Bermudan Swaption Model Risk for an Insurer
An insurer has sold a ten-year Bermudan swaption to a bank. The option is exercisable annually after year two and, if exercised, enters a GBP 500m swap paying fixed 3.5% and receiving SONIA. The insurer uses a one-factor Hull-White model calibrated to swaptions up to ten years for pricing and hedging. That model gives a liability of GBP 42m. A displaced-diffusion Libor market model, calibrated to the same swaptions, gives GBP 55m. The insurer’s internal capital model uses the Hull-White mark. The risk committee asks whether the GBP 13m difference is acceptable. Rates have been volatile; the SONIA forward curve is inverted at the short end; the insurer has no Bermudan swaption hedging desk and hedges only delta with linear swaps. The insurer must decide which model to use for reserving, capital, and hedging.
1. Explain why two models calibrated to the same swaptions can produce materially different Bermudan values.
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A Bermudan swaption is an option on an option: its value depends on the exercise boundary and on the joint distribution of rates across exercise dates. A one-factor Hull-White model imposes a restrictive correlation structure and a deterministic volatility term structure. It can match the calibration swaptions but may not reproduce the co-movement of rates that determines whether exercising early is optimal. A displaced-diffusion Libor market model has more factors and can capture a richer correlation and volatility structure, so it can produce a different exercise boundary and a different value.
Calibration to the same swaptions is not enough if those swaptions do not span the Bermudan exercise dates, or if the calibration weights and interpolation differ. Negative rates and the choice of displacement also matter. The GBP 13m difference, about 31% of the Hull-White liability, is therefore model risk, not a data error. It should be quantified and disclosed rather than treated as acceptable simply because both models calibrate.
2. Recommend which model to use for reserving, capital, and hedging, and what controls to implement.
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For reserving, the insurer should not book the lower Hull-White value without a reserve. A prudent approach is to reserve at the higher of the two model values, or at Hull-White plus a model-risk reserve sized to cover the GBP 13m difference. For capital, the internal model should include model risk and should be recalibrated to coterminal swaptions where possible; using a single Hull-White mark understates the liability.
For hedging, Hull-White delta with linear swaps is inadequate because it ignores vega and the correlation exposure embedded in the Bermudan. The insurer should add swaption hedges, at least for the coterminal swaptions, and monitor the hedge under both models. Controls should include independent validation, a documented model-risk policy, a limit on the difference between models, and regular stress testing of mean reversion and volatility. If the insurer cannot hedge the optionality, it should reduce the position or hold additional capital rather than rely on the lower model value.