Methods for Calculating Volatility

Methods for calculating volatility define how the system finds volatility for a combination of parameters (moneyness, term of option, and term of underlying) for which there is no value saved in the SAP system. The following methods can be used:

· Nearest Neighbor Search

· Linear Interpolation Using Delaunay Triangulation

You make the settings for the relevant methods in Customizing for Market Data under  Volatilities  Edit Volatility Types .

• Nearest Neighbor Search

  For the required parameter combination, the system finds the nearest point and takes its volatility as the volatility required. To find the nearest point, the system first calculates the relative distances between the given points and the required point for each parameter. It then adds up these relative distances for the three parameters. The nearest point is the one with the lowest cumulative value.

  To minimize system performance problems, the system initially only finds volatility that has the same external number as the option that is required. However, if the system does not find anything during this restricted search, the search is extended to all volatility.

• Linear Interpolation Using Delaunay Triangulation

  The distance between two points x and y is calculated using the equation illustrated in the figure below:

  
  
  

  Point x has the coordinates x(1), x(2), x(3), and point y has the coordinates y(1), y(2), y(3). You make settings for the scaling factors g(i) in Customizing for Market Data under  Volatilities  Edit Volatility Types .

  For g(1) = g(2) = g(3) = 1, the defined distance function is equivalent to the distance between two points in Euclidean space. The system assumes g(1) = g(2) = g(3) = 1 as default values when you create a new volatility type. You can weight certain dimensions differently using different scaling factors. For example moneyness (numeric values usually much smaller than the remaining term of the option in days) can be given more weight.

• Preprocessing for the Interpolation (Only Once After Data is Selected)

  1. The system reads the volatility structure from the database.

  2. Calculating the average for points that are multiply delivered: The average value is calculated and transferred for data points that are consistent in the parameters moneyness, residual maturity of the option, and residual maturity of the option. The original data points are deleted from the set of points.

  3. The space that was defined by the coordinate axes of moneyness, residual maturity of the option, and residual maturity of the underlying is divided into simplexes. The vertices of these simplexes are the data points that were previously calculated (Delaunay triangulation). See also: Additional Mathematical Explanation of Linear Interpolation

• Interpolation (As Often As Required)

  1. The system looks for the simplex that contains the point with the required coordinates for moneyness, residual maturity of the option, and residual maturity of the underlying. There is one simplex at the most that contains the required point.

  2. The system interpolates the volatility using the vertices of the simplex found as interpolation grid points.

• Extrapolation (As Often As Required)

  If the system cannot find a simplex for the required volatility, the point is outside the volatility structure. In this case, the system looks for the nearest point on the outer hull of the set of points delivered. The volatility of this nearest point is then used.