Skip to main content

Decentralized Governance of Stablecoins with Closed Form Valuation

  • Conference paper
  • First Online:
Mathematical Research for Blockchain Economy (MARBLE 2022)

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Note that for the purposes of debt valuation the no-arbitrage theory of option pricing is not relevant. Only the Black and Scholes formulas are needed, i.e. the closed form solution for the expectation in (5) when the random return is log-normal. Moreover, while the valuation of corporate debt can be achieved in a dynamic model, the same formulas govern our one period case where the end of the period can be seen as the bond maturity.

References

  1. Beck, R., Müller-Bloch, C., & King, J. L. (2018). Governance in the blockchain economy: A framework and research agenda. Journal of the Association for Information Systems, 19(10), 1.

    Google Scholar 

  2. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.

    Article  Google Scholar 

  3. Bullmann, D., Klemm, J., & Pinna, A. (2019). In search for stability in crypto-assets: are stablecoins the solution? ECB Occasional Paper, 230.

    Google Scholar 

  4. Daian, P., Kell, T., Miers, I., & Juels, A. (2018). On-chain vote buying and the rise of dark DAOs. https://hackingdistributed.com/2018/07/02/on-chain-vote-buying/.

  5. Dybvig, P. H., & Zender, J. F. (1991). Capital structure and dividend irrelevance with asymmetric information. The Review of Financial Studies, 4(1), 201–219.

    Article  Google Scholar 

  6. Foxley, W. (2020). \$10.8m stolen, developers implicated in alleged smart contract ‘rug pull’. CoinDesk. https://www.coindesk.com/compounder-developers-implicated-alleged-smart-contract-rug-pull.

  7. Gudgeon, L., Perez, D., Harz, D., Livshits, B., & Gervais, A. (2020). The decentralized financial crisis. arXiv preprint arXiv:2002.08099.

  8. Klages-Mundt, A. (2019). Vulnerabilities in maker: oracle-governance attacks, attack daos, and (de)centralization. https://link.medium.com/VZG64fhmr6.

  9. Klages-Mundt, A., Harz, D., Gudgeon, L., Liu, J.Y., & Minca, A. (2020). Stablecoins 2.0: Economic foundations and risk-based models. In Proceedings of the 2nd ACM Conference on Advances in Financial Technologies (pp. 59–79).

    Google Scholar 

  10. Klages-Mundt, A., & Minca, A. (2019). (in) stability for the blockchain: Deleveraging spirals and stablecoin attacks. arXiv preprint arXiv:1906.02152.

  11. Klages-Mundt, A., & Minca, A. (2020). While stability lasts: A stochastic model of stablecoins. arXiv preprint arXiv:2004.01304.

  12. Lee, B. E., Moroz, D. J., & Parkes, D. C. (2020). The political economy of blockchain governance. Available at SSRN 3537314.

    Google Scholar 

  13. Lee, L., & Klages-Mundt, A. (2021). Governance extractable value. https://ournetwork.substack.com/p/our-network-deep-dive-2.

  14. MakerDAO (2020). Black thursday response thread. https://forum.makerdao.com/t/black-thursday-response-thread/1433.

  15. Merton, R. C. (1970). A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm.

    Google Scholar 

  16. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of finance, 29(2), 449–470.

    Google Scholar 

  17. Myers, S. C., & Majluf, N. S. (1984). Corporate financing and investment decisions when firms have information that investors do not have. Journal of financial economics, 13(2), 187–221.

    Article  Google Scholar 

  18. Reijers, W., O’Brolcháin, F., & Haynes, P. (2016). Governance in blockchain technologies & social contract theories. Ledger, 1, 134–151.

    Article  Google Scholar 

  19. Rekt. (2021). Paid network–rekt. https://rekt.eth.link/paid-rekt/.

  20. Shreve, S. E., et al. (2004). Stochastic calculus for finance II: Continuous-time models, vol. 11. Springer.

    Google Scholar 

  21. Werner, S. M., Perez, D., Gudgeon, L., Klages-Mundt, A., Harz, D., & Knottenbelt, W. J. (2021). Sok: Decentralized finance (defi). arXiv preprint arXiv:2101.08778.

  22. Zoltu, M. (2019). How to turn $20m into $340m in 15 seconds. https://link.medium.com/k8QTaHzmr6.

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andreea Minca .

Editor information

Editors and Affiliations

Appendices

A Derivative Analysis

1.1 A.1 Sensitivity of the Expected Collateral Shortfall

Note the following relationship,

$$\begin{aligned} Fe^\delta \phi (d_2)=N\phi (d_1) \end{aligned}$$
(12)

With this, we have the following derivatives,

$$\begin{aligned} \frac{\partial P}{\partial F}&=e^\delta \varPhi (-d_2)+Fe^\delta \cdot \phi (-d_2)\cdot \left( -\frac{d d_2}{dF}\right) -N\cdot \phi (-d_1)\cdot \left( -\frac{d d_1}{dF}\right) \nonumber \\&=e^\delta \varPhi (-d_2) \end{aligned}$$
(13)
$$\begin{aligned} \frac{\partial P}{\partial \delta }&=Fe^\delta \varPhi (-d_2)+Fe^\delta \cdot \phi (-d_2)\cdot \left( -\frac{d d_2}{d\delta }\right) -N\cdot \phi (-d_1)\cdot \left( -\frac{d d_1}{d\delta }\right) \nonumber \\&=Fe^\delta \varPhi (-d_2) \end{aligned}$$
(14)

