When should I use a lognormal interest rate model instead of Gaussian?

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AcadiFi · Accepted Answer

Lognormal models like Black-Derman-Toy (BDT) and Black-Karasinski (BK) evolve the logarithm of the short rate: d ln(r) = (theta(t) - a*ln(r))*dt + sigma*dW. Because r = exp(X) where X is normal, rates remain strictly positive, matching the historical reality that nominal rates rarely go negative (except briefly in Europe and Japan).

Compare simulated rate paths over twenty years starting from r0 = 4%. A Gaussian Hull-White with sigma = 1.2% produces a 5% probability of rates below zero at year five. A BK model with the same unconditional volatility never crosses zero and produces positively skewed rate distributions that better match observed caplet implied volatilities.

Tradeoffs: lognormal models do not admit closed-form bond prices, requiring numerical methods like trinomial trees. Calibration is slower. Meanwhile, when rates spend long periods near zero, lognormal volatility shrinks proportionally, which can underpredict volatility in low-rate regimes. Practitioners blending these concerns use shifted lognormal or displaced diffusion models: d ln(r + alpha) follows a lognormal process with a shift alpha that allows rates to go slightly negative while preserving most of the lognormal character. This became standard in European swap markets after 2015.

When should I use a lognormal interest rate model instead of Gaussian?

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