Amortizing Perpetual Options

This paper introduces amortizing perpetual options (AmPOs), a fungible variant of installment options that replaces explicit payments with notional decay, enabling their valuation to be reduced to equivalent vanilla perpetual American options with derived analytical formulas for exercise boundaries, Greeks, and risk-neutral pricing.

The Gist

The Big Idea: The "Self-Consuming" Infinite Subscription

Imagine you want to buy a subscription to a video game server that never closes. You want to keep playing forever.

In the old way (Traditional Installment Options), you had to pay a monthly fee to keep the server open.

• The Problem: If you ran out of money or forgot to pay, the server shut down for you specifically. But here's the catch: because everyone paid at different times and stopped at different times, every player's "server access" was unique. You couldn't sell your subscription to someone else easily because your server was "older" or "cheaper" than theirs. It wasn't fungible (interchangeable).

The Paper's Solution (AmPOs):
The author, Zachary Feinstein, invents a new type of subscription called an Amortizing Perpetual Option (AmPO).

Instead of you writing a check every month, the subscription has a built-in "self-eating" feature.

• The Analogy: Imagine your subscription is a candle.
  • In the old model, you had to buy a new candle every month to keep the light on. If you stopped buying, the light went out.
  • In the new AmPO model, the candle is designed to melt down automatically. The "payment" is the candle getting smaller. You don't pay cash; you just accept that your "amount of light" (the size of your bet) slowly shrinks over time.

Because every candle melts at the exact same programmed speed, every candle is identical to every other candle. You can sell your candle to a friend, and they can sell it to someone else. It is now fungible and can be traded on a stock exchange.

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How It Works: The "Melting Ice Cube"

Let's break down the mechanics using the paper's logic:

1. The Setup: You buy an option (a bet on whether a stock price will go up or down).
2. The Cost: Instead of paying a fee to the bank, the bank "takes back" a tiny piece of your bet every second.
   • If you bet on $1,000 worth of stock, the bank might take back 1% of your bet every day.
   • Tomorrow, your bet is only worth $990. The next day, $980.
3. The Trade-off:
   • Old Way: You pay cash, your bet stays at $1,000, but if you stop paying, you lose everything.
   • New Way (AmPO): You pay nothing in cash, but your bet slowly shrinks. If the stock price goes up, you win on the remaining amount. If the stock price crashes, your bet shrinks so fast that you lose less money than you would have if you had kept a full-sized bet.

Why Is This a Big Deal?

The paper argues that this simple change solves three massive problems:

1. The "Lapsing" Nightmare

In the old system, if you stopped paying, your contract died. This meant no two contracts were exactly alike. You couldn't trade them on an exchange (like the NYSE or Binance) because the exchange needs identical items to match buyers and sellers.

• The Fix: Since AmPOs melt automatically, no one ever "stops paying." The contract never lapses; it just gets smaller. This makes every contract identical, allowing them to be traded on an exchange.

2. The "Infinite" Mystery

Perpetual options (options that never expire) are popular in crypto, but they are hard to price.

• The Fix: The author proves a mathematical magic trick. He shows that an AmPO is mathematically identical to a standard option on a stock that pays a special dividend.
  • Analogy: Imagine a stock that pays you a dividend, but instead of giving you cash, the dividend is "eating" your share of the stock. By using this trick, the author can use existing, simple formulas to price these complex new options.

3. The "Safety" Factor

The paper runs simulations (like a video game) to see what happens when you change the "melting speed" (the amortization rate).

• Fast Melting (High Rate): Your bet shrinks very quickly. The option becomes cheaper, but it also becomes less sensitive to wild market swings. It's like a "safe" bet that doesn't panic when the market crashes.
• Slow Melting (Low Rate): Your bet stays big for a long time. It acts more like a traditional option, reacting strongly to market changes.

Real-World Applications

The paper suggests two main places where this will be useful:

• Traditional Finance (Wall Street): Big institutions (like pension funds) often want to hold "protective" bets forever to guard against crashes. Currently, they have to constantly buy and sell (roll over) options as they expire, which is risky and expensive. With AmPOs, they can buy a "forever" option and just let it slowly melt, avoiding the stress of expiry dates.
• Decentralized Finance (Crypto/Blockchain): Crypto markets love "perpetual" contracts. However, current crypto options are often clunky and hard to trade because they aren't identical. AmPOs allow these options to be traded on automated market makers (like Uniswap) because every token is exactly the same. This makes the market liquid and efficient.

The Bottom Line

The author has invented a new financial tool that replaces cash payments with shrinking bet sizes.

• Before: You pay cash to keep a big bet alive. If you stop, you die.
• Now: You let your bet slowly shrink to pay for itself. It never dies, it just gets smaller.

This makes the bets identical to one another, allowing them to be traded freely on exchanges, while giving investors a new way to manage risk and volatility. It turns a complex, custom-made financial product into a standardized, easy-to-trade commodity.

Technical Summary

1. Problem Statement

The paper addresses a critical limitation in the design of perpetual options, specifically Continuous-Installment (CI) options.

• The Issue: Traditional CI options require the holder to make explicit, continuous cash payments to keep the contract alive. If the holder stops paying, the contract "lapses" (terminates).
• The Consequence: Because the decision to lapse depends on the specific holder's payment history, the notional units of these contracts are not identical. One unit might have 10 days of payments remaining, while another has 5. This lack of uniformity destroys fungibility, making it impossible to list these options on a centralized exchange or trade them via Automated Market Makers (AMMs).
• The Gap: While perpetual futures are widely traded in DeFi and crypto markets, perpetual options have struggled to gain traction due to this structural incompatibility with exchange-based trading.

