11-APR-2025
We price interest rate derivatives — caplets, floorlets, bond options, and swaptions — under the Langevin-Zamrik (LZ) model, using its representation as a Gaussian Heath-Jarrow-Morton (HJM) model in forward-rate space. Since the LZ forward rate is normally distributed (not lognormal), caplets are priced exactly as put options on zero-coupon bonds — whose log-prices are Gaussian — giving a Black formula on the lognormal bond price with variance $\sigma_P^2(\tau_0, \delta)$. Swaptions use the Bachelier (normal) formula on the approximately-Gaussian swap rate. We derive closed-form expressions for the bond option variance $\sigma_P^2(\tau_0, \tau_S)$, the HJM variance profile $V(\tau) = \int_0^\tau [\sigma^{\mathrm{HJM}}(s)]^2\,ds$, and the swaption cross-covariance $C_{ij}(\tau_0)$ in all three LZ damping regimes and in 1FHW. A key structural finding is that in the underdamped regime the bond option volatility is non-monotone in bond tenor $\tau_S$ — it can decrease as tenor increases near resonance periods $n\pi/\omega$ — a phenomenon impossible in any Gaussian HJM model with positive $\sigma^{\mathrm{HJM}}$. A unified LZ-HJM pricer algorithm computes all instruments in a single pass.