Comparing Market Model Accuracy to the Volatility Smile

Following up our post on LIBOR Market Models in Numerix, I thought it would be valuable to show how the incorporation of shifts and stochastic volatility enables more accurate pricing for instruments with high market skew.

To measure the calibration fit of various models in use today, we tested the standard BGM model, a shifted BGM (SBGM) model and a LMM with stochastic volatility (LMM-SV) for a grid of 55 call swaptions at various expiry/maturity combinations (from 1Y/1Y to 25Y/5Y) and relative strikes (-2% to +2%). (Note: The plotted implied model vols are obtained from Monte Carlo simulation pricing of the swaptions.) Ideally, a model should track the market swaption volatility (representing the market-quoted price of the swaption) as close as possible across expirations, maturities and relative strikes. The better the fit, the more accurate the calculations for pricing and risk analysis.

Findings:

• LMM-SV provided the best overall fit to the market volatility smile across all expiry/maturity combinations
• SBGM accuracy declined for shorter-dated maturities
• BGM was generally only effective for ATM options (0% relative strike)

Average Relative Error Over the Full Swaption Grid

Source: Numerix

Settings: Factors: 2; Solver: Fast; Vol Type: Flat Correlations; Shift Type (SBGM/LMM-SV): Flat Shift; Stoch Vol Type (LMM-SV): CIR Flat; Stoch Vol Vol (LMM-SV): 100%; Stoch Vol Reversion (LMM-SV): 15%