Abstract Cryptocurrencies and Bitcoin, in particular, are prone to wild swings resulting in frequent jumps in prices, making them historically popular for traders to speculate. It is claimed in recent literature that Bitcoin price is influenced by sentiment about the Bitcoin system. Transaction, as well as the popularity, have shown positive evidence as potential drivers of Bitcoin price. This study introduces a bivariate jump-diffusion model to capture the dynamics of Bitcoin prices and the Bitcoin sentiment indicator, integrating trading volumes or Google search trends with Bitcoin price movements. We derive a closed-form solution for the Bitcoin price and the associated BlackāScholes equation for Bitcoin option valuation. The resulting partial differential equation for Bitcoin options is solved using an artificial neural network, and the model is validated with data from highly volatile stocks. We further test the modelās robustness across a broad spectrum of parameters, comparing the results to those obtained through Monte Carlo simulations. Our findings demonstrate the modelās practical significance in accurately predicting Bitcoin price movements and option values, providing a reliable tool for traders, analysts, and risk managers in the cryptocurrency market.
We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. We then show that the value function can be reduced to a new function with a finite dimensional argument by a so called branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.
Constant function market makers (CFMMs) such as Uniswap, Balancer, Curve, and mStable, among many others, make up some of the largest decentralized exchanges on Ethereum and other blockchains. Because all transactions are public in current implementations, a natural next question is if there exist similar decentralized exchanges which are privacy-preserving; i.e., if a transaction's quantities are hidden from the public view, then an adversary cannot correctly reconstruct the traded quantities from other public information. In this note, we show that privacy is impossible with the usual implementations of CFMMs under most reasonable models of an adversary and provide some mitigating strategies.
Charles Bertucci, Louis Bertucci, JeanāMichel Lasry, PierreāLouis Lions
We present an analysis of the Proof-of-Work consensus algorithm, used on the Bitcoin blockchain, using a Mean Field Game framework. Using a master equation, we provide an equilibrium characterization of the total computational power devoted to mining the blockchain (hashrate). From a simple setting we show how the master equation approach allows us to enrich the model by relaxing most of the simplifying assumptions. The essential structure of the game is preserved across all the enrichments. In deterministic settings, the hashrate ultimately reaches a steady state in which it increases at the rate of technological progress. In stochastic settings, there exists a target for the hashrate for every possible random state. As a consequence, we show that in equilibrium the security of the underlying blockchain is either $i)$ constant, or $ii)$ increases with the demand for the underlying cryptocurrency.