Valuation of Interest Rate Derivatives › Limitations and Practical Challenges

Black's model, while industry-standard, carries significant limitations that you must understand.

The assumption of lognormal rate distributions becomes problematic when rates approach zero or turn negative, as occurred in Europe and Japan. Lognormal models assign zero probability to negative rates, yet reality has shown otherwise. This led to the development of shifted lognormal and normal (Bachelier) models that handle negative rates appropriately. The choice of model affects both pricing and Greeks, particularly for low-strike options.

Model calibration presents ongoing challenges. Volatility surfaces evolve continuously, and models calibrated to today's market may misprice tomorrow's trades. The volatility cube contains thousands of points, each requiring consistent and stable calibration. You must balance fit quality against model stability and computational efficiency. We'll explore calibration techniques in depth in Chapter 21.

Correlation assumptions matter significantly when pricing instruments that depend on multiple forward rates. The LMM requires specifying a correlation matrix between forward rates, and different correlation structures can produce materially different prices for instruments like spread options or CMS products. Historical correlations may not persist, and there's no unique way to specify the correlation structure from market observables.

Liquidity risk poses practical challenges for hedging. While vanilla caps, floors, and swaptions trade actively in major currencies, exotic variations and longer tenors may have limited liquidity. Bid-ask spreads widen during market stress, exactly when hedging becomes most important.