7.2 Position Sizing — Kelly, Fractional, Fixed-Risk
Module 07 · Lesson 7.2 · Estimated read 9 min
Lesson 7.1 established that position sizing determines whether positive expectancy translates into long-run wealth or into ruin. This lesson catalogs the four sizing frameworks worth knowing — Kelly, fractional Kelly, volatility-targeted, and fixed-fractional — and addresses the practical complications that options trading introduces. Each framework targets a different objective: Kelly maximizes expected log wealth, fractional Kelly trades growth for drawdown control, volatility-targeted equalizes risk per position across instruments, and fixed-fractional is the simplest practical implementation that most traders can execute without error. The right answer is rarely “the most sophisticated framework.” It is the framework the trader will actually follow for several thousand trades.
1. Kelly criterion: full Kelly and the math behind growth-optimal
Kelly sizing answers the question: what fraction of bankroll, bet on each opportunity, maximizes the expected logarithm of terminal wealth? The log objective is not arbitrary — it corresponds to the only sizing rule that, applied repeatedly, converges to the highest long-run growth rate while having zero probability of true ruin in the unit-bet limit.
For a binary bet with win probability p, win amount b times the stake, and loss of the full stake, the Kelly fraction is:
For asymmetric trading where wins and losses can be different multiples of risk, the generalized form uses the win amount W and loss amount L (both as positive numbers, expressed as fractions of bankroll):
Worked example: a system with 55% win rate, 1.5R wins, and 1.0R losses where R is “risk per trade”. In Kelly terms, b = 1.5. Full Kelly fraction:
Full Kelly says risk 25% of bankroll on every trade. This is theoretically growth-maximizing — and operationally insane. Full Kelly produces drawdowns that no human can hold through. Empirically, Kelly-sized strategies produce expected drawdowns near 50% even with healthy edge. The math is right; the math is not what limits practical sizing.
2. Fractional Kelly: why half-Kelly is the practical standard
Fractional Kelly sizes at f = k × f* for some k between 0 and 1. The key result is that growth rate is roughly quadratic in fraction-of-Kelly: half-Kelly gives roughly 75% of full-Kelly growth with about half the drawdown variance. Quarter-Kelly gives roughly 44% of growth with about a quarter of drawdown.
| Fraction | Growth rate (% of full) | Expected max drawdown |
|---|---|---|
| Full Kelly (1.0) | 100% | ~50% |
| Half Kelly (0.5) | ~75% | ~25% |
| Quarter Kelly (0.25) | ~44% | ~12% |
| Tenth Kelly (0.1) | ~19% | ~5% |
The right fraction depends on three factors. First, edge confidence: if win rate and reward-to-risk are uncertain, full Kelly is dangerous because the formula is sensitive to inputs — a 5-percentage-point error in win rate can move full Kelly by 20% of bankroll. Second, drawdown tolerance: real human tolerance is typically 15-25% before behavioral interference begins. Third, trade independence: Kelly assumes independent trades, but most trading involves correlated bets, which inflates effective Kelly fraction.
The practical heuristic: half-Kelly when edge is documented over hundreds of trades; quarter-Kelly when edge is plausible but newer; tenth-Kelly when scaling a strategy from paper to live. Most professional traders settle near 25-30% of full Kelly across positions, well below what the formula suggests.
3. Volatility-targeted sizing
Kelly sizes based on edge. Volatility-targeted sizing sizes based on instrument variance — the idea is to take roughly equal expected risk per position regardless of how volatile the instrument is. The formula:
Concrete example: target portfolio risk contribution per position of $500 daily standard deviation. SPY has a daily standard deviation of roughly 0.8% on a $500 share price — about $4 per share per day. A vol-targeted SPY position would size at 500 / 4 = 125 shares. NVDA at $900 with 3% daily standard deviation is $27 per share per day; the vol-targeted NVDA position is 500 / 27 = 18-19 shares.
The benefit is that risk is comparable across instruments. The trader is not implicitly taking 7× the risk on NVDA versus SPY just because both positions look the same on a P&L screen. The cost is operational complexity: vol estimates require maintenance, and overnight gap risk in single names is not captured by intraday volatility.
For options, the analog is to size by dollar delta or dollar gamma exposure rather than contract count. Two long-call positions of identical contract count but different underlyings can carry vastly different risk because delta scales with underlying price and option moneyness. Targeting dollar delta of $X per position is a practical analog to vol-targeting in equities.
4. Fixed-fractional and the 1-2% rule
The simplest framework: risk a fixed fraction of bankroll on every trade. The classical rule is 1-2% per trade, where “risk” means the dollar loss if the stop is hit. For a $100,000 account at 1%, the dollar loss budget per trade is $1,000. If the planned stop is 5% below entry on a stock, the position size is $1,000 / 0.05 = $20,000. If the stop is 10% below entry, the position size halves to $10,000.
