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Decentralized Governance of Stablecoins with Closed Form Valuation

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Mathematical Research for Blockchain Economy (MARBLE 2022)


We model incentive security in non-custodial stablecoins and derive conditions for participation in a stablecoin system across risk absorbers (vaults/CDPs) and holders of governance tokens. We apply option pricing theory to derive closed form solutions to the stakeholders’ problems, and to value their positions within the capital structure of the stablecoin. We derive the optimal interest rate that is incentive compatible, as well as conditions for the existence of equilibria without governance attacks, and discuss implications for designing secure protocols.

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The authors thank the Center for Blockchains and Electronic Markets at University of Copenhagen for support.

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Correspondence to Andreea Minca .

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A Derivative Analysis

1.1 A.1 Sensitivity of the Expected Collateral Shortfall

Note the following relationship,

$$\begin{aligned} Fe^\delta \phi (d_2)=N\phi (d_1) \end{aligned}$$

With this, we have the following derivatives,

$$\begin{aligned} \frac{\partial P}{\partial F}&=e^\delta \varPhi (-d_2)+Fe^\delta \cdot \phi (-d_2)\cdot \left( -\frac{d d_2}{dF}\right) -N\cdot \phi (-d_1)\cdot \left( -\frac{d d_1}{dF}\right) \nonumber \\&=e^\delta \varPhi (-d_2) \end{aligned}$$
$$\begin{aligned} \frac{\partial P}{\partial \delta }&=Fe^\delta \varPhi (-d_2)+Fe^\delta \cdot \phi (-d_2)\cdot \left( -\frac{d d_2}{d\delta }\right) -N\cdot \phi (-d_1)\cdot \left( -\frac{d d_1}{d\delta }\right) \nonumber \\&=Fe^\delta \varPhi (-d_2) \end{aligned}$$

1.2 Vault Objective Sensitivities


$$\begin{aligned} V:=\quad Ne^\frac{\sigma ^2}{2}+F(e^b-e^\delta )-P(\delta ,F)(e^b-1) \end{aligned}$$

Note the following derivatives,

$$\begin{aligned} \frac{\partial V}{\partial F}&=\left( e^b-e^\delta \right) -\left( e^b-1\right) e^\delta \varPhi (-d_2) \end{aligned}$$
$$\begin{aligned} \frac{\partial V}{\partial \delta }&=-Fe^\delta -(e^b-1)\frac{\partial P}{\partial \delta }\nonumber \\&=-Fe^\delta -(e^b-1)Fe^\delta \varPhi (-d_2)\nonumber \\&=-Fe^\delta (1+(e^b-1)\delta \varPhi (-d_2))<0\quad \text {always} \end{aligned}$$

1.3 GOV Objective Sensitivities


$$\begin{aligned} G:=\quad F\left( e^\delta -1\right) \end{aligned}$$

Note the following derivatives,

$$\begin{aligned} \frac{\partial G}{\partial \delta }&=Fe^\delta >0\quad \text {always} \end{aligned}$$
$$\begin{aligned} \frac{\partial G}{\partial F}&=e^\delta -1 \end{aligned}$$


Proposition 1


Since V is concave in F, we set (16) equal to zero to obtain a maximum for V:

$$\begin{aligned} \varPhi (-d_2)&=\frac{e^b-e^\delta }{e^\delta \left( e^b-1\right) }\nonumber \\ \frac{\log \left( \frac{F}{N}\right) +\delta +\frac{\sigma ^2}{2}}{\sigma }&=\varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \nonumber \\ F^*=\varphi (\delta )&= N\cdot \exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) -\delta -\frac{\sigma ^2}{2}\right] , \end{aligned}$$

with (8) implicitly requiring that \(\delta \in [0,b]\).

Together with the leverage constraint, this implies

$$\begin{aligned} F^*(\delta )=\min (\varphi (\delta ),\beta N) \end{aligned}$$

since the leverage constraint imposes a cap on amount of stablecoins issued. Substitute (9) into (18) and obtain

$$\begin{aligned} G= F^*(\delta )\left( e^\delta -1\right) , \end{aligned}$$

thus transforming GOV’s optimization into finding the optimum for (23).

Proposition 2


We begin by establishing a lower bound for \(\delta \): There exists a \(\delta _{\beta }\in [0,b]\) such that \(\varphi ( \delta _{\beta })=\beta N\), i.e.

$$\begin{aligned} \frac{F^{*}}{N}=\exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b- \delta _{\beta }}-1}{e^b-1}\right) - \delta _{\beta }-\frac{\sigma ^2}{2}\right] =\beta . \end{aligned}$$

The quantity \(\delta _{\beta }\) is the interest rate for which vaults’ leverage constraint is hit, i.e. for \(\delta < \delta _{\beta }\) the optimal stablecoin issuance is given by \(\beta N\).

Indeed, by comparing \(\varphi (\delta )\) to \(\beta N\) we obtain

$$\begin{aligned} \frac{d\varphi }{d\delta }&=-\varphi (\delta )\cdot \left[ 1+\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^{b-\delta }}{e^b-1}\right] <0\quad \text {always} \end{aligned}$$

Thus \(\exists \delta _{\beta }\in [0,b]\) such that \(\varphi ( \delta _{\beta })=\beta N\), i.e.

$$\begin{aligned} \frac{F^{*}}{N}=\exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b- \delta _{\beta }}-1}{e^b-1}\right) - \delta _{\beta }-\frac{\sigma ^2}{2}\right] =\beta , \end{aligned}$$

which effectively is setting a lower bound to \(\delta \), such that \(\forall \delta \in ( \delta _{\beta },b]\), \(\varphi (\delta )<\beta N\).

