Kelly Criterion: Optimal Betting Strategy

• Kelly Criterion is a rule for allocating wealth that maximizes the expected logarithmic growth of wealth in repeated risky bets.
• It applies across binary, continuous-time, and multivariate models, providing unique optimal fractions even under variable return frameworks.
• The method integrates risk control, volatility trade-offs, estimation risk, and information-theoretic concepts to guide practical betting strategies.

The Kelly criterion is a rule for choosing the fraction of wealth allocated to a risky opportunity so as to maximize expected logarithmic growth, equivalently the long-run exponential growth rate of wealth. In its classical form it applies to repeated favorable bets and prescribes a fixed fraction of current wealth; in broader formulations it becomes a log-optimal portfolio rule for general return distributions, continuous-time models, multivariate portfolios, and markets with serial dependence. Across these settings, the central object is the expected log-growth function, while the main technical questions concern admissibility, uniqueness, asymptotic optimality, volatility, estimation risk, and computational tractability (Hsieh, 2020, Lototsky et al., 2020).

1. Classical binary formulation

In the canonical repeated binary gamble, wealth evolves multiplicatively. If a fraction $f$ of current wealth is staked each round, then one-step wealth is multiplied by $1+fb$ on a win and by $1-f$ on a loss when the odds are $b:1$. The corresponding expected log-growth is

$g(f)=p\log(1+fb)+q\log(1-f),$

with $q=1-p$. Maximizing $g(f)$ yields the classical Kelly fraction

$f^*=p-\frac{q}{b},$

and in the even-money Bernoulli case this reduces to

$f^*=2p-1.$

This is the standard statement that a positive fraction is wagered only when the game has positive edge (Hsieh, 2020, Lototsky et al., 2020).

The same logic appears in the explicit Bernoulli wealth recursion

$$\mathscr{W}(N)=\mathscr{W}(0)\prod_{I=1}^{N}(1+\mathcal{F}\mathscr{Z}(I)),$$

with $1+fb$0. The associated utility function

$1+fb$1

is strictly concave on $1+fb$2, and its unique maximizer is

$1+fb$3

The concavity statement is central: it makes the Kelly fraction a unique global maximizer of expected log-growth rather than merely a stationary point (Miller, 24 Feb 2025).

The operational interpretation is long-run rather than one-period expected value. If wealth after $1+fb$4 rounds is $1+fb$5, then

$1+fb$6

under the i.i.d. assumption. This is why the criterion is usually described as maximizing almost-sure exponential growth, or maximizing expected log wealth, rather than maximizing expected terminal wealth (Hsieh, 2020).

2. General return models and continuous-time limits

The binary model is only a special case of a more general discrete-time problem with i.i.d. returns $1+fb$7 satisfying $1+fb$8 almost surely. For a fixed fraction $1+fb$9, wealth is

$1-f$0

and the growth function is

$1-f$1

Under the integrability conditions stated in the random-walk formulation, $1-f$2 is the almost-sure long-run growth rate, and

$1-f$3

Accordingly, the general Kelly problem remains a concave maximization problem with a unique optimizer under mild conditions (Lototsky et al., 2020).

This formulation accommodates non-Bernoulli return laws. For the Bernoulli loss-win model with $1-f$4 and $1-f$5, the maximizer becomes

$1-f$6

For exponential-return models and heavy-tailed examples such as the Cauchy-based construction discussed in the general random-walk treatment, the same log-growth functional remains the organizing object, but the optimizer need no longer have a simple closed form (Lototsky et al., 2020).

In continuous time, the diffusion limit of high-frequency Kelly betting under geometric Brownian motion produces the familiar Merton-type fraction

$1-f$7

with long-run growth rate

$1-f$8

The same paper extends this to Lévy processes. If $1-f$9 is a Lévy return process with drift $b:1$0, diffusion variance rate $b:1$1, and Lévy measure $b:1$2, then the continuous-time Kelly objective becomes

$b:1$3

again leading to a unique optimizer under suitable edge and boundary conditions (Lototsky et al., 2020).

A related stock-market framework expresses the approximate multivariate Kelly rule through first and second moments alone. For a single stock with future price $b:1$4 and current price $b:1$5, the approximate fraction is

$b:1$6

and under Gaussian or lognormal specifications this recovers the standard small-parameter approximation $b:1$7 while remaining slightly more conservative away from that limit (Byrnes et al., 2018).

