Perpetual American Put Option Pricing

Updated 30 August 2025

Perpetual American put option is a financial derivative that grants the right to sell an asset at a fixed strike price at any time without expiration.
Its pricing is established via an optimal stopping framework using free-boundary formulations and scale function techniques in diverse models such as GBM and Lévy processes.
Model extensions include asset-dependent discounting, random horizons, nonlinear volatility, and delivery lags, addressing key market frictions and execution constraints.

A perpetual American put option is a financial derivative conferring the right, but not the obligation, to sell an underlying asset at a specified strike price at any time, with no expiry. Analytically, the option’s value is determined by solving a time-homogeneous optimal stopping problem, typically for an underlying following either a diffusion, jump-diffusion, or more general Lévy process. The structure of the optimal stopping region, the regularity of value functions, and the explicit or semi-analytical form of the price depend sensitively on model specifics, boundary features (e.g., exercise constraints, random horizons, filtrations, or delivery lags), and the presence of additional market frictions such as state-dependent discounting, transaction costs, or constraints on the frequency or nature of exercise.

1. Pricing and Free-Boundary Formulation

The perpetual American put option price, denoted $V(s)$ when the underlying asset is $S_t=s$ , solves

$V(s) = \sup_{\tau \geq 0} \mathbb{E}_s\left[ e^{-r \tau} (K - S_\tau)^+ \right],$

with the process $(S_t)_{t\geq0}$ endowed with a risk-neutral dynamics, e.g., geometric Brownian motion, exponential Lévy process, CEV, or more general jump-diffusion.

For classical models (e.g., geometric Brownian motion), the optimal stopping rule is of threshold type: exercise when $S_t \leq a^*$ . The price function $V(s)$ then solves a stationary variational inequality with a free boundary at $a^*$ , enforcing “value matching” and “smooth fit”: $\begin{cases} \mathcal{L} V(s) - r V(s) = 0, & s > a^*, \ V(a^*) = K - a^*, \ V'(a^*) = -1, \ V(s) = K - s, & s \leq a^*, \end{cases}$ where $\mathcal{L}$ is the infinitesimal generator of the underlying process (Xu et al., 2010).

In jump-diffusion or general Lévy settings, the same character of optimal stopping persists; the boundary $a^*$ , however, must be determined with respect to the process's scale function or relevant fluctuation identities (Kim, 2018, Palmowski et al., 2022).

2. Extensions: Lévy Models, State-Dependent Discounting, and Nonlinear Volatility

When the underlying is modeled as a (possibly spectrally negative) Lévy process, the value function and optimal stopping threshold can often be expressed in terms of the scale function $W^{(r)}$ and its companion $Z^{(r)}$ . For $X_t = \log S_t$ , with Laplace exponent $\psi$ ,

$\mathbb{E}_x\left[ e^{-r \tau_a} \right] = \frac{W^{(r)}(x - a)}{W^{(r)}(0)}.$

The explicit value function typically takes the form

$V(s) = (K - a^*) \frac{Z^{(r)}(\log(s/a^*))}{Z^{(r)}(0)},$

with $a^*$ determined by a smooth-pasting condition (Palmowski et al., 2022, Palmowski et al., 28 Aug 2025).

For options with asset-dependent discounting, the value function becomes

$V^{\omega}_A(s) = \sup_{\tau \geq 0} \mathbb{E}_s\left[ \exp\left(-\int_0^\tau \omega(S_w) dw\right) (K - S_\tau)^+ \right],$

which requires solving an associated Hamilton–Jacobi–BeLLMan (HJB) equation: $\mathcal{A} V^{\omega}_A(s) - \omega(s) V^{\omega}_A(s) = 0, \quad s > a^*,$ with boundary values enforced as above. The stopping region may be interval- or half-line–valued depending on the sign and structure of $\omega$ ; explicit pricing employs so-called omega scale functions, arising from a renewal-type Volterra equation (Al-Hadad et al., 2020, Al-Hadad et al., 2021).

When volatility is a nonlinear function of the second derivative of the price, as in feedback models capturing transaction costs or market illiquidity, the pricing PDE becomes nonlinear: $\frac{1}{2}\sigma(H)^2 S^2 V''(S) + r S V'(S) - rV(S) = 0,\ H = S V''(S).$ Analytic tractability is achieved by reducing to an ODE using transformations in the logarithmic variable, and the optimal boundary is found via an integral equation involving the (generalized) scale function (Grossinho et al., 2016, Grossinho et al., 2017).

3. Structural Variations: Random Horizons, Delivery Lags, Filtration Enlargement

Perpetual American puts with random time horizons (e.g., cancellation at the last passage above a barrier or termination at a drawdown) require incorporating random “time-caps” in the gain functional: $V(x,\bar{x}) = \sup_{\tau} \mathbb{E}_{x,\bar{x}}\left[ e^{-r(\tau \wedge \tau_D)} (K - S_{\tau \wedge \tau_D})^+ \right],$ with $\tau_D$ the first drawdown time past a threshold or a last exit time above a level (Wu et al., 2022, Palmowski et al., 2022, Palmowski et al., 28 Aug 2025). Here, smooth fit may involve local time terms due to singularities in the gain function, and the value function is solved via explicit martingale arguments, scale functions, and careful fluctuation theory.

