Calibration of the Hull-White Model

Use

You use this function to calculate the volatility parameters of the Hull-White model from the current implied volatilities of the Black-Scholes model for swaptions or caplets.

You use this function if you want to use the Hull-White model to price interest rate options, and you have only the volatilities of the Black-Scholes model. Volatility values for the Black-Scholes model are provided by external sources such as Reuters or Bloomberg, and usually exist in the system. This is not normally the case for the volatility parameters of the Hull-White model.

The calibration of the Hull-White interest rate model is largely an optimization in which the system finds values for the Hull-White volatility parameters sigma σ and reversion rate a , in which the option prices, calculated using the Hull-White model or Black-Scholes model, match as far as possible.

You have the following options for calibrating the Hull-White model:

Calibrating the model manually for a group of transactions

You call the calibration function manually using a separate transaction, or you can schedule the job in the Schedule Manager.
Using the price calculator to calibrate each transaction

If the central volatility database does not contain any Hull-White volatility parameters, the price calculator calls the calibration function automatically when it prices interest-rate options.

Integration

The calibration of the Hull-White model is used to prepare the data for valuation runs in Market Risk Analysis. The model is calibrated automatically in the price calculation immediately before a transaction is priced. You call the function for manual calibration independently of the valuation runs.

You can call the function for calibrating the model at any time manually in the SAP Easy Access screen (transaction RMHWCAL) or you can schedule the calibration in the Schedule Manager so that it takes place regularly. The name of the calibration program is RFTBB_HWCALIBRATION2.

Note Note

The calibration process takes a relatively long time. In particular, the automatic calibration can lead to performance problems, as it is called for each transaction and each valuation method, and not for each potential exercise date. If the results of your analysis are to be exact to 5-10 percent of the NPV, then you should calibrate the model manually for a group of transactions. The system saves the Hull-White volatility values that are calculated to the database so that you can use these for multiple evaluations and transactions.

End of the note.

Prerequisites

In Customizing under SEM Banking Data Pool Price Parameters Interest Rate Tables , you need to have already entered the settings for yield curves.

Since only current interest rates should be used to calibrate the Hull-White model, assign the read procedure Read back directly to the yield curve type.
In Customizing under SEM Banking Data Pool Price Parameters Statistical Data Define Volatility Type , you need to have already defined a volatility type, and under Statistical Data Volatilities with Moneyness Master Data for Volatilities , you need to have stored values for the Black-Scholes volatility. For more information, see the document Central Volatility Database . We recommend that you create a separate volatility type for Hull-White volatility values.

Note the following properties of the calibration function:
- In order to find the Hull-White volatility parameters σ and a , the system needs at least two Black-Scholes volatility values for different option terms and underlying terms.
- You can calculate the volatility parameters σ and a for the Hull-White model from the Black-Scholes volatility values for swaptions or caplets. If you want to price swaptions and caplets using just the relevant values for swaptions and caplets, then you need to store the volatilities for the caplets and swaptions under a separate volatility type. You use the valuation rule to control how the volatilities are used. The reason for this is that the system cannot distinguish between the volatility values of swaptions and caplets without additional information.
- The volatilities of swaptions and caplets are usually shown in a two-dimensional grid. You use the term of the option and the term of the underlying as the coordinates.
The following prerequisites apply for automatic calibration:
- In Customizing for SEM Banking under Common Settings for Market Risk and Asset/Liability Management Valuation Valuation Rule Define Valuation Rule, you have defined a valuation rule. Under Common Settings for Market Risk and Asset/Liability Management Valuation Define Evaluation Type , you have assigned the volatility name for the implied Black-Scholes volatilities to the valuation rule.
- You have defined an evaluation type, and assigned the volatility type for the implied Black-Scholes volatilities to it. You assign the volatility type to the evaluation type in the Interest Rate Volatility Types area. During the valuation process, the system usually interpolates the volatility values. The values are interpolated at different points on the level of the term of the option and the term of the underlying. If the standard setting is used, the system uses the nearest neighbor search to interpolate the values. However, we recommend that you use a smooth (at least linear) interpolation method. The interpolation is contained in a BAdI. You store your own implementation for this BAdI in Customizing for SEM Banking under Data Pool Price Parameters Statistical Data Volatilities with Moneyness Define Interpolation of Volatilities.
Note
To trigger the function for automatic calibration, define a Hull-White volatility type but do not assign it any volatility values. You assign this dummy Hull-White volatility type to the evaluation type in the Yield Curve Volatility Type area. In the valuation process, these settings prevent the system from using the volatility values already stored in the database.