3. Entropy, information, and growth

One of the most persistent features of Kelly theory is its information-theoretic structure. In the Bernoulli even-money model, the optimized utility satisfies

$b:1$8

This makes the maximal log-growth equal to binary entropy deficit: the fair game $b:1$9 gives zero optimal log-growth, whereas deviations from fairness create exploitable structure (Miller, 24 Feb 2025).

The horse-race formulation sharpens this connection. If odds induce “track probabilities” $g(f)=p\log(1+fb)+q\log(1-f),$0 and true outcome probabilities are $g(f)=p\log(1+fb)+q\log(1-f),$1, then the optimal per-race log-growth can be written in terms of relative entropy. In that representation, maximizing over bet allocations amounts to minimizing $g(f)=p\log(1+fb)+q\log(1-f),$2, so the optimal allocation is $g(f)=p\log(1+fb)+q\log(1-f),$3, and the optimal growth depends on $g(f)=p\log(1+fb)+q\log(1-f),$4. This is the precise sense in which Kelly growth can be interpreted through Shannon information rather than through expected payoff alone (0903.2243).

When side information is introduced, the increase in doubling rate is bounded by the pragmatic information of the message process, and in the horse-race case the difference between the optimal growth rate with side information and the rate without it equals the mutual information between winners and messages. The same paper uses this perspective to define an efficient market as one in which the pragmatic information of the “tradable past” with respect to current prices is zero; under that definition, a GARCH(1,1) return process is not efficient (0903.2243).

Kullback–Leibler structure also appears in recent prediction-market analysis. In a biased-coin asset-price model, the paper shows using the Kullback-Leibler divergence how the misjudgment of the bias and the miscalculation of the investment fraction influence the portfolio growth rate. A related implication is that bounded-payoff market prices need not equal mean beliefs even under logarithmic utility, so “price as probability” is not a general Kelly conclusion (Meister, 2024).

4. Martingales, risk control, and approximation

The Kelly criterion is growth-optimal, but the same papers emphasize that it is not variance-minimizing. In the explicit Bernoulli model, there exists a threshold $g(f)=p\log(1+fb)+q\log(1-f),$5 such that

$g(f)=p\log(1+fb)+q\log(1-f),$6

For $g(f)=p\log(1+fb)+q\log(1-f),$7, the wealth process is a submartingale when $g(f)=p\log(1+fb)+q\log(1-f),$8, a martingale when $g(f)=p\log(1+fb)+q\log(1-f),$9, and a supermartingale when $q=1-p$0. The formal consequence is that over-betting can turn a favorable game into a supermartingale regime in long-run log-growth terms (Miller, 24 Feb 2025).

The same Bernoulli analysis derives explicit small-$q=1-p$1 approximations for second moments: $q=1-p$2

$q=1-p$3

This is the basis for the usual trade-off between full Kelly and fractional Kelly: reducing the fraction reduces both growth and volatility (Miller, 24 Feb 2025).

A distinct control-theoretic treatment approximates the exact Kelly objective by a second-order Taylor expansion: $q=1-p$4 This approximation turns the problem into quadratic programming and yields closed-form expressions for expected cumulative gain, variance of cumulative gain, variance of log-wealth, and survivability. It also exposes a limitation: the unconstrained approximate solution can lie outside the survivable interval

$q=1-p$5

so saturation is required if one wants pathwise positivity of wealth (Hsieh, 2020).

Risk control can also be built directly into the objective. In the horse-race model with growth-rate mean $q=1-p$6 and fluctuations $q=1-p$7, the utility

$q=1-p$8

produces a family of optimal strategies interpolating between Kelly’s point and a null risk-free strategy. The resulting frontier exhibits a first-order-like coexistence between a risky-growth phase and a null strategy, and on the positive-growth branch the paper derives the uncertainty-like bound

$q=1-p$9

This suggests that “fractional Kelly” can be formalized as a mean-fluctuation trade-off rather than only as an ad hoc reduction of the Kelly stake (Dinis et al., 2020).