In models with filtration enlargement (e.g., insider models where the global maximum or minimum of the asset process is known in advance), the state variable must be augmented to include running extrema and auxiliary indicator variables. The candidate value function $V(x, s, \Xi)$ (for price $x$ , running max $s$ , indicator $\Xi$ ) satisfies a three-dimensional free-boundary problem with normal-reflection and normal-entrance conditions along the diagonal ( $x = s$ or $x = m$ ), with closed forms in terms of exponents and hypergeometric functions, and the optimal exercise boundary is a stochastic function of the running max or min (Gapeev et al., 4 Jul 2025).

For delivery lags, the American put’s value decomposes as

$V^{(\delta)}(t,X) = P(T-\delta, X) + U^{(\delta)}(t,X),$

where $P$ is the European put price at shifted maturity and $U^{(\delta)}$ solves an optimal stopping problem with the Greek Theta of the European option as running payoff. The exercise boundary is strictly increasing with respect to time and analyzed via variational inequalities (Liang et al., 2018).

4. Exercise Strategies: One-sided, Double-Continuation, and Discrete Exercise

The optimal exercise strategy is typically a threshold rule—exercise when $S_t \leq a^*$ . However, nonmonotone and double-continuation regions can arise:

With negative effective discount rates ( $q = r - \gamma < 0$ as encountered in real options or stock loans), the stopping region is an interval $[\ell^*, u^*]$ , and early exercise is optimal only when the underlying is inside this band. Strategies must be formulated via entrance times to intervals (Donno et al., 2017, Palmowski et al., 2020).
When exercise is restricted to random discrete times, specifically the arrival times of a Poisson process (i.e., periodic or randomly spaced Bermudan structures), the optimal strategy remains of barrier type (albeit at discrete times): exercise at the first observation when $S_t \leq a^*$ (put) or $S_t \geq b^*$ (call). Explicit formulas in terms of scale functions accommodate one-sided or two-sided jumps in the underlying's Lévy process (Pérez et al., 2017, Palmowski et al., 2020).
In models integrating delivery lags, filtration enlargement, or drawdown caps, the stopping region may involve state-dependent boundaries or additional pathwise information, modifying the straightforward barrier characterization (Wu et al., 2022, Palmowski et al., 28 Aug 2025, Gapeev et al., 4 Jul 2025).

5. Numerical and Analytical Methods

Analytical solutions are possible in cases with sufficient symmetry (constant coefficients, known scale functions, or tractable jump structures). When not, tractable numerical methods include:

Non-standard finite difference schemes with non-uniform (quasi-uniform) grids, enabling exact enforcement of boundary conditions at infinity and yielding high convergence rates with rigorous a posteriori error control via Richardson extrapolation (Fazio, 2014).
Explicit computation (and calibration) of scale functions for spectrally negative Lévy processes via inversion of Laplace transforms or, in the tempered stable class, via the characteristic function of first passage times and the root-finding solution of the associated “martingale condition” equations (Kim, 2018).
Free-boundary ODEs/PDEs are solved either analytically, when possible, or numerically using root-finding and smooth fit to pin down the optimal exercise boundary; for asset-dependent discounting, solution requires solving renewal-type (Volterra) equations for omega-scale functions and, for selected $\omega$ , recasting the problem as standard hypergeometric or Bessel ODEs (Al-Hadad et al., 2020, Al-Hadad et al., 2021).

6. Practical Implications, Hedging, and Model-Independence

The structure of the free-boundary, and its dependence on parameters (volatility, jump intensity/size, interest and dividend rates, asset-dependent discounting), has direct implications for risk management and hedging.

For low interest rates or high volatility (e.g., in the CEV model), the optimal exercise boundary is exponentially close to zero; thus, perpetual American put values become highly sensitive to model parameters and option values can be dominated by rare, extreme events (Xu et al., 2010).
In negative discount rate regimes, options should not be exercised "too deep in the money", as value grows without exercise (the waiting premium dominates); this is crucial in contexts like stock loans or real options (Donno et al., 2017).
Discrete-time hedging with restrictions (e.g., a single trade) can be optimized by solving joint optimal stopping and control problems, with free-boundary characterization for trading boundaries and explicit formulae minimizing the tracking error variance (Cai et al., 2020).

Model-independent arbitrage bounds (based on duality relations and Legendre–Fenchel transforms between American and European options) provide structural upper and lower pricing constraints, ensuring arbitrage-free prices for American puts regardless of the precise dynamic specification (Cox et al., 2013).

Table 1: Key Modelling Extensions and the Character of the Stopping Region

Model Feature	Exercise Region Structure	Solution Method
GBM / standard Lévy, const. discount	One-sided (threshold)	Closed-form/barrier via scale func
Asset-dependent discount, ω(s) ≥ 0	One-sided or interval	Omega-scale functions, renewal eq.
Negative discount rate	Double continuation (interval)	Scale/exit theory, interval opt.
Random horizon, path-dependent features	Threshold, modified via survival Z	Filtration enlargement, local time
Delivery lag	Smooth, time-increasing boundary	Variational inequality, free-bdry
Insider, filtration enlarged with extrema	Stochastic, state-dependent boundary	Free-bdry ODEs, reflection, fit

The rich diversity of perpetual American put option models, spanning classical Black–Scholes and Lévy processes, asset-dependent discounting, filtration enlargement, drawdown or cancellation features, and exercise constraints, is unified by the optimal stopping framework and, at the technical level, by the critical role of martingale/superharmonic arguments, free-boundary analysis, and scale functions. This spectrum of results forms the modern theoretical and computational basis for pricing, hedging, and risk assessment of such derivatives under realistic market features and operational constraints.