30-NOV-2024
We introduce the Langevin-Zamrik partial differential equation, a zero-coupon bond pricing equation derived from the full underdamped Langevin dynamics of the short interest rate. The state space is two-dimensional, comprising the short rate $r_t$ and its velocity $v_t = \dot{r}_t$. For a quadratic (OU-type) potential the equation admits a closed-form affine solution $P = \exp\{A(\tau) + B_1(\tau)r + B_2(\tau)v\}$, where the coefficient pair $(B_1, B_2)$ satisfies a $2\times 2$ linear ODE system solved explicitly via the matrix exponential of the inertia-friction-spring matrix $M$. The three damping regimes — overdamped, critically damped, and underdamped — produce qualitatively distinct yield curve shapes: monotone, single-humped, and oscillatory respectively. The Vasicek formula is recovered exactly in the overdamped limit $m\to 0$. A central result is that the Langevin-Zamrik yield curve in the overdamped regime is structurally equivalent to the Nelson-Siegel-Svensson specification, providing its first no-arbitrage structural derivation. Extensions to nonlinear potentials, state-dependent volatility, multi-factor dynamics, and Lévy noise are discussed.