Extension of the Black-Scholes-Merton Model to Include Supply Change Rate for Ethereum Options

Ethereum (ETH) is a cryptocurrency that resembles a combination of a currency, a stock, and a commodity. It is a non-dividend-paying crypto asset with a dynamic supply change parameter. Ethereum options have been traded since 2019. Deribit is the largest ETH options exchange by volume, with a market share of approximately 80%.

A particularity of ETH is the changing nature of its supply. Specifically, the supply change rate can be expressed as follows,

Supply change rate = Net issuance rate = Issuance rate − Burn rate

Reference [1] generalized the Black-Scholes-Merton (BSM) formalism to include ETH’s supply change rate. The authors pointed out,

The net issuance rate introduces skewness in volatility structures. This skewness is weighted for deep-in-the-money options, which is consistent with the behavior of option prices. As strike prices increase significantly, the implied volatilities asymptotically approach each other.

Moreover, the sensitivity of the results to even minor changes in the net issuance rate parameter is noteworthy. Figures 12 - 13 demonstrate this effect. This indicates that the inclusion of the net issuance rate can cause significant changes in option pricing. Consequently, options could be fundamentally mispriced if the parameter is completely ignored.

The main contribution of the thesis is the identification of a deterministic factor in the pricing of crypto asset options, the supply change rate, which is not taken into account in the traditional BSM model. The extended BSM model, or alternatively the crypto asset BSM, presented in this thesis includes this rate in the model. The supply change rate can take both positive and negative values within its mathematically defined limits. Moreover, the crypto asset BSM can be used for any other crypto asset that has a supply change parameter, preferably with low block times.

Basically, the author employed the formula used for incorporating stock dilution effects and extended it to ETH options.

Another interesting insight from the paper is that it shows the volatility smirk of ETH options, where, unlike equity options, out-of-the-money call options have higher implied volatility than at-the-money calls.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Teemu Laurikainen, An extension of the Black-Scholes-Merton options pricing model to Ethereum, Aalto University, 2025

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