2. Methodology

The author proposes a novel financial instrument called the Amortizing Perpetual Option (AmPO) and develops a rigorous mathematical framework for its valuation.

A. Structural Innovation: Implicit Amortization

Instead of requiring explicit cash payments from the holder, the AmPO replaces the installment mechanism with an implicit payment scheme:

• Mechanism: The claimable notional ($N_t$) of the option decays exponentially over time.
• Mathematical Formulation: If $V_t$ is the option premium per unit and $c_t$ is the required installment cost, the notional decays according to $dN_t = -q_t N_t dt$, where the amortization rate is $q_t = c_t / V_t$.
• Result: The holder does not "cancel" the option; rather, their exposure naturally shrinks. This eliminates the "lapse" logic, ensuring all units of the contract evolve identically, thereby restoring fungibility.

B. Valuation Framework

The paper establishes a theoretical equivalence between AmPOs and standard derivatives:

• Equivalence Lemma: The author proves that an AmPO with amortization rate $q_t$ is mathematically identical to a Vanilla Perpetual American Option on a dividend-paying asset, but with modified market parameters:
  • Effective Risk-Free Rate: $r_{eff} = r + q$
  • Effective Dividend Yield: $\delta_{eff} = \delta + q$
• Risk-Neutral Valuation: Under the risk-neutral measure $\mathbb{Q}$, the value of the AmPO is derived as the solution to an optimal stopping problem involving these augmented rates.
• Black-Scholes Application: The paper derives closed-form analytical solutions for Call and Put AmPOs under the Black-Scholes framework with a constant amortization rate $q$.

3. Key Contributions

1. Introduction of AmPOs: The first design for a perpetual option that is fungible and possesses a positive cost-of-carry, making it suitable for exchange-based trading (central limit order books and AMMs).
2. Analytical Pricing Formulas: Derivation of explicit formulas for the price, optimal exercise boundaries, and Greeks (Delta, Gamma, Theta, Vega) for both Call and Put AmPOs.
3. Proof of Non-Cancellability: A formal proof (Proposition 2.3) showing that a rational holder will never voluntarily cancel an AmPO position, as the "cost" is internalized via notional decay rather than external cash outflow.
4. Comparative Statics: A detailed analysis of how the amortization rate ($q$) influences option behavior, effectively acting as a proxy for "time to maturity" in a perpetual setting.

4. Key Results & Findings

A. Pricing and Exercise Boundaries

• Price Sensitivity: The premium of an AmPO is strictly decreasing in the amortization rate $q$. As $q \to 0$, the AmPO converges to a standard perpetual American option; as $q \to \infty$, the price converges to the intrinsic value.
• Exercise Boundaries: Higher amortization rates make the optimal exercise strategy more conservative.
  • For Calls, the optimal exercise boundary decreases toward the strike price $K$.
  • For Puts, the optimal exercise boundary increases toward $K$.

B. The Greeks and "Effective Maturity"

• Theta (Time Decay): Unlike vanilla perpetual options (which have zero Theta), AmPOs exhibit an economic time decay equal to $-q \times \text{Premium}$. This creates a realized time exposure.
• Vega Sensitivity: Higher amortization rates reduce the Vega (volatility sensitivity) of the option.
• Effective Maturity ($T_{eff}$): The paper defines an "effective maturity" for an AmPO as the maturity $T$ of a dated American option that would have the same price.
  • Finding: Even at high amortization rates, the AmPO retains a significant portion of its notional (e.g., >65% at $q=1$) at the time its "effective maturity" date arrives. Thus, $T_{eff}$ is a pricing equivalence, not a termination point.
• Safety Ratio: The ratio of AmPO Gamma to that of a dated option is always less than 1 (typically <80%), indicating AmPOs are more stable against underlying price fluctuations than their dated counterparts.

C. Strategic Implications

• Volatility Strategies: The paper analyzes how varying $q$ affects volatility strategies (Calls, Puts, Straddles). It finds that while higher $q$ reduces the absolute Vega, it also reduces the premium cost. For a fixed budget, there is an optimal amortization rate that maximizes the Vega-to-Premium ratio (leverage), particularly for bearish volatility strategies.

5. Significance and Applications

A. Decentralized Finance (DeFi)

• Liquidity & Fungibility: AmPOs solve the liquidity fragmentation problem in DeFi. Because they are fungible, they can be traded on Automated Market Makers (AMMs) and centralized exchanges, unlike non-fungible installment options.
• Hedging AMM Risk: The paper notes that CI options can hedge the "Loss-Versus-Rebalancing" (LVR) risk faced by liquidity providers in AMMs. AmPOs extend this hedging capability to a liquid, exchange-tradable format.

B. Traditional Finance

• Institutional Adoption: The deterministic cost-of-carry allows institutional investors (e.g., pension funds) to hold perpetual protective positions without the operational risk of "rolling" options near expiration.
• Market Structure: The paper suggests market operators should define standard, fixed amortization rates (e.g., 3, 10, 50 bps/day) to create standardized contract series, similar to how expiration dates are standardized today.

C. Robustness

• Anti-Manipulation: The paper highlights that using a fixed amortization rate (rather than one dependent on the option price) is crucial. If the rate were endogenous (e.g., fixed dollar cost), a temporary price manipulation could cause the amortization rate to explode, wiping out open interest. Fixed rates prevent this fragility.

Conclusion

Zachary Feinstein's work provides the theoretical foundation and practical design for Amortizing Perpetual Options. By converting explicit installment payments into implicit notional decay, AmPOs bridge the gap between the flexibility of perpetual contracts and the liquidity requirements of exchange-traded markets. The paper demonstrates that these instruments can be priced analytically, offer unique risk profiles distinct from dated options, and are essential for the next generation of liquid derivatives in both traditional and decentralized finance.