Basket Options and Average Spot Options 
Use
The price calculator uses one pricing model to price the following exotic options:
Basket Options (BO)
Basket options are options that usually have more than one underlying. The spot of a basket option is the total of the spots of the individual underlyings that make up the basket option. A basket option that has just one underlying is the same as a plain vanilla option.
Average Spot Options (ASpO)
Average spot options are options that have one underlying and whose value is based on an average, rather than on the current spot of the underlying. The average value is based on a time series; the spot of the underlying is fixed for each time point in this time series. The fixed spot values are totaled, and divided by the number of spot values. The calculation of the average in SAP's pricing model weights the spot values equally.
Average Spot Basket Options (ASpO)
Average spot basket options are options that usually have more than one underlying, and whose value is calculated as an average. The average value is based on a time series; the spot of the underlying basket option is calculated for each time point in this time series. The resulting spot values are totaled, and divided by the number of spot values. The calculation of the average in SAP's pricing model weights the spot values equally.
Average spot options and basket options can be seen as variants of the average spot basket options. Therefore, it is possible to use the same pricing model for these three types of option. An average spot basket option with just one underlying is an average spot option. If only one date is entered in the calculation of the average value for an average spot basket option, then in this case the option can be treated as a basket option. If there is only one underlying and one date, the average spot basket option can be treated as a plain vanilla option in the calculation of the average.
Caution
The price calculator can currently price only basket options and average spot options whose underlyings are foreign currency transactions.
The following constraints apply to the pricing model:
Calculation of the Discrete Average
The average is calculated only at certain specified time points.
Assumption of Equidistant Time Points in the Calculation of the Average
In the pricing process, the system has to calculate moments for the distribution of future spot values (in other words, spot values that are not fixed). To do this, the system has to assume that the time points used to calculate the average are equidistant.
European Exercise Type
Since the option pricing is based on the Black-Scholes model, only European-style options can be priced.
Correlation options have components of basket options and average spot basket options. For information about how these options are priced, see the document Correlation Options .
Integration
You create average spot basket options, average spot options, and basket options as generic transactions. For more information about this, see the documents Basket Options and Average Spot Options in the bank transaction data documentation.
Prerequisites
In order to price average spot basket options, the system needs the correlations between the underlyings contained in the basket, as well as the usual market data. The correlation matrix must be positive semidefinite. You can use the statistics calculator to calculate correlations and adjust the correlation matrix if required so that it is positive semidefinite.
Features
The payment amount of an average spot basket option as a call option c ASBO or as a put option p ASBO is calculated as follows:
where SP={t i |1=i=N} is the number of time points used to calculate the average, M is the number of underlyings, X is the strike of the average spot basket option, and s j (t i ) is the spot of the j th underlying at time point t i . The call option has a positive value, the put option has a negative value.
There is no explicit formula for the pricing of average spot basket options, average spot options or basket options. SAP's pricing model is a more general version of the approximation method that Levy (1998) developed for average spot options. [see David F. DeRosa: Currency Derivatives. Pricing Theory, Exotic Options, and Hedging Applications, John Wiley & Sons: New York 1998]
The Levy approach is based on the principle that although the spot values s j (t i ) of average spot options are log normally distributed, this does not apply for their totals. However, it is assumed that the total of the spot values are distributed in a way that corresponds approximately to log normal distribution. Therefore, the system calculates the first two moments for the distribution of the total, and uses these as the moments of the approximation of log normal distribution. Then the system applies the Black-Scholes formula for the actual pricing of the options.
SAP's pricing model is a more general version of the Levy approach for average spot basket options. In particular, it considers the interrelationships between the two underlyings at different points in time. For average spot options, SAP's pricing model uses the Levy approach.