
- Underlying down by 3/3 of price scanning range; volatility down
- Underlying up extreme move, double the price scanning range (cover 35% of loss)
- Underlying down extreme move, double the price scanning range (cover 35% of loss)

SPAN uses the risk arrays to scan probable underlying market price changes and probable volatility changes for all contracts in a portfolio, in order to determine value gains and losses at the portfolio level. This is the single most important calculation executed by the system.

As shown above in the sixteen standard risk scenarios, SPAN starts at the last underlying market settlement price and scans up and down three even intervals of price changes ('price scan range').

At each 'price scan point', the program also scans up and down a range of probable volatility from the underlying market's current volatility ('volatility scan range'). SPAN calculates the probable premium value at each price scan point for volatility up and volatility down scenario. It then compares this probable premium value to the theoretical premium value (based on last closing value of the underlying) to determine profit or loss.

Deep-out-of-the-money short options positions pose a special risk identification problem. As they move towards expiration, they may not be significantly exposed to "normal" price moves in the underlying. However, unusually large underlying price changes may cause these options to move into-the-money, thus creating large losses to the holders of short option positions. In order to account for this possibility, two of the standard risk scenarios in the Risk Array (sr. no. 15 and 16) reflect an "extreme" underlying price movement, currently defined as double the maximum price scan range for a given underlying. However, because price changes of these magnitudes are rare, the system only covers 35% of the resulting losses.

After SPAN has scanned the 16 different scenarios of underlying market price and volatility changes, it selects the largest loss from among these 16 observations. This "largest reasonable loss" is the 'Scanning Risk Charge' for the portfolio - in other words, for all futures and options contracts.

**Price Scan Range**

The price scan range, as explained above, is the probable price change over a one-day period. In case of index products and stock products the price scan range is taken as three standard deviations (3 sigma ) and three and a half standard deviations (3.5 sigma) respectively as calculated for VaR purpose for the underlying index and underlying security or other price scan range as may be prescribed. The price scan range for options and futures on individual securities is also linked to liquidity. This is measured in terms of impact cost for an order size of Rs 5 lakh calculated on the basis of order book snapshots in the previous six months as per defined methodology. Accordingly if the mean value of the impact cost exceeds 1%, the price scanning range is scaled up by square root of three. This is in addition to the requirement on account of look ahead period as may be applicable.

The mean impact cost as stipulated by SEBI is calculated on the 15th of each month on a rolling basis considering the order book snap shots of previous six months. If the mean impact cost of a security moves from less than or equal to 1% to more than 1%, the price scan range in such underlying is scaled by square root of three and scaling is dropped when the impact cost drops to 1% or less. Such changes are made applicable on all existing open positions from the third working day from the 15th of each month.

**Impact Cost**

- Impact cost is computed for an order size of Rs.5 lakh per security.
- For the purpose of computation of impact cost, four snapshots at different times are taken everyday from the order book in the capital market segment for all securities.
- Impact cost is the percentage price movement caused by an order size of Rs.5 Lakh from the average of the best bid and offer price in each order book snapshot.
- The closing price of the last day of the six month period is reckoned to determine the order size for the purpose of impact cost computation.
- Impact cost is calculated separately for the buy side and the sell side, in each order book snapshot, for the past 6 months.
- Where sufficient numbers of shares are not available on either the buy side or the sell side in a snapshot, the impact cost for such observations is given a penal value of 5%.
- The buy side impact cost (or sell side impact cost) is the simple average of the buy side impact cost (or sell side impact cost) computed for all the snapshot observations in the past 6 months.

Impact cost reckoned for the purpose of all computations is the mean of such buy side impact cost and sell side impact cost.

**Composite Delta**

SPAN uses delta information to form spreads between futures and options contracts. Delta values measure the manner in which a future's or option's value will change in relation to changes in the value of the underlying instrument. Futures deltas are always 1.0; options deltas range from -1.0 to +1.0. Moreover, options deltas are dynamic: a change in value of the underlying instrument will affect not only the option's price, but also its delta.

