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arxiv: 2512.06505 · v4 · pith:K753XX3Pnew · submitted 2025-12-06 · 💱 q-fin.PR · q-fin.MF

Amortizing Perpetual Options

Zachary Feinstein This is my paper

Pith reviewed 2026-05-17 01:07 UTC · model grok-4.3

classification 💱 q-fin.PR q-fin.MF
keywords amortizing perpetual optionsperpetual American optionsdividend-paying assetexercise boundariesrisk-neutral valuationGreeksamortization ratecontinuous installment options
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The pith

Amortizing perpetual options reduce to equivalent vanilla perpetual American options on dividend-paying assets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces amortizing perpetual options as a fungible variant of continuous-installment options suitable for exchange-based trading. Traditional installment options lose fungibility because they lapse when payments stop. AmPOs avoid this by using decay of the claimable notional as an implicit payment method so that all units evolve identically. The central move is to show that this construction makes AmPO valuation identical to that of a vanilla perpetual American option on a dividend-paying asset. A reader would care because the reduction immediately supplies analytical formulas for exercise boundaries and risk-neutral prices of calls and puts.

Core claim

We demonstrate that AmPO valuation can be reduced to an equivalent vanilla perpetual American option on a dividend-paying asset. This enables analytical expressions for the exercise boundaries and risk-neutral valuations for calls and puts. These formulas and relations allow us to derive the Greeks and study comparative statics with respect to the amortization rate. Illustrative numerical case studies demonstrate how the amortization rate shapes option behavior and reveal the resulting tradeoffs in the effective volatility sensitivity.

What carries the argument

The reduction of amortizing perpetual option valuation to an equivalent vanilla perpetual American option on a dividend-paying asset via the implicit notional decay payment scheme.

If this is right

  • Analytical expressions for the exercise boundaries become available for calls and puts.
  • Risk-neutral valuations are obtained in closed form.
  • The Greeks can be derived explicitly from the equivalent model.
  • Comparative statics with respect to the amortization rate can be performed.
  • Numerical studies show how the amortization rate changes the option's effective volatility sensitivity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Traders could reuse existing perpetual American option pricing code to handle AmPOs without new implementations.
  • The amortization rate functions as an adjustable parameter that lets issuers tune the option's risk profile and holding period characteristics.
  • The same decay approach to preserving fungibility might be applied to other installment or path-dependent contracts to make them exchange-tradable.

Load-bearing premise

The implicit payment scheme via decay of the claimable notional produces an economically equivalent claim to a dividend-paying asset without introducing unmodeled frictions, path-dependence, or inconsistencies in the risk-neutral measure.

What would settle it

A direct numerical solution of the amortizing notional dynamics or a Monte Carlo simulation of the decay process could be compared against the analytical exercise boundaries and values from the reduced model to test whether they match.

Figures

Figures reproduced from arXiv: 2512.06505 by Zachary Feinstein.

Figure 1
Figure 1. Figure 1: Example 3.7: Comparison of AmPO and dated at-the-money [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 3.12: Vega of a $100 position in call, put, and straddle strategies. 4 Discussion and Applications 4.1 Setting the Amortization Rate In Section 3.2, we imposed a fixed qt ≡ q > 0 amortization rate. While the general theory permits endogenous amortization (i.e., dependent on the option value), such a construction can introduce novel manipulations in AmPOs in practice. For instance, consider an AmPO … view at source ↗
read the original abstract

In this work, we introduce amortizing perpetual options (AmPOs), a fungible variant of continuous-installment options suitable for exchange-based trading. Traditional installment options lapse when holders cease their payments, destroying fungibility across units of notional. AmPOs replace explicit installment payments and the need for lapsing logic with an implicit payment scheme via the decay of the claimable notional. This amortization ensures all units evolve identically, preserving fungibility. We demonstrate that AmPO valuation can be reduced to an equivalent vanilla perpetual American option on a dividend-paying asset. This enables analytical expressions for the exercise boundaries and risk-neutral valuations for calls and puts. These formulas and relations allow us to derive the Greeks and study comparative statics with respect to the amortization rate. Illustrative numerical case studies demonstrate how the amortization rate shapes option behavior and reveal the resulting tradeoffs in the effective volatility sensitivity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces amortizing perpetual options (AmPOs) as a fungible variant of continuous-installment options that replaces explicit installment payments and lapsing logic with an implicit scheme based on decay of the claimable notional. The central result is that AmPO valuation reduces to pricing a vanilla perpetual American option on a dividend-paying asset, with the amortization rate serving as the continuous dividend yield. This mapping yields analytical expressions for the early-exercise boundary (via the characteristic equation from value-matching and smooth-pasting) and risk-neutral valuations for calls and puts, from which Greeks and comparative statics with respect to the amortization rate are derived. Numerical case studies illustrate the effects of varying the amortization rate on option values and volatility sensitivity.

