Kelly Sizing for Options Carry: How Much of the Notional Should the Overlay Cover?

Abstract

A $2M micro-cap position. A $1.7M liquid proxy basket that replicates 78% of its variance. Net options carry of $52,500 per month from writing 30Δ calls against a put-spread hedge. The question is not whether the carry is positive. It is how much of the $2M should carry it. Full 85–100% coverage maximizes expected dollars and nearly maximizes drawdowns too. Half-Kelly coverage at 60–70% keeps roughly three-quarters of the growth rate with half the volatility. Here is the math, the variance input most operators get wrong, and the liquidity ceiling that overrides the formula when the proxy’s own options market is thin.

1. The Sizing Math: Full Kelly vs. Half-Kelly

Start with the continuous Kelly criterion. For an overlay with expected monthly return μ on deployed notional and monthly standard deviation σ, the coverage fraction maximizing long-run log growth is:

c* = μ / σ²

where c* is overlay notional divided by bankroll. Bankroll here is the $2M micro-cap notional — the capital the overlay exists to monetize.

Measured inputs from the live structure:

• Net carry yield μ = 3.09% per month on overlay notional. That is $52,500 on the $1.7M basket, after the $11,500 monthly put-spread cost, before transaction costs.
• Monthly σ = 15% on overlay notional. This is not backtested call-premium wiggle. It is realized P&L volatility of the combined overlay including premium variance, tracking error against the micro-cap, and gap months. Section 3 breaks it down.

Full Kelly:

c* = 0.0309 / 0.0225 = 1.37, or 137% coverage

That is a $2.74M overlay against a $2M position. Untradeable on this proxy — more on that in section 4 — and a drawdown machine even if it were tradeable.

Half-Kelly is half of that:

c_half = 0.685, or ~68% coverage, ~$1.37M overlay

Expected log growth at half-Kelly is 75% of the full-Kelly maximum, with half the volatility and roughly half the drawdown depth. The arithmetic:

Full 85% replication collects $10,200 more per month than half-Kelly. It also adds $49,000 of monthly standard deviation and widens the 2σ loss by four points of bankroll. That is the trade overlay sizing always forces: linear carry against quadratic variance drag. The Kelly criterion exists to price that exchange. Full replication ignores it.

Operating rule: size to half-Kelly, then apply the liquidity cap below. Take the lower of the two.

2. Why Fractional Kelly, Not Full Kelly

Full Kelly assumes μ and σ are known exactly. They never are. Three discounts apply.

Estimation error. μ = 3.09% is a trailing average from a normal-to-elevated IV regime. Forward edge runs 20–40% lower after roll slippage, assignment friction, and one low-vol quarter. Halving input μ halves output c*. Half-Kelly admits the overstatement without pretending to know its exact size.

Drawdown tolerance. At 137% coverage, a 2σ month loses ~37% of bankroll — on top of the illiquid micro-cap underneath. At 68%, the same shock costs ~18%. Same edge, survivable outcome. A carry trade layered on a concentrated position cannot be sized as if the bankroll were cash.

Fat left tails. Short-call overlays have a clipped right tail and a gap-prone left one: assignment in a melt-up, tracking failure in a selloff, IV collapse that zeroes premium for months. Variance understates these. Half-Kelly keeps ~75% of full-Kelly growth at ~50% of the variance. Quarter-Kelly (35–40%) is the fallback with fewer than 24 months of σ history.

3. What σ to Use: Variance of Monthly Premium Income

Most options position sizing errors come from σ, not μ. Three components, summed — not the largest one alone.

Premium variance (∼7% monthly). A 5-point IV swing moves collection by ±$8,000 on a $1M ETF sleeve at 30% IV, ±$12,000–$15,000 per $500K single-name sleeve at 40–45% IV.

Tracking and basis variance (∼9% monthly). The overlay pays on the proxy; the risk lives in the micro-cap. Divergence — earnings gaps, sector rotation, a liquidity air pocket — produces flat overlay months against combined-book losses. This leg exceeds the premium leg. Sizing on premium variance alone understates σ by nearly half and oversizes by ~2×.

Gap and assignment jumps. A 15% overnight gap through the short 30Δ strike on one $500K sleeve costs ~$45,000 net of the put spread — three months of that sleeve’s carry. Hold 10–15% of trailing twelve-month carry as an unencumbered assignment reserve; exclude it from μ.

Combined σ: √(0.07² + 0.09² + cross-term) ≈ 12–13% normally, rounded to 15% for one gap event per year. Re-estimate quarterly on 24 months. Above 18% for two quarters, cut to quarter-Kelly.

4. The Market-Impact Ceiling: When Size Moves the Proxy’s Market

Kelly gives an upper bound from risk. Liquidity gives a harder one from the market. The overlay writes options on the proxy — and at scale, it becomes the proxy’s options flow.

Constraints for this basket:

1. Equity leg: overlay per name < 5% of 30-day ADV. A $500K sleeve on $30M ADV is 1.7%. Above 5%, assignment delta-hedging prints against your own marks.
2. Options size: written contracts < 5–10% of strike OI. The ETF leg shows OI > 2,000 near the money; 100–150 contracts is absorbable. At 300+ (needed for 100% coverage), the roll moves IV 1–2 points against the position — ~$4,000–$6,000/month in lost forward premium per point on a $1.7M overlay.
3. Roll concentration. Above ~200 contracts per expiry per name, split across 30 and 45 DTE. Same theta, half the footprint.

For this specific proxy, the ceiling binds at roughly $1.4–$1.5M overlay notional — 70–75% coverage. That is just above half-Kelly ($1.37M), so half-Kelly governs. On a thinner proxy — single-name ADV under $15M, strike OI under 800 — the ceiling drops to $600–$800K (30–40% coverage) and liquidity governs regardless of what Kelly says. Check both, run the smaller.

5. What to Run

For the $2M / 78%-R² / $1.7M-proxy structure in this note:

• Coverage: 65–70% (~$1.3–$1.4M overlay). Half-Kelly, inside the liquidity ceiling.
• Expected carry at 68%: ~$42,000/month, ~25% APY on the $2M notional, net of the scaled put-spread cost. Lower than the $52,500 full-replication figure. Drawdown depth lower by a wider margin.
• Review triggers: re-run μ and σ quarterly; cut to 40% on any single-month loss above 12% of bankroll; halt new writes if sector IV drops below 18 for two consecutive rolls — the edge is gone and no sizing formula restores it.

Full replication is the reference case, not the target. The Kelly criterion, applied to net carry yield with a variance that includes tracking error and capped by the proxy’s own options liquidity, lands 15–25 points below it. That gap is the cost of surviving long enough for the carry to compound.