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.
This paper makes four contributions to the Langevin-Zamrik (LZ) interest rate model. **(1) HJM identification:** we show that LZ is a Gaussian Heath-Jarrow-Morton model with closed-form volatility $\sigma^{\mathrm{HJM}}(\tau) = -(\sigma/m)B_2'(\tau)$, where no-arbitrage is automatic rather than imposed. **(2) Three-regime volatility:** the overdamped, critically damped, and underdamped regimes produce humped, Nelson-Siegel, and oscillatory volatility shapes respectively; the underdamped volatility changes sign, a structural feature impossible in any existing Gaussian HJM family. The no-arbitrage drift is fully closed-form in all three regimes. **(3) Inertial Musiela SPDE:** LZ is lifted to an infinite-dimensional second-order SPDE on the space of forward curves; it collapses back to the 2D LZ system via the Björk-Christensen finite-dimensional realisation theorem, with the critically damped case yielding the no-arbitrage Nelson-Siegel family. **(4) Forward Rate LZ PDE:** the central new result. A change of coordinates from $(r_t,v_t)$ to any two market-observable forward rates $(F_1, F_2)$ transforms the LZ PDE into one whose state variables are directly quoted in swap markets and whose coefficients are exactly the HJM drift and volatility. The original LZ PDE is the special case $x_1=0$, $x_2\to 0$.
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.
We study finite-maturity Israeli put options in the Black-Scholes framework when the writer faces Knightian uncertainty about the risk-neutral drift. Modelling this ambiguity as a bounded drift distortion controlled adversarially by the writer, the valuation problem becomes a three-player zero-sum game whose value satisfies a Hamilton-Jacobi-Isaacs (HJI) equation coupled with a double obstacle. The bang-bang structure of the optimal drift control is established in closed form, and the resulting nonlinear PDE is solved numerically via an explicit upwind finite-difference scheme. We find that drift ambiguity strictly lowers the option value, contracts the early-exercise boundary, and generates a positive ambiguity discount $V^0 - V^\kappa$ that peaks near the exercise boundary and vanishes throughout the cancellation interval centered at $s = K$. The cancellation region is a bounded interval $[s_l(\tau), s_r(\tau)]$ around $K$ that is absent near expiry and grows with time-to-maturity, shrinking and eventually disappearing as $\kappa$ increases.
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American option whose payoff is the running maximum $M_\tau = \sup_{s \leq \tau} X_s$ of the underlying stock price, discounted to the stopping time $\tau$. We analyse the perpetual problem for a geometric Brownian motion with drift $\mu < r$, and exploit the homogeneity of the payoff to reduce the two-dimensional state space $(X_t, M_t)$ to the one-dimensional ratio process $Y_t = X_t/M_t \in (0,1]$. In the continuation region $(y^*, 1)$ the reduced value function $v(y)$ solves an Euler--Cauchy ODE whose general solution is $v(y) = Ay^{\beta_+} + By^{\beta_-}$, and the four constants $(A, B, y^*)$ are determined by a Neumann reflection condition at $y = 1$ and smooth-pasting conditions at $y^*$. We present Peskir's maximality principle, which characterises $y^*$ as the largest candidate threshold for which the ODE solution dominates the obstacle, and we identify the two Doob--Meyer compensators of the discounted value process as the local times of $Y$ at $y^*$ and at the reflecting barrier $y = 1$.
The Israeli option, introduced by Kifer (2000), is a financial contract in which both the holder and the writer possess the right to terminate the contract at any time, making it a zero-sum Dynkin game. We analyse the perpetual Israeli put on a geometric Brownian motion under the risk-neutral measure, and show that its value function satisfies a double obstacle variational inequality with two free boundaries $x_1^* < x_2^*$. In the continuation region $(x_1^*, x_2^*)$ the value solves the Black–Scholes ODE, yielding the explicit form $V(x) = Ax + Bx^{-\alpha}$ with $\alpha = 2r/\sigma^2$. We derive the smooth-pasting system that determines all four unknowns $(A, B, x_1^*, x_2^*)$, recover the perpetual American put as the limit $\delta \to \infty$, and identify the two Doob–Meyer compensators of the discounted value process as the local times of the stock price at each free boundary.