Significance. If the reduction holds, the work supplies a practical analytical toolkit for a new class of exchange-tradable perpetual options by leveraging established perpetual American formulas rather than requiring bespoke numerical methods. Credit is given for the explicit demonstration that, after factoring out the deterministic notional decay, the value function satisfies the Black-Scholes PDE and free-boundary conditions of the mapped model, as well as for the consistency checks through Greeks and numerical illustrations of amortization-rate tradeoffs.

minor comments (3)
  1. [Abstract] Abstract: the statement that the reduction 'enables analytical expressions' would be strengthened by a one-sentence mention of the characteristic equation or the explicit form of the boundary condition.
  2. [Numerical Illustrations] Numerical case studies: the specific parameter values (volatility, risk-free rate, amortization rates) used to generate the Greeks and sensitivity plots should be stated explicitly to support reproducibility.
  3. [Model Setup] Notation: the distinction between the original notional and the decaying claimable notional could be clarified with a short table or explicit symbols in the model setup to avoid any ambiguity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work on amortizing perpetual options. We appreciate the recognition of the reduction to vanilla perpetual American options and the practical implications. Since the report recommends minor revision but does not list specific major comments, our responses below address the key points from the referee's summary. We are prepared to make minor adjustments if needed.

read point-by-point responses

  1. Referee: The central result is that AmPO valuation reduces to pricing a vanilla perpetual American option on a dividend-paying asset, with the amortization rate serving as the continuous dividend yield. This mapping yields analytical expressions for the early-exercise boundary and risk-neutral valuations for calls and puts.

    Authors: We confirm that this is the core contribution of the paper. By setting the amortization rate as the dividend yield, the AmPO satisfies the same PDE and boundary conditions as the standard perpetual American option, leading to the closed-form solutions presented. revision: no

  2. Referee: Credit is given for the explicit demonstration that, after factoring out the deterministic notional decay, the value function satisfies the Black-Scholes PDE and free-boundary conditions of the mapped model, as well as for the consistency checks through Greeks and numerical illustrations of amortization-rate tradeoffs.

    Authors: We are pleased that the referee acknowledges the rigor in our derivation. The factoring out of the notional decay is shown in Section 2, leading to the standard perpetual American problem. The Greeks and numerical studies in Section 5 further validate the model. revision: no

Circularity Check

0 steps flagged

No significant circularity; reduction to standard model is self-contained

full rationale

The paper establishes the AmPO valuation equivalence by factoring the deterministic notional decay out of the claim value function, after which the resulting process satisfies the Black-Scholes PDE and the usual value-matching/smooth-pasting free-boundary conditions for a perpetual American option with continuous dividend yield set equal to the amortization rate. This mapping is derived directly from the risk-neutral dynamics and the implicit payment scheme definition; the subsequent analytical exercise boundary and option price formulas are the independently known closed-form solutions for the mapped perpetual American problem. The amortization rate enters as an exogenous model parameter rather than a fitted quantity, and no self-citation, ansatz smuggling, or redefinition of inputs as outputs occurs in the central derivation chain. The numerical illustrations simply vary this parameter within the mapped model.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on standard risk-neutral pricing for perpetual American options plus the new modeling choice that notional decay exactly replicates the economic effect of continuous installments.

free parameters (1)
  • amortization rate
    Controls the speed of notional decay and is the central parameter varied in the comparative statics and numerical studies.
axioms (1)
  • domain assumption Risk-neutral valuation applies directly to the mapped perpetual American option on a dividend-paying asset.
    Invoked to obtain closed-form exercise boundaries and risk-neutral valuations.
invented entities (1)
  • Amortizing perpetual option (AmPO) no independent evidence
    purpose: Fungible continuous-installment claim achieved via notional decay
    Newly defined financial instrument whose properties are derived in the paper.

pith-pipeline@v0.9.0 · 5432 in / 1357 out tokens · 50922 ms · 2026-05-17T01:07:03.615062+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Designing On-Chain Options: Amortizing Perpetual Options

    q-fin.MF 2026-05 unverdicted novelty 5.0

    Introduces amortizing perpetual options as a blockchain-native primitive that supports decentralized risk management including endogenous collateralization and de-peg insurance.