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Trading under the Proof-of-Stake Protocol -- a Continuous-Time Control Approach

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  • Wenpin Tang
  • David D. Yao

Abstract

We develop a continuous-time control approach to optimal trading in a Proof-of-Stake (PoS) blockchain, formulated as a consumption-investment problem that aims to strike the optimal balance between a participant's (or agent's) utility from holding/trading stakes and utility from consumption. We present solutions via dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equations. When the utility functions are linear or convex, we derive close-form solutions and show that the bang-bang strategy is optimal (i.e., always buy or sell at full capacity). Furthermore, we bring out the explicit connection between the rate of return in trading/holding stakes and the participant's risk-adjusted valuation of the stakes. In particular, we show when a participant is risk-neutral or risk-seeking, corresponding to the risk-adjusted valuation being a martingale or a sub-martingale, the optimal strategy must be to either buy all the time, sell all the time, or first buy then sell, and with both buying and selling executed at full capacity. We also propose a risk-control version of the consumption-investment problem; and for a special case, the ''stake-parity'' problem, we show a mean-reverting strategy is optimal.

Suggested Citation

  • Wenpin Tang & David D. Yao, 2022. "Trading under the Proof-of-Stake Protocol -- a Continuous-Time Control Approach," Papers 2207.12581, arXiv.org, revised Jun 2023.
    • Handle: RePEc:arx:papers:2207.12581
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    References listed on IDEAS

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    1. Jonathan Chiu & Thorsten V. Koeppl, 2017. "The Economics Of Cryptocurrencies - Bitcoin And Beyond," Working Paper 1389, Economics Department, Queen's University.
    2. Jiri Chod & Nikolaos Trichakis & Gerry Tsoukalas & Henry Aspegren & Mark Weber, 2020. "On the Financing Benefits of Supply Chain Transparency and Blockchain Adoption," Management Science, INFORMS, vol. 66(10), pages 4378-4396, October.
    3. Ioanid Roşu & Fahad Saleh, 2021. "Evolution of Shares in a Proof-of-Stake Cryptocurrency," Management Science, INFORMS, vol. 67(2), pages 661-672, February.
    4. Camilo Mora & Randi L. Rollins & Katie Taladay & Michael B. Kantar & Mason K. Chock & Mio Shimada & Erik C. Franklin, 2018. "Bitcoin emissions alone could push global warming above 2°C," Nature Climate Change, Nature, vol. 8(11), pages 931-933, November.
    5. Wenpin Tang & David D. Yao, 2022. "Polynomial Voting Rules," Papers 2206.10105, arXiv.org, revised Jan 2024.
    6. Charles Bertucci & Louis Bertucci & Jean-Michel Lasry & Pierre-Louis Lions, 2020. "Mean Field Game Approach to Bitcoin Mining," Papers 2004.08167, arXiv.org.
    7. Wenpin Tang, 2022. "Stability of shares in the Proof of Stake Protocol -- Concentration and Phase Transitions," Papers 2206.02227, arXiv.org.
    8. Chiu, Jonathan & Koeppl, Thorsten V, 2017. "The economics of cryptocurrencies – bitcoin and beyond," Working Paper Series 6688, Victoria University of Wellington, School of Economics and Finance.
    9. Fahad Saleh & Wei Jiang, 2021. "Blockchain without Waste: Proof-of-Stake [Proof of Work vs Proof of Stake]," The Review of Financial Studies, Society for Financial Studies, vol. 34(3), pages 1156-1190.
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    Cited by:

    1. Wenpin Tang & David D. Yao, 2023. "Transaction fee mechanism for Proof-of-Stake protocol," Papers 2308.13881, arXiv.org, revised Aug 2023.
    2. Wenpin Tang, 2023. "Trading and wealth evolution in the Proof of Stake protocol," Papers 2308.01803, arXiv.org, revised Aug 2023.

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