*Yotam Bar-On (April 27, 2025)*
Constant-Product Market Maker positions can be modelled as zero-premium option instruments. In this analysis we examine the “Premium Loss” metric, which tracks a CPMM pool’s loss suffered from not collecting premia. We compare this metric to the loss versus the “Rebalancing strategy”[1] (known as “LVR”) and the loss versus “Holding” (which we refer to as “LVH”). Our model provides a quantitative method for assessing the viability of delta-hedging CPMM pools. We do not introduce any new ideas to the literature but rather attempt to empirically analyze and compare existing models for CPMM performance.
The CPMM can be modeled as a “Straddle” option, which is both a call and a put option with the same strike. A straddle gives the buyer the option to either buy or sell the stock at a certain strike. A CPMM effectively acts as a premium-free straddle, as it does not charge for the option. We can immediately derive that the CPMM is consistently loss-making:
Figure 1: The payoff of a premia-free straddle; Red indicates put ITM, blue indicates call ITM
To price the straddle we use the well-known Black-Scholes option pricing model:
$$ \begin{align} C(t, S(t)) &= N(d_1) S(t) - N(d_2)K e^{-rt} \\ P(t, S(t)) &= N(-d_2) K e^{-rt} - N(-d_1)S(t) \end{align} $$
The Black-Scholes option pricing model gives us a way to decide what should be the price (premium) of a specific option given its strike price $K$, current stock price $S$, its duration $t$ and the “implied volatility” $\sigma$ of the market (the volatility is included in the $d_1$, $d_2$ terms). $N(d)$ denotes the standard normal cumulative distribution function. For simplicity, we assume the risk-free rate $r$ is zero. The expiry $t$ is effectively the block time of the blockchain on which the CPMM is deployed.
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To put this intuitively: Every 12 seconds (on Ethereum), the CPMM gives out the option for anyone to either buy or sell the asset at a certain strike. The CPMM provides this option at no initial cost, meaning it does not charge a premium.
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For the purpose of this research we assume that we can estimate instrument prices over slightly longer periods of time, rather than block times, specifically 5 minutes.
We use the CEX price for the stock price, $S(t)$, and the CPMM price for the strike price, $K(t)$. We can infer the “implied volatility”, $\sigma$, using well known methods, from either the options market or the funding rate of a perpetuals exchange. Using these parameters we can, at any point in time $t$, estimate the expected loss over the forward period $T$—by marking the uncollected premia as a loss.
CPMMs typically hold 2 reserves: the “risky reserve” and the “stable reserve”. We refer to the amount of risky/stable reserves held by the CPMM at any point in time $t$, using the notation of $x(t)$ and $y(t)$ respectively.
A CPMM at time $t_0=0$ had the following reserves:
$$ x(t_0) = 4.139173696\\ y(t_0) = 9201.953864 $$
With strike price of the pool $K(t_0) = 2229.845$.