Extending Black–Scholes for Crypto Assets

Extending Black–Scholes for Crypto Assets

Pricing Options in the Age of Tokenomics

The Black–Scholes model has been the backbone of option pricing for over 50 years. It provides a clean, closed-form solution for valuing European options, built on the assumption that the underlying asset follows a lognormal diffusion with constant volatility. But while this framework works reasonably well for equities and traditional markets, it needs rethinking when applied to crypto tokens, where the supply schedule itself is dynamic.

Unlike equities, where the share count is relatively stable, many tokens have programmed inflation, periodic emissions, or burn mechanics that constantly alter the effective supply. Ethereum after EIP-1559, for example, introduced a burn rate linked to transaction activity, while other protocols issue new tokens through staking rewards or liquidity mining. These mechanisms create an entirely new dimension to pricing derivatives: how do you incorporate tokenomics into a valuation model?

Tokenomics as an Effective Dividend Yield

The bridge between traditional finance and crypto lies in the dividend adjustment of Black–Scholes. In equities, when a stock pays a continuous dividend yield q, the call option formula becomes:

Under

where q reduces the expected growth of the asset because dividends are siphoned off to shareholders.

This intuition translates directly to crypto. Instead of dividends, tokens face dilution or concentration of supply:

• Inflation (emissions, staking rewards): increases supply, dilutes existing holders, functions like a positive dividend yield.

• Burns (EIP-1559, protocol fees destroyed): reduces supply, concentrates value among holders, functions like a negative dividend yield.

We can therefore define an effective tokenomic yield:

If q{token} > 0, supply is expanding (net inflation). This dilutes holders, just like a dividend yield would. Options priced under such conditions will be cheaper than they otherwise would be. If q{token} < 0, supply is contracting (net deflation). This concentrates value, boosting forward prices. Options under this regime are more expensive.

The Black–Scholes call price then becomes:

Where

If a token has high inflation (say +10%/yr), option values are depressed relative to a fixed-supply asset. If a token has a strong burn mechanism (say –2%/yr), option values are higher because the forward curve is effectively lifted.

Beyond Constants: Dynamic Tokenomics

Most tokenomics are not constant. Ethereum’s burn rate depends on gas fees, Solana’s inflation rate decreases on a schedule, and Bitcoin undergoes discrete supply shocks at each halving.

To capture this, we replace the constant dividend yield with a time-dependent term:

Where q(t) can be a deterministic function or a mean-reverting stochastic function:

Therefore the option price incorporates the cumulative effect of tokenomics:

This form allows us to plug in any deterministic or a stochastic supply schedule. If a network burns more tokens during periods of high activity, then option values are effectively tied to usage metrics, embedding fundamental adoption into the derivatives market.

Tokenomics as Jumps

Not all supply changes are smooth. Some are sudden, event-driven, and entirely predictable in timing. Large token unlocks release billions in supply at once. Bitcoin halving cuts issuance by 50% on a single block. Protocol upgrades can instantly change burn mechanics or reward schedules.

These cannot be modeled simply as continuous drift adjustments. Instead, we treat them as scheduled jumps with a deterministic or random magnitude:

Where

The final term is a Dirac delta function to ensure jumps occur at scheduled times.

q(t) captures net inflation or burn, tau is the known event date (e.g. unlock), and J is a random or deterministic jump factor describing the market impact of that event. This structure mirrors how equity options are priced around earnings, where the date is known, but the magnitude of repricing is not. For crypto, it explains why implied volatilities often spike into large token unlocks or halving events. Traders price in the possibility of discontinuous moves, even when the supply event itself is fully anticipated.

Practical Implications

Incorporating tokenomics into option pricing produces markedly different valuations across crypto asset classes. Tokens with high inflation tend to trade with depressed option premiums, as their forward curves are naturally discounted by ongoing supply expansion. Conversely, deflationary tokens, such as ETH post-EIP-1559 or BNB with buybacks and burns, support higher option values because the forward curve benefits from expected supply contraction. Event-driven assets, like BTC around halvings or governance-driven emissions, exhibit jump risk that is reflected in implied volatility, often producing characteristic skew and term structure effects.

Conclusion

Black–Scholes transformed finance by linking continuous hedging and risk-neutral valuation into a single closed-form solution. In crypto, however, supply is a programmable parameter, not a fixed assumption, and option pricing must evolve accordingly. Incorporating tokenomics, whether as inflation, burns, or scheduled events, allows models to reflect the true economic realities of blockchain assets.

For traders, this means recognizing that implied volatilities are context-dependent: a 90% volatility token with 15% annual inflation behaves very differently from a 90% volatility deflationary ETH derivative. For protocol designers, it highlights how tokenomics influence not only spot supply but also derivative pricing, liquidity, and capital formation. Ultimately, in the age of tokenomics, option pricing extends beyond volatility alone. It must embed supply mechanics and event-driven dynamics into the model itself.