Calibration risk in pricing excess interest options

Abstract

Due to renewals in the financial reporting standards, financial instruments on the balance of insurance companies are reported at their market consistent, fair, values. Since there is not always an active market for these financial instruments, these fair values are often computed by particular models. One can imagine that the choice of the model will have impact on the fair value of the financial instrument. But when a model is chosen, it has to be adapted to the actual market situation in order to produce market consistent prices. This process of fitting is called calibration. Model calibration is not a straightforward process, several choices have to be made during the calibration process. We investigate the effect of these choices, in other words, we determine the calibration risk.

We look at the value of a particular embedded option in an insurance contract: an excess interest option. The value of this embedded option is calculated with two commonly used interest rate models: the two-factor Hull-White model and the Libor Market Model. We calibrate these two models to market values of interest rate swaptions and to market values of interest rate caps. The market values of these instruments can be either prices or implied volatilities. Cap data consist of market values for a whole range of caps or market values for only at-themoney caps. The choice between these collections of cap data is also investigated.

Actually, calibrating boils down to minimizing the ‘difference’ between model and market values over a set of model parameters. This difference can be measured in several ways. One can minimize absolute differences or one can minimize relative differences between market and model values.

In pricing the embedded option, two risks can be identified, see [DH07]. One is model risk. This is the impact of the model choice on the value of the option. We quantify model risk by the fraction between two model prices of the embedded option. Calibration risk is the other risk which arises from the methods chosen in the calibration proces. Calibration risk is measured as the fraction between the option values within one model, but with different calibrations applied to that model.

The goal of this thesis is formulated as: Investigate the impact on pricing an excess interest option of different calibration methods by valuing the option with the two interest rate models, where each model is calibrated in different ways:

a) with respect to absolute or relative differences of swaption prices or swaption implied volatilities;

b) with respect to absolute or relative differences of all cap prices or only at-the-money cap prices.

Combining these possibilities results in eight different ways to calibrate both interest rate models, which we will investigate.

Two important conclusions we draw from the investigation are: • If one decides to calibrate to swaptions, then the impact of the model choice on the option price is larger than the impact of the calibration choice, i.e. model risk is larger than calibration risk. • Calibration to caps must be performed with respect to all caps by minimizing absolute price differences, in order to obtain proper model parameters.