Fixed-fractional is the rule most professional traders actually follow, even when they could in principle use Kelly. Three reasons. First, it is mechanical: position size is a deterministic function of stop distance and risk budget, no edge estimation required. Second, it caps catastrophic loss at a known fraction per trade; ruin probability is tractable. Third, it adapts to volatility automatically — wider stops produce smaller positions.
The 1% rule is the conservative working number for new traders or new strategies. The 2% rule is appropriate when edge is documented over several hundred trades and the trader can demonstrate emotional tolerance for the resulting drawdowns (a 5-loss streak at 2% sizing is a 10% drawdown, which sounds tolerable until experienced live).
5. Options sizing complications: defined-risk, naked, spreads
Equity sizing is straightforward because the dollar risk is bounded by stop placement. Options introduce three structural complications that make sizing rules instrument-specific.
Defined-risk long premium (long calls, long puts, debit spreads). The maximum loss is the debit paid — the position cannot lose more than 100% of its cost. Sizing is straightforward: position size such that debit equals the chosen risk fraction. The 1-2% rule applies cleanly. Caveat: the 100% loss is more probable on long premium than equity stops, because options decay through time even when the underlying is flat. Effective hit rate matters more than headline risk.
Naked short premium (short puts, short calls, short strangles). Maximum loss is undefined (uncapped on short calls; bounded by strike on short puts but typically much larger than premium received). Sizing must be done on margin requirement and stress-test loss, not premium received. A common professional rule: limit naked short premium positions such that a 2-standard-deviation adverse move would not exceed 5% of bankroll. This is far more conservative than premium-based sizing and is the only safe approach for naked structures.
Defined-risk credit spreads (credit put spreads, iron condors). Maximum loss is the strike width minus credit received. Sizing is straightforward on max loss but requires accounting for early assignment risk and pin risk near expiration. Position size such that maximum loss equals the chosen risk fraction; do not size on premium received because the loss-to-premium ratio is asymmetric (typical credit spread risks $4 to make $1).
Across all options structures, the sizing input is the dollar amount at risk in the realistic worst case, not the headline contract count or premium. Two long-call positions with identical premium can have very different effective risk because of time decay, gamma, and underlying volatility.
Key takeaways
- Full Kelly is theoretically optimal but operationally unhinged. It produces 50% drawdowns and is sensitive to input errors. Real-world sizing sits well below Kelly.
- Half-Kelly captures most of the growth at half the drawdown. Fractional Kelly between 25% and 50% is the practical sweet spot for traders with documented edge.
- Vol-targeting equalizes risk across instruments. Same dollar risk per position regardless of asset volatility. Critical for multi-instrument portfolios where headline contract count is misleading.
- Fixed-fractional 1-2% is what professionals actually follow. Mechanical, robust to edge estimation error, and adapts to stop distance automatically. The 1% number is the safe default.
- Options sizing depends on structure, not premium. Long premium sizes on debit; naked short sizes on stress-test loss; credit spreads size on max loss. Premium-based sizing is wrong for naked structures.
Check your understanding
- A system has 60% win rate, 2R wins, 1R losses. What is full Kelly fraction? Half Kelly?
Show answer
f* = (b × p – q) / b with b = 2, p = 0.6, q = 0.4. f* = (2 × 0.6 – 0.4) / 2 = (1.2 – 0.4) / 2 = 0.40. Full Kelly says risk 40% per trade. Half-Kelly: 20%. Even half-Kelly is aggressive — expected drawdown above 25%. Most traders would size at quarter-Kelly (10%) or below for any real-money implementation, particularly given that the 60% win rate and 2R wins are estimates with their own sampling error. - You want to enter a long-call position on a $200 stock with $500 risk budget per trade. The call costs $3.50 per contract. How many contracts?
Show answer
The maximum loss on a long call is the debit paid. At $3.50 per contract = $350 per contract risked. Risk budget $500 / $350 per contract = 1.43, so 1 contract. Two contracts would be $700 risked, exceeding the budget. The rule is to round down to stay within budget, never up — the asymmetry of ruin is too costly to violate the cap. - Why is sizing a naked short put on premium received a dangerous practice?
Show answer
Premium received is not the relevant risk metric. A 30-delta short put might collect $2 in premium but have $30 of strike-width-to-zero exposure. A move to the strike, much less below it, can produce losses 10-15x premium received. Margin-aware sizing or stress-test sizing (loss given a 2-sigma adverse move) is the only defensible approach. Premium-based sizing of naked shorts is the most common mechanism by which intermediate options traders blow up: many small wins followed by a single large loss that wipes out months of premium.