Trading Under the Proof-of-Stake Protocol – A Continuous-Time Control Approach

To appear in Mathematical Finance https://onlinelibrary.wiley.com/doi/full/10.1111/mafi.12403

24 Pages Posted: 30 Jul 2022 Last revised: 11 Jun 2023

See all articles by Wenpin Tang

Wenpin Tang

Columbia University - Department of Industrial Engineering and Operations Research

David Yao

Columbia University

Date Written: July 26, 2022

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.

Keywords: Consumption-investment, Proof of Stake (PoS) protocol, cryptocurrency, dynamic programming, HJB equations, continuous-time control, risk control

Suggested Citation

Tang, Wenpin and Yao, David, Trading Under the Proof-of-Stake Protocol – A Continuous-Time Control Approach (July 26, 2022). To appear in Mathematical Finance https://onlinelibrary.wiley.com/doi/full/10.1111/mafi.12403, Available at SSRN: https://ssrn.com/abstract=4172486 or http://dx.doi.org/10.2139/ssrn.4172486

Wenpin Tang (Contact Author)

Columbia University - Department of Industrial Engineering and Operations Research ( email )

500 W. 120th Street #315
New York, NY 10027
United States

David Yao

Columbia University ( email )

New York, NY
United States

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