1.2 Vault Objective Sensitivities

Denote

$$\begin{aligned} V:=\quad Ne^\frac{\sigma ^2}{2}+F(e^b-e^\delta )-P(\delta ,F)(e^b-1) \end{aligned}$$
(15)

Note the following derivatives,

$$\begin{aligned} \frac{\partial V}{\partial F}&=\left( e^b-e^\delta \right) -\left( e^b-1\right) e^\delta \varPhi (-d_2) \end{aligned}$$
(16)
$$\begin{aligned} \frac{\partial V}{\partial \delta }&=-Fe^\delta -(e^b-1)\frac{\partial P}{\partial \delta }\nonumber \\&=-Fe^\delta -(e^b-1)Fe^\delta \varPhi (-d_2)\nonumber \\&=-Fe^\delta (1+(e^b-1)\delta \varPhi (-d_2))<0\quad \text {always} \end{aligned}$$
(17)

1.3 GOV Objective Sensitivities

Denote

$$\begin{aligned} G:=\quad F\left( e^\delta -1\right) \end{aligned}$$
(18)

Note the following derivatives,

$$\begin{aligned} \frac{\partial G}{\partial \delta }&=Fe^\delta >0\quad \text {always} \end{aligned}$$
(19)
$$\begin{aligned} \frac{\partial G}{\partial F}&=e^\delta -1 \end{aligned}$$
(20)

Proofs

Proposition 1

Proof

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

$$\begin{aligned} \varPhi (-d_2)&=\frac{e^b-e^\delta }{e^\delta \left( e^b-1\right) }\nonumber \\ \frac{\log \left( \frac{F}{N}\right) +\delta +\frac{\sigma ^2}{2}}{\sigma }&=\varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \nonumber \\ F^*=\varphi (\delta )&= N\cdot \exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) -\delta -\frac{\sigma ^2}{2}\right] , \end{aligned}$$
(21)

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

Together with the leverage constraint, this implies

$$\begin{aligned} F^*(\delta )=\min (\varphi (\delta ),\beta N) \end{aligned}$$
(22)

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

$$\begin{aligned} G= F^*(\delta )\left( e^\delta -1\right) , \end{aligned}$$
(23)

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

Proposition 2

Proof

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.

$$\begin{aligned} \frac{F^{*}}{N}=\exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b- \delta _{\beta }}-1}{e^b-1}\right) - \delta _{\beta }-\frac{\sigma ^2}{2}\right] =\beta . \end{aligned}$$
(24)

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

$$\begin{aligned} \frac{d\varphi }{d\delta }&=-\varphi (\delta )\cdot \left[ 1+\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta }-1}{e^b-1}\right) \right) }\cdot \frac{e^{b-\delta }}{e^b-1}\right] <0\quad \text {always} \end{aligned}$$
(25)

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

$$\begin{aligned} \frac{F^{*}}{N}=\exp \left[ \sigma \cdot \varPhi ^{-1}\left( \frac{e^{b- \delta _{\beta }}-1}{e^b-1}\right) - \delta _{\beta }-\frac{\sigma ^2}{2}\right] =\beta , \end{aligned}$$
(26)

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

Proof

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

Proof

$$\begin{aligned}&1-\sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta _\text {th}}-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta _\text {th}}}{e^b-1}>0\nonumber \\ \Leftrightarrow \quad&1-\frac{\sigma }{2\phi (0)}>0\nonumber \\ \Leftrightarrow \quad&\sigma <2\phi (0) \end{aligned}$$
(27)

Proposition 3

Proof

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

Proof

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

$$\begin{aligned} \sigma \cdot \frac{1}{\phi \left( \varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) \right) }\cdot \frac{e^b-e^{b-\delta ^{*}}}{e^b-1}=1 \end{aligned}$$
(28)

Substitute (28) into (15),

$$\begin{aligned} \begin{aligned} V(\delta ^{*})&=Ne^\frac{\sigma ^2}{2}+\varphi (\delta ^{*})(e^b-e^{\delta ^{*}})\nonumber \\&-P(\delta ^{*},\varphi (\delta ^{*}))(e^b-1) \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} P(\delta ^{*},\varphi (\delta ^{*}))&=\varphi (\delta ^{*})e^{\delta ^{*}}\varPhi (-d_2)-N\varPhi (-d_1) \end{aligned} \end{aligned}$$
(29)
$$\begin{aligned} \begin{aligned} d_1(\delta ^{*})&=\frac{1}{\delta }\cdot \left( \log (\frac{N}{\varphi (\delta ^{*})e^\delta })+\frac{\sigma ^2}{2}\right) \nonumber \\&=-\varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) +\sigma \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} d_2(\delta ^{*})&=d_1-\sigma =-\varPhi ^{-1}\left( \frac{e^{b-\delta ^{*}}-1}{e^b-1}\right) \end{aligned} \end{aligned}$$
(30)

We obtain

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

and

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

such that we must assume

$$\begin{aligned} u\le Ne^\frac{\sigma ^2}{2}+N\varPhi (-d_1(\delta ^{*}))(e^b-1), \end{aligned}$$
(31)

in order for vault participation to hold.

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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. https://doi.org/10.1007/978-3-031-18679-0_4

Download citation

Publish with us

Policies and ethics