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Amortizing Perpetual Options

Zachary Feinstein

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. In this way, analytical expressions are possible 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.

Amortizing Perpetual Options

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. In this way, analytical expressions are possible 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.

Paper Structure

This paper contains 13 sections, 8 theorems, 12 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Amortizing Options
  3. Risk-Neutral Valuation
  4. General Setting
  5. Constant Amortization in the Black-Scholes Framework
  6. Pricing and Hedging
  7. Comparative Statics on Amortization Rate
  8. Discussion and Applications
  9. Setting the Amortization Rate
  10. Applications
  11. Traditional Finance
  12. Decentralized Finance
  13. Conclusion

Key Result

Proposition 2.3

Consider an AmPO with amortization rate $(q_t)_{t \geq 0}$ and payoff function $\Phi: \mathbb{R}_+ \to \mathbb{R}_+$ on some underlying $(S_t)_{t \geq 0}$. A rational option holder, who purchased this AmPO at time $t_0 \geq 0$, will never cancel the option position.

Figures (2)

  • Figure 1: Example \ref{['ex:maturity']}: Comparison of AmPO and dated at-the-money call options.
  • Figure 2: Example \ref{['ex:amort']}: Vega of a $100 position in call, put, and straddle strategies.

Theorems & Definitions (23)

  • Example 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 13 more