We can no conclude the proof of Proposition 2. Suppose \(\varphi (\delta )\ge \beta N\), i.e. \(\delta \in [0, \delta _{\beta }]\)

$$\begin{aligned} G&=\beta N\cdot \left( e^\delta -1\right) \\ \frac{dG}{d\delta }&=\beta Ne^\delta >0\quad \text {always}. \end{aligned}$$

Thus, GOV will choose \(\delta ^*= \delta _{\beta }\).

Lemma 1


Suppose \(\varphi (\delta )<\beta N\), i.e. \(\delta \in ( \delta _{\beta },b]\)

$$\begin{aligned} G&=\varphi (\delta )\cdot \left( e^\delta -1\right) \\ \frac{dG}{d\delta }&=\varphi (\delta )\cdot e^\delta + \frac{\partial \varphi }{\partial \delta }\cdot (e^\delta -1)\\&=\varphi (\delta )\cdot e^\delta -\varphi (\delta )\cdot \left[ 1+\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^{b-\delta }}{e^b-1}\right] \cdot (e^\delta -1)\\&=\varphi (\delta )\left[ 1-\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta }}{e^b-1}\right] \end{aligned}$$

from which the lemma follows. Consider threshold value \(\delta _\text {th}\) such that

$$\begin{aligned} \frac{e^{b-\delta _\text {th}}-1}{e^b-1}&=0.5\quad \Rightarrow \quad \delta _\text {th}=b-\log \left( \frac{e^b+1}{2}\right) . \end{aligned}$$

When \(\delta >\delta _\text {th}\), we have as \(\delta \) increases

  • \(\varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \) decreases from 0 to \(-\infty \)

  • \(\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) \) hence decreases from \(\phi (0)\) to 0

  • \(\frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\) increases from \(\frac{1}{\phi (0)}\) to \(\infty \)

  • \(\frac{e^b-e^{b-\delta }}{e^b-1}\) increases from 0.5 to \(\frac{e^b}{e^b-1}\)

  • Overall, \(\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta }}{e^b-1}\) is increasing.

Assumption 1


$$\begin{aligned}&1-\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta _\text {th}}-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta _\text {th}}}{e^b-1}>0\nonumber \\ \Leftrightarrow \quad&1-\frac{\sigma }{2\phi (0)}>0\nonumber \\ \Leftrightarrow \quad&\sigma <2\phi (0) \end{aligned}$$

Proposition 3


Under Assumption 1, we have that G is locally increasing at \(\delta =\delta _{th}\) and we have that \(\frac{dG}{d\delta }\) is non-increasing with \(\delta \) for \(\delta > \delta _{th}\).

Therefore, when setting \(\frac{dG}{d\delta }=0\), i.e. \(1-\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta }}{e^b-1}=0\), we implicitly obtain a \(\delta ^{*}\), at which level G is maximized.

Theorem 1


At \(\delta ^{*}\), we have

$$\begin{aligned} \sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta ^{*}}}{e^b-1}=1 \end{aligned}$$

Substitute (28) into (15),

$$\begin{aligned} \begin{aligned} V(\delta ^{*})&=Ne^\frac{\sigma ^2}{2}+\varphi (\delta ^{*})(e^b-e^{\delta ^{*}})\nonumber \\&-P(\delta ^{*},\varphi (\delta ^{*}))(e^b-1) \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} P(\delta ^{*},\varphi (\delta ^{*}))&=\varphi (\delta ^{*})e^{\delta ^{*}}\varPhi (-d_2)-N\varPhi (-d_1) \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} d_1(\delta ^{*})&=\frac{1}{\delta }\cdot \left( \log (\frac{N}{\varphi (\delta ^{*})e^\delta })+\frac{\sigma ^2}{2}\right) \nonumber \\&=-\varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) +\sigma \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} d_2(\delta ^{*})&=d_1-\sigma =-\varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) \end{aligned} \end{aligned}$$

We obtain

$$\begin{aligned} P(\delta ^{*},\varphi (\delta ^{*}))&=\varphi (\delta ^{*})\frac{e^{b}-e^{\delta ^{*}}}{e^b-1}-N\varPhi (-d_1) \end{aligned}$$


$$\begin{aligned} V(\delta ^{*})&=Ne^\frac{\sigma ^2}{2}+N\varPhi (-d_1)(e^b-1) \end{aligned}$$

such that we must assume

$$\begin{aligned} u\le Ne^\frac{\sigma ^2}{2}+N\varPhi (-d_1(\delta ^{*}))(e^b-1), \end{aligned}$$

in order for vault participation to hold.

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Huo, L., Klages-Mundt, A., Minca, A., Münter, F.C., Wind, M.R. (2023). Decentralized Governance of Stablecoins with Closed Form Valuation. In: Pardalos, P., Kotsireas, I., Guo, Y., Knottenbelt, W. (eds) Mathematical Research for Blockchain Economy. MARBLE 2022. Lecture Notes in Operations Research. Springer, Cham.

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