5. Multivariate portfolios and computational scaling

For multiple simultaneous bets or assets, Kelly becomes a multivariate log-optimal portfolio problem. One formulation starts from continuous price distributions and derives a matrix equation

$g(f)$0

where the entries of $g(f)$1 and $g(f)$2 depend only on first and second moments of the future price distribution. This provides a practical approximation for one or many stocks, including correlated multivariate assets, and recovers the single-stock $g(f)$3 rule in the appropriate limit (Byrnes et al., 2018).

A separate portfolio-optimization line distinguishes a “decoupled” Kelly return function,

$g(f)$4

from the more difficult coupled form

$g(f)$5

The decoupled model combines Kelly-style return with a variance penalty and solves the resulting nonlinear problem using differential evolution under cardinality constraints (Peterson, 2017).

The computational bottleneck becomes severe in the fully multivariate binary setting because naive evaluation scales as $g(f)$6 in the number of simultaneous wagers. For independent bets, an integral-transform formulation eliminates explicit enumeration of outcomes and reduces objective evaluation from $g(f)$7 to $g(f)$8. Together with a decomposition-based lower/upper bound method, this makes it possible to study problems with hundreds of bets and reveals a sigmoidal scaling law for the shortfall ratio between lower and upper bounds as a function of relative subproblem size (Tepelyan et al., 27 Apr 2026).

The same growth-optimal logic has also been generalized to markets with serial dependence. In a block-wise i.i.d. model of random return matrices, the appropriate object is a $g(f)$9-cyclic constant rebalancing strategy. The generalized Kelly criterion becomes

$f^*=p-\frac{q}{b},$0

and the corresponding $f^*=p-\frac{q}{b},$1-log-optimal strategy asymptotically grows to the highest rate among all strategies. A universal learning algorithm then learns such serial dependence online and can eventually exceed the cumulative wealth of the best constant rebalancing portfolio (Lam, 8 Jul 2025).

6. Variable payoffs, estimation risk, and frontier extensions

A common simplification is to replace variable payoffs by their mean and then apply the classical formula. In the variable-payoff model, however, the expected log-growth is

$f^*=p-\frac{q}{b},$2

and the optimal fraction is the unique solution of the fundamental integral equation

$f^*=p-\frac{q}{b},$3

Pérez Marco’s comparison result states that this $f^*=p-\frac{q}{b},$4 is smaller than the classical Kelly fraction computed at the constant average payoff $f^*=p-\frac{q}{b},$5, with equality only when the payoff is constant. The direct implication is that substituting average payoff into the classical formula overstates the optimal fraction when payoffs are variable (Pérez-Marco, 2014).

Estimation risk is a second major limitation. In a simple binomial stock–bond model, the classical stock fraction depends sensitively on the true up factor, down factor, probability, and risk-free rate, and mis-specification can lead to greatly suboptimal growth. One recent proposal adds a European put option to construct growth-optimal portfolios that are robust to estimation risk. In the correctly specified model, the option does not improve the Kelly growth rate because the option-augmented strategy is payoff-equivalent to the classical Kelly strategy; under parameter mis-specification, however, convex combinations of two option-based Kelly strategies can asymptotically dominate the classical stock–bond Kelly strategy (Lillo et al., 26 Aug 2025).

Prediction-market analysis adds another correction to naive intuition. In the all-or-nothing contract setting, logarithmic utility implies a specific optimal fraction, but the resulting market-clearing prices generally differ from the capital-weighted mean beliefs of investors. The same work studies a modified payout structure parameterized by $f^*=p-\frac{q}{b},$6 to adjust liquidity and the gap between prices and beliefs. This suggests that Kelly aggregation in bounded-payoff markets is a risk-adjusted equilibrium object rather than a direct probability estimator (Meister, 2024).

Recent applications and extensions are correspondingly diverse. Kelly-based mechanisms have been proposed for invoice discounting in a “Reverse Kelly AMM,” where the formula is inverted to compute a premium rather than a stake fraction (Esteva et al., 2023). Quantum variants replace wealth by ergotropy or by spin-measurement payoffs, preserving the asymptotic log-growth logic while altering the state and measurement structure of the problem (Tirone et al., 2020, Meister et al., 2023). A plausible implication is that the Kelly criterion is better understood as a growth-optimal principle that survives substantial changes in the underlying state space, provided wealth dynamics remain multiplicative and admissibility is enforced.