When Kelly Meets Options: A Robust Growth Strategy for Uncertain Markets

Financial theory loves to promise the best long-run growth with elegant math. The Kelly criterion is a famous example: it tells you how to bet (or invest) so your wealth grows the fastest on a log scale over time. But there’s a catch. That perfection hinges on knowing the right inputs—expected returns, probabilities of outcomes, and payoffs. In the real world, our estimates are noisy. A small miscue can turn a seemingly unbeatable plan into a suboptimal one.

A recent study by Fabrizio Lillo, Piero Mazzarisi, and Ioanna-Yvonni Tsaknaki explores exactly this problem in a simple, transparent setting. They ask: can we make Kelly-style investing more robust when we don’t know the inputs perfectly? Their answer, surprisingly practical: yes—by adding a European option to the mix. In short, stock-plus-option strategies can be more forgiving of estimation errors than stock-only strategies, and a smart blend of the two can be especially resilient over the long run.

In this blog post, I’ll unpack the key ideas in plain terms, highlight the most useful takeaways, and point to how you might think about applying these ideas in a real-world, uncertain world.

A light introduction to the Kelly ideaThe Kelly criterion asks you to maximize the long-run growth rate of your wealth by choosing how much of your wealth to invest in risky assets versus safe ones.In a simple setup, you invest a fraction f of your wealth in a stock (risky asset) and the rest in a bond (risk-free). Your future wealth depends on whether the stock goes up or down. The “best” f is the one that, in the long run, gives you the highest exponential growth of your wealth.The math is clean in the paper’s binomial model: at each step, the stock can go up (by a factor u) or down (by a factor d). The bond grows deterministically at a rate R. The optimal f, usually denoted f*, depends on the probabilities and the up/down factors (roughly: how likely you think the stock will rise and how big the moves might be).The main lure of Kelly is long-run efficiency. The catch is sensitivity: if your estimates of those probabilities and payoffs are off, your actual growth can be far lower than planned. That’s the estimation risk we’re talking about.The simple market used in the study

Imagine a very down-to-earth market with:

A stock S that can move up by a factor u or down by a factor d each period.A bond B that grows at a known rate R.A probability p that the stock goes up (and 1−p that it goes down).No-arbitrage requires d < R < u.

In this world, your wealth can be allocated between stock and bond (the classic Kelly setup). The twist in the study is: what if we also buy European put options on the stock? How does that change the long-run growth, especially when our estimates of p, u, d, and R aren’t perfect?

Kelly with options: how the setup changes

In addition to the stock, you can also purchase put options. The study considers one-period European puts with a strike K0, and you can allocate fractions of wealth among:

f: the fraction in the stock,g: the fraction in put options,1 − f − g: the fraction in the bond.

Two practical points:

The put options act as a hedging tool. Their payoff at the end of the period is (K0 − S1)+, which cushions you if the stock falls.There’s a useful relationship that ties together stock and option positions into a parameter c, which you can think of as a hedging parameter: c = f − (S0 / P0) · g. This links how many options you buy with how much stock you hold.

Intuitively, the option position adds a corrective force: it protects against downside moves that would otherwise drag your log-growth down, especially when your input estimates are not perfect.

Two key takeaways from the paper’s math (without diving into the full formulas):

If you could perfectly estimate the market, adding options does not change the best possible long-run growth rate. The Kelly growth rate remains the same as in the stock-only world.If estimation is uncertain (which is typical in the real world), the two strategies differ in how they perform when your inputs are misspecified.The big findings: robustness to estimation riskWithout estimation risk (perfect knowledge of p, u, d, R), the option is basically a free addition: you can include puts, but the optimal long-run growth rate you could achieve with stock alone can be matched by a stock-plus-options strategy. In other words, no guaranteed extra growth from adding options in the perfect-knowledge world.With estimation risk (the real world), neither the stock-only Kelly strategy nor the stock-plus-options Kelly strategy consistently outperforms the other across all ways your inputs might be wrong. Depending on which inputs are mis-specified, one approach can do better and the other worse.The most practical and striking result: a proper convex combination (a weighted blend) of two Kelly portfolios can be robust to estimation risk in the long run. In plain terms, mixing a stock-only Kelly strategy with a stock-plus-options Kelly strategy tends to be more resilient when your parameter guesses are off.

Think of it as not putting all your eggs in one well-tuned basket. By combining two different growth-optimized strategies, you dampen the risk that misestimates derail your long-run growth.

Practical implications: what this could mean for investorsIf you’re worried about estimation risk (and who isn’t?), consider not just the “best guess” Kelly plan, but a blend of two growth-optimized plans:A stock-only Kelly plan (the classic approach).A stock-plus-options Kelly plan (stock with a hedging layer via puts).The blend doesn’t have to be complicated. A simple weighted average of the two strategies, tuned to how uncertain you feel about the inputs, can yield more robust growth over the long run than either approach on its own.The use of options adds a layer of protection against downside moves when your estimates are uncertain. While you might sacrifice some upside in certain parameter settings, the payoff is a more stable long-run growth path when inputs are noisy.For practitioners who already use fractional Kelly (to limit drawdowns), this work suggests an additional knob: diversify across growth-optimized strategies and mix them. It’s a principled way to hedge estimation risk, grounded in a clear mathematical framework.

A few practical cautions:

The analysis is in a simple, one-period binomial world. Real markets are multi-period, with more complicated dynamics and a broader menu of options. The core intuition, though, carries over: hedging can reduce sensitivity to misestimated inputs.Implementing a mix requires some calibration: what weights to put on the stock-only vs. stock-plus-options strategies? The best mix depends on your degree of estimation risk and risk tolerance, and it’s a good candidate for a robust, age-old approach—backtest and stress-test.Conclusion

The Kelly criterion remains a cornerstone of growth-optimal investing, but its practical application can be fragile in the face of estimation risk. By incorporating European options into a simple binomial framework, Lillo, Mazzarisi, and Tsaknaki show a compelling path to robustness: don’t rely on a single, perfectly estimated plan. Instead, blend two Kelly-like strategies—one stock-only and one stock-plus-options—and you gain resilience in the long run.

As markets continue to surprise us and data remains imperfect, this approach offers a clear, actionable mindset: design diversification not just across assets, but across growth-optimizing strategies themselves. The result is a more stable, patient path to growth—exactly the kind of strategy that appeals to long-term, wealth-building enthusiasts.

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