Perpetual Options | Freeport Markets

From The Logbook. The original article is reproduced in full, with minor grammar corrections; references to market conditions reflect the context in which it was written. Read the editorial clarification below.

Imagine you’ve got a derivative that’s got plenty of curvature (gamma), yet, by definition, doesn’t decay with the passage of calendar time (no theta). How can convexity exist without time decay? That’s the riddle of perpetual options.

The puzzle: Options have curvature (gamma). That curvature “costs” something. For ordinary, finite-maturity options, the cost shows up as calendar theta (time decay). But a perpetual option never gets closer to expiry, so its calendar theta is, by definition, zero. How can you have gamma with no theta?

Start from the Black–Scholes PDE with dividend yield q. Rearranging for theta shows how gamma and carry fund time decay:

\Theta \;=\; \frac{\partial V}{\partial t} \;=\; -\frac{1}{2}\sigma^2 S^2 \Gamma \;-\; (r-q)\,S\,\Delta \;+\; rV.

In plain English, convexity (gamma) needs funding. In finite-T, some (often most) of that funding appears as calendar theta; the rest is rate carry. Perpetual options have no expiry, so the value is time-stationary with ∂V/∂t=0. The pricing equation in the continuation region becomes an ODE:

\frac{1}{2}\sigma^2 S^2 V \;+\; (r-q)\,S\,V \;-\; rV \;=\; 0.

This is naturally an American optimal-stopping problem (exercise boundary pinned by value-matching and smooth-pasting). We note that with no calendar decay, the balance is entirely between gamma/delta terms and rate terms. Setting Θ=0 in the PDE shows that carry alone funds convexity in stationarity:

\Theta=0 \quad\Longrightarrow\quad \frac{1}{2}\sigma^2 S^2 \Gamma \;=\; -\Big[(r-q)\,S\,\Delta \;-\; rV\Big].

If r=0, then q > 0 is necessary such that lost dividend carry pays for gamma. If q=0 the RHS is zero, so Γ= 0, giving a contradiction. More broadly, a zero rate curve forever can’t exist in equilibrium. If every maturity had r=0 then government-issued perpetuity prices diverge, implying that governments can issue a single bond to finance all its purchases (which is clearly impossible).

In general, for finite-time options, gamma is funded by calendar theta plus carry, for perpetuals, carry alone funds gamma. The sensitivities behave intuitively where higher r or higher q generally pulls exercise earlier (waiting is costlier) and lower σ reduces convexity, pushing values toward intrinsic and moving boundaries closer.

Disclaimer

The information provided on TheLogbook (the "Substack") is strictly for informational and educational purposes only and should not be considered as investment or financial advice. The author is not a licensed financial advisor or tax professional and is not offering any professional services through this Substack. Investing in financial markets involves substantial risk, including possible loss of principal. Past performance is not indicative of future results. The author makes no representations or warranties about the completeness, accuracy, reliability, suitability, or availability of the information provided.

This Substack may contain links to external websites not affiliated with the author, and the accuracy of information on these sites is not guaranteed. Nothing contained in this Substack constitutes a solicitation, recommendation, endorsement, or offer to buy or sell any securities or other financial instruments. Always seek the advice of a qualified financial advisor before making any investment decisions.