End of the note.

Features

The system first selects the underlying yield curve, and the volatility values from the Black-Scholes model as grid points. The system proceeds as follows:
- Calibrating the model manually for a group of transactions
  
  The system selects only those values that actually exist in the central volatility database. It does not interpolate or extrapolate any volatility values.
  
  You can use selection parameters to define how many volatility values or option prices the system uses as a basis for calibrating the Hull-White model.
- Using the price calculator to calibrate each transaction
  
  For Bermuda options, the system selects a Black-Scholes volatility value for each potential exercise date. If there is only one potential exercise date left, or the options to be priced are European or American, the system uses an additional grid point. This grid point is in the middle of the evaluation date and the maturity date of the option.
  
  If required, the system interpolates the selected values.
The system uses the Black-Scholes model to calculate the option prices. It interprets the selected Black-Scholes volatility values as interest rate volatilities, and prices the swaptions and caplets accordingly as interest rate options, and not as bond options. The system uses the forward interest rate for the time span between the maturity of the option and the maturity of the underlying for the swap rate or cap rate.
The system optimizes these values using the Simplex method:
- In the automatic calibration, the system takes the values σ=0.01 and a=0.1 as the starting values. In the manual calibration, you enter the values for σ and a , or the system uses historical values, which it reads from the central volatility database.
- The system minimizes the following merit function:
  
  ( )
  
  where n i s the number of option prices calculated by the Black-Scholes model, and σ _B/S,i is the associated implied volatility, which the system calculated from the selected option terms, underlying terms, and moneyness. V _H/W,i (σ,a) is the option prices calculated by the Hull-White model, σ _B/S (V _{H/W, i} (σ,a)) is the implied volatility of the Black-Scholes model calculated from this value.
- The system stops the calibration process when one of the following criterion is met:
  1. The maximum number of iteration steps is reached
  2. The values for χ and for the change in χ have not yet reached the maximum values
    
    In the automatic calibration, the following values are predefined: χ=1.0, ∆χ=1*10 ^-8 . The maximum number of iteration steps is 250.

If you call the manual calibration function for a group of transactions, if required the system saves on the database the volatility values σ and a that it calculated. The values it calculated for volatility parameters σ and a are annualized to 365 days. Unlike in the Black-Scholes model, the system saves the Hull-White volatility values as absolute values, and not as percentages. When these values are saved, the system uses the following key fields:

Field	Use for Hull-White Parameters
Validity date	Evaluation date
Volatility type	Volatility type that you used as a selection criterion
Term of the option	Largest value specified for the term of the option
Term of the underlying	Largest value specified for the entire term
Moneyness	Average of the specified strikes

Activities

To use the function for calibrating the Hull-white model manually for a group of transactions, do the following:

On the SAP Easy Access screen, choose Accounting Bank Applications SEM Banking Data Pool Price Parameters Calibrate Hull-White Interest Rate Structure Model.

The system displays a selection screen.
Use selection criteria to define which Black-Scholes volatilities the system is to read from the database in order to calibrate the Hull-White model.
Decide whether the system is to use historical values as starting parameters for the calibration, and, if appropriate, enter the starting parameters for σ and a manually.
Decide whether the system is to use a fixed value for the sigma σ parameter or for reversion rate a during the calibration process, or whether these parameters should vary.
Define the criteria for ending the calibration process by specifying the maximum value for merit function χ, change in χ, and the number of iteration steps.
Decide whether the system is to start the calibration process as a test run, or whether it should save the results on the database.
Choose Execute .

The system displays a list of the parameters and the results of the calibration of the model.

If you did not start the calibration as a test run, the system saves the volatility parameters σ and a that were calculated to the central volatility database.