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The hybrid HW2F-CIR model is a very promising, straightforward short-rate model for risk-management purposes. It has a number of original and very important features:

1. Is a short-rate model suitable for low-dimensional simulation of portfolio exposure. 2. Allows for analytic calibration to market OIS, IRS and semi-analytic to ATM (or any

fixed strike) caps.

3. Allows calibration to multiple tenors of LIBOR-OIS spreads and pricing of caps on illiquid tenors.

4. And allows simulating IRS, cap exposures on different tenors for CVA valuation.

6.2.1

Calibration

The model can be calibrated to a single-day of market data, yield curves and a series of market caps. In the calibration, the information about OIS term-structure goes into the (base) HW2F model, the LOIS spread term-structure is fitted to CIR fixed parameters and the cap volatility information is fed into both: Hull-White volatility and mean reversion, as well as CIR parameters. Hence, the fitting is done jointly to forward rates and market cap volatilities.

We have successfully calibrated the model to market data on 22-Nov-2013. However, we note that in this global-calibration exercise, multiple local minima exist in the error function and the optimizer can fall into one, ignoring the absolute global solution (see example in Appendix E.4). To avoid this, we propose to calibrate a simple Hull-White model with constant-basis spread assumption to market curves and caps and then use these Hull-White volatility and mean-reversion parameters as a starting point for full HCIR model calibration.

We further note that the model requires a smooth, convergent term-structure of the forward rates and cap volatilities. Any irregularities could disrupt the calibration (see Ap- pendix4.5). In the current formulation of the model, there may be cases where the calibra- tion is not successful.

6.2.2

Cap Pricing

We have performed a number of tests with the HCIR model, including illiquid cap pricing and comparison to market data, exposure and CVA valuation with HCIR model and bench- marking against pure HW1F and HW2F models. Our analysis has shown that calibration to market forward curves works within 5bp error for HCIR models1, with exception of the first few IRS points - the difference is up to 20bp, does not fit well as the market LOIS spread increases too fast for the HCIR model. Then, the calibration of HW2F-CIR model to market caps for a single-strike works well, within a 1bp volatility error for maturities

>3years. The shorter maturity volatilities are hard to match: the 2-year vol is 5bp higher in the model, the 1-year market volatility is too low to be represented by HW2F or HCIR models. The HW1F model is not flexible enough to price a single multi-maturity series of

market caps and falls behind both previous models. What is more, as the HCIR model is based on HW and CIR models, however, the hybrid model has a very small volatility smile and cannot be used reliably for smile extrapolation.

6.2.3

CVA Impact Evaluation

In our tests for CVA and exposure assessment, we have benchmarked the HCIR model ver- sus the fixed-basis HW1F and HW2F models in market and synthetic-data situations. We have found that the HW2F and HCIR models show comparable IRS exposure profiles and CVA values in market-data tests, with differences around2bp on an IRS fixed rate adjust- ment. The HW1F model results in overall higher volatility and higher CVA/DVA values for ATM swaps than the other two models. Whenever the LOIS spread is small or the calibrated CIR model has low-volatility and the HCIR and HW2F models give very close results. The main difference between the models arises in synthetic situation of rapidly de- creasing LOIS spread term-structure. This happens in a liquidity squeeze when short-term unsecured borrowing is very expensive when compared to collateralized lending (using OIS rates). In this case the HCIR model yields higher CVA/DVA values than HW2F model by even7bp change on fixed leg.

Comparison of exposures for LOIS swap as well as caps did now show any changes in CVA/DVA figures, as these derivatives stay in or out-of the money throughout the life- time of the derivative, depending on the form of the forward curve, and therefore yield matching CVA/DVA mean exposures. The95%-quantiles of HW2F and HCIR models are slightly different, as the HCIR model in high-volatility spread case has much larger short- term volatility. A full analysis is included in AppendixE. We have also tested the WWR situation, when LOIS spread (or CIR state variablec(t)) is correlated with the conditional default probabilityCDP(t). In high-vol spread environment the CVA/DVA values are in- flated/deflated whenρ = 100% andρ = ₋100%by up to 3bp on fixed IRS leg, which is very small when compared to the adjustment from exposure-default correlation.