Using the Hull-White Model to Price Options 
Use
The system contains a model, which is based on the Hull-White model, for pricing options on interest rate instruments and Bermuda options. This model reproduces the current yield curve resulting from the market, and reflects the fact that volatility depends upon the term of the option. It also assumes that the variance of the interest rate is limited for transactions with long terms.
The Hull-White model can be particularly useful for caps and floors, and OTC options on bonds, loans, fixed-term deposits, forward rate agreements, and swaps.
Integration
The model requires the following market data:
Reference interest rates
The system uses reference interest rates to generate a yield curve for the evaluation date. In doing so, it uses the yield curve type defined in Customizing for the evaluation type or valuation rule. The yield curve is generated for the transaction currency.
Volatility
The system requires the following parameters for interest rate volatilities: volatility parameter s , and reversion rate a . It has to be able to read both parameters from the volatility database. The system reads the volatility type defined in Customizing for the evaluation type or valuation rule.
Prerequisites
The function for using the Hull-White model to price options is not a standard function. To be able to use the Hull-White model, you need to make the following Customizing settings:
You need to have already defined suitable yield curve types, and assigned a Hull-White volatility to them.
The system must contain master data and market data for the volatilities. If appropriate, you have calibrated the Hull-White model (see Calibration of the Hull-White Model ).
You need to have defined and set up a suitable valuation rule.
To define a valuation rule, in the Customizing for Risk Analysis choose Common Settings for Market Risk and ALM → Valuation → Valuation Rule → Define Valuation Rule.
To set up a valuation rule, in the Customizing for Risk Analysis choose Common Settings for Market Risk and ALM → Valuation → Define Evaluation Type.
In the Market Data Categories tab page, under the Yield Curve Volatility Type area specify the volatility types for the yield curve for the bid, ask, and middle rates.
On the Evaluation Control tab page, choose the Hull-White valuation model, and enter a default step number, and a maximum time step.
The model assumes that the interest rate options contain options on fixed-rate transactions. Therefore, you have to map options on variable-rate transactions as options on fixed-rate transactions. This is done in the system before the model is called. For more information about this see Using the Hull-White Model to Price Interest Rate Options .
Features
The Hull-White model contains two methods. European options are priced using an analytical formula. American options and Bermuda options are priced using a trinomial tree method.
Analytical Solution for European Interest Rate Options
The system uses the value c b (t,T,S,X) as the basis for valuing a call option, and p b (t,T,S,X) for a put option with a zero bond as the underlying:
where t is the horizon, T is the expiry date of the option, S is the end of the term of the underlying bond, NV is the nominal volume, X is the strike of the option, and F (x) the cumulated normal distribution. Volatility s p is calculated from volatility parameterσand reversion rate a.
The following relationships also apply:
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.
where d M (t,T) is the discount factor from time point T to time point t , and f M is the short rate, which is defined as follows:
The system uses the following rule for the actual calculation:
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where X i is the strike at time point t i for cash flow CF i . The system uses the following relationships to calculated X i . The first equation is solved iteratively for spot price R at time point T :
Trinomial Tree Method (Bermuda and American Interest Rate Options)
The system generates a trinomial tree, which describes how the underlying may develop. The nodes (i, j) of the tree form a right-angled grid with the co-ordinates time and interest rate . The evaluation date is the root node. All other nodes fill the time period from the evaluation date through to the end of the notification period for the last possible date on which the option can be exercised.
Each node has three secondary nodes, which, when compared with the primary node, represent a constant interest rate, a rising interest rate, or a falling interest rate. The system generates the trinomial tree so that it reproduces a valid yield curve in the evaluation. (This means that at any point in time, the weighted total of the interest rates on the nodes belonging to this time point is equal to the relevant interest rate in the yield curve when the node probability is used as the weighting.) The system stores the following figures on the nodes of the trinomial tree: Discounting factor, and the three transition probabilities from the primary node.
The system then calculates prices for the nodes of the trinomial tree. In doing so, it first discounts the cash flows that are after the exercise time point or, in the case of Bermuda options, the end of the notification period, to the last (in terms of dates) node. Then the prices for the previous nodes are calculated by discounting the prices of the subsequent nodes, and the cash flows that fall between the nodes. The end of the notification period and the exercise time point of a possible exercise date can be different. This means that the expected price for the future exercise date is also projected to the nodes. The system calculates the transition probabilities, discount factors, and the evaluation profile according to the Hull-White model as depicted in Brigo/Mercurio (2001) (Damiano Brigo und Fabio Mercurio: Interest Rate Models: Theory and Practice, New York 2001)
The algorithm used to generate the grid points in the trinomial tree recognizes that it is the number of grid points between the evaluation date and the end of the notification period that is critical for the calculation. The accuracy of the valuation hardly varies, regardless of whether you divide a term of 10 days into 10 intervals, or a term of three years into 10 intervals. The process in the system is as follows:
The system divides the period from the evaluation date to the end of the notification period into equal intervals. The system takes the default number of steps you defined in the valuation rule as the number of intervals. If you did not specify a default number of steps, the system takes 60 as the default number.
It repeats the first step for all the subsequent possible dates on which the option can be exercised, but continues using the prior grid points.
If the maximum time step, as defined in the valuation rule, is exceeded, the system adds more grid points. It does so by adding a grid point at the current point in the calculation so that the respective interval has the maximum time step. The system checks the length of the intervals only if you have defined a maximum time step in the valuation rule.
Therefore, the grid points of the trinomial tree are not always equidistant, and there can be more grid points than you defined in the valuation rule by specifying the default step number.