Kelly Staking Plans

The Kelly Criterion is a formula used to determine the optimal size of a series of bets or investments. It was derived by John L. Kelly Jr. in 1956 while working at Bell Labs. In the context of financial investments and particularly algorithmic trading, the Kelly Criterion has been adapted to determine the proportion of capital to allocate to a particular trading strategy to maximize long-term growth.

Foundation of the Kelly Criterion

The Kelly Criterion maximizes the logarithm of wealth, ensuring long-term growth based on several outcomes with varying probabilities. The basic formula for the Kelly Criterion in its simplest form is:

f* = (bp - q) / b

Where:

This formula calculates the optimal bet size that maximizes the expected logarithmic growth of capital.

Illustration

Imagine a scenario where a trader is considering a stock investment:

Assume the return on success is 1:1 (b = 1), so the investment either doubles or loses its stake. Using the Kelly formula:

f* = (1 * 0.6 - 0.4) / 1 = 0.2 or 20%

This result suggests that, to maximize the expected logarithmic growth of capital, 20% of the current bankroll should be allocated to this stock.

Application in Algorithmic Trading

Algorithmic trading involves the use of computer programs to automatically make trading decisions and execute trades. The Kelly Criterion fits well with this approach, given that these programs are designed to process large amounts of data and identify statistically significant trading opportunities.

Implementing Kelly Staking in Algorithms

Developing an algorithm that uses the Kelly Criterion involves several key steps:

  1. Modeling Probabilities: Estimate the probability of various outcomes for trades. This could be done through historical data analysis, machine learning models, or other predictive analytics techniques.
  2. Calculating Returns: Define the potential returns (b) for each trade. This involves understanding market dynamics and how much profit or loss is expected.
  3. Computing Fraction: Apply the Kelly formula to compute the fraction of the bankroll to be allocated to each trade.
  4. Execution: Implement trade execution strategies within the algorithm that respect the calculated fraction, adjusting for transaction costs, slippage, and other market factors.

Practical Considerations

  1. Overbetting and Risk of Ruin: Strictly following the Kelly Criterion can lead to large bet sizes, which might be impractical or excessively risky in real-world trading. To mitigate this, traders often use a “fractional Kelly” approach by wagering only a portion (such as half) of the Kelly-recommended amount.

  2. Market Frictions: Real-world markets introduce complexities such as transaction costs, bid-ask spreads, and slippage, which can affect the accuracy of Kelly-based strategies. Algorithms must account for these factors to avoid overestimating the optimal bet size.

  3. Diversification: The Kelly Criterion typically addresses betting on a single event. In a diversified trading strategy involving multiple assets, the formula needs to be adapted to allocate the total capital across different assets simultaneously. This requires understanding the co-movement between different assets and balancing the portfolio accordingly.

  4. Recalibration: Probabilities and potential returns are not static. Continuous monitoring and recalibration of probabilities and returns are necessary to keep the staking plan aligned with the current market conditions.

Kelly Criterion Variants

Beyond the basic Kelly formula, several variants and extensions exist to address different investment scenarios:

Continuous Kelly Criterion

In continuous markets, such as stocks and currencies, the discrete nature of the Kelly formula can be extended to continuous models. This approach involves using stochastic calculus and differential equations to determine optimal investment strategies over time.

Kelly for Multiple Outcomes

For investments with multiple possible outcomes beyond binary win/loss, the Kelly formula can be expanded to account for these multiple states. The generalized formula considers the expected returns and probabilities for each possible outcome.

Fractional Kelly

A common adaptation in practice is to use a fraction of the Kelly recommendation to reduce risk and volatility. For instance, multiplying the Kelly fraction by a factor (like 0.5) to get a “half-Kelly” wager introduces a more conservative approach while still capturing some benefits of the original criterion.

Bayesian Kelly

Incorporating Bayesian inference allows traders to update probabilities dynamically as new information arrives. Bayesian Kelly methods combine prior beliefs about probabilities with new market data to refine investment decisions over time.

Historical Use and Criticism

While the Kelly Criterion has strong theoretical foundations, its practical use in financial markets varies:

Modern Implementations

Several fintech companies and trading platforms integrate the Kelly Criterion into their trading algorithms and portfolio management tools:

Conclusion

The Kelly Criterion offers a robust framework for determining bet sizing in trading and investment, balancing risk and reward to maximize long-term capital growth. While its theoretical foundations are strong, practical implementation requires careful consideration of market realities, such as transaction costs, diversification, and probability estimation accuracy. Adaptations like fractional Kelly and Bayesian approaches provide additional flexibility to align the criterion with real-world investment scenarios. As algorithmic trading continues to evolve, the Kelly Criterion remains a valuable tool in the arsenal of quantitative finance and investment professionals.