In the interest of simplicity, SPAN employs only one delta value per contract, called the "Composite Delta." It is the weighted average of the deltas associated with each underlying 'price scan point'. The weights associated with each 'price scan point' are based upon the probability of the associated price movement, with more likely price changes receiving higher weights and less likely price changes receiving lower weights. Please note that Composite Delta for an options contract is an estimate of the contract's delta after the look ahead - in other words, after one trading day has passed.

**Calendar Spread or Intra-commodity or Inter-month Risk Charge**

As SPAN scans futures prices within a single underlying instrument, it assumes that price moves correlate perfectly across contract months. Since price moves across contract months do not generally exhibit perfect correlation, SPAN adds a Calendar Spread Charge (also called the Inter-month Spread Charge) to the Scanning Risk Charge associated with each futures and options contract. To put it in a different way, the Calendar Spread Charge covers the calendar (inter-month etc.) basis risk that may exist for portfolios containing futures and options with different expirations.

For each futures and options contract, SPAN identifies the delta associated with each futures and option position, for a contract month. It then forms spreads using these deltas across contract months. For each spread formed, SPAN assesses a specific charge per spread which constitutes the Calendar Spread Charge.

The margin for calendar spread shall be calculated on the basis of delta of the portfolio in each month. Thus a portfolio consisting of a near month option with a delta of 100 and a far month option with a delta of -100 would bear a spread charge equivalent to the calendar spread charge for a portfolio which is long 100 near month futures contract and short 100 far month futures contract.

The calendar spread position shall be granted calendar spread treatment till the expiry of the near month contract.

**Short Option Minimum Charge**

Short options positions in extremely deep-out-of-the-money strikes may appear to have little or no risk across the entire scanning range. However, in the event that underlying market conditions change sufficiently, these options may move into-the-money, thereby generating large losses for the short positions in these options. To cover the risks associated with deep-out-of-the-money short options positions, SPAN assesses a minimum margin for each short option position in the portfolio called the Short Option Minimum charge, which is set by the NSCCL. The Short Option Minimum charge serves as a minimum charge towards margin requirements for each short position in an option contract.

For example, suppose that the Short Option Minimum charge is Rs. 50 per short position. A portfolio containing 20 short options will have a margin requirement of at least Rs. 1,000, even if the scanning risk charge plus the inter month spread charge on the position is only Rs. 500.

**Net Option Value (only for option contracts)**

In the above scenario only sell positions are margined and offsetting benefits for buy positions are given to the extent of long positions in the portfolio by computing the net option value.

**Computation of Initial Margin - Overall Portfolio Margin Requirement**

The total margin requirements for a member for a portfolio of futures and options contract are computed as follows:

- SPAN will add up the Scanning Risk Charges and the Intracommodity Spread Charges.
- SPAN will compare this figure (as per i above) to the Short Option Minimum charge
- It will select the larger of the two values between (i) and (ii)
- Total SPAN Margin requirement is equal to SPAN Risk Requirement (as per iii above), less the 'net option value', which is mark to market value of difference in long option positions and short option positions.

Black-Scholes Option Price calculation model

The options price for a Call, computed as per the following Black Scholes formula:

C = S * N (d_{1}) - X * e^{- rt} * N (d_{2})

and the price for a Put is: P = X * e^{- rt} * N (-d_{2}) - S * N (-d_{1})

where :

d_{1} = [ln (S / X) + (r + ?^{2} / 2) * t] / σ * sqrt(t)

d_{2} = [ln (S / X) + (r - ?^{2} / 2) * t] / σ * sqrt(t)

= d_{1} - σ * sqrt(t)

C = price of a call option

P = price of a put option

S = price of the underlying asset

X = Strike price of the option

r = rate of interest

t = time to expiration

σ = volatility of the underlying

N represents a standard normal distribution with mean = 0 and standard deviation = 1

ln represents the natural logarithm of a number. Natural logarithms are based on the constant e (2.71828182845904).

Rate of interest may be the relevant MIBOR rate or such other rate as may be specified.

SPAN^{®} is a registered trademark of the Chicago Mercantile Exchange, used herein under License. The Chicago Mercantile Exchange assumes no liability in connection with the use of SPAN by any person or entity.