Cryptocurrency is one of the earliest and the most successful applications of blockchain, and it utilizes the distributed ledger, which is a commonly used technique in blockchain, to make a decentralized transaction within the blockchain of a cryptocurrency. However, how to make a decentralized transaction of cryptocurrencies between parties on different blockchains, that is, the cross-chain exchange, is not well-studied. In this paper, we develop a new method to make cross-chain exchanges based on the classical atomic swap. We first study the optionality embedded into the atomic swap and propose to add a premium into the atomic swap, and then design a new procedure with the premium to guarantee the fairness of the cross-chain exchange. We also provide an algorithm based on the least-squares Monte Carlo method to estimate the premium and analyze the convergence of the algorithm. Moreover, we study the cross-chain exchange with margin trading. We propose an adapted exchange procedure to make a fair cross-chain exchange and an algorithm to estimate the fair premium under the margin trading. Numerical experiments are provided to show the effectiveness of the algorithms.

中文翻译：

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基于蒙特卡罗模拟的加密货币跨链交易公平设计

加密货币是区块链最早、最成功的应用之一，它利用区块链中常用的分布式账本技术，在加密货币的区块链内进行去中心化的交易。然而，如何在不同区块链上的各方之间进行加密货币的去中心化交易，即跨链交易，还没有得到很好的研究。在本文中，我们开发了一种基于经典原子交换进行跨链交换的新方法。我们首先研究嵌入到原子交换中的可选性，并提出在原子交换中加入溢价，然后用溢价设计一个新的程序来保证跨链交换的公平性。我们还提供了一种基于最小二乘蒙特卡罗方法的算法来估计溢价并分析算法的收敛性。此外，我们研究了保证金交易的跨链交易。我们提出了一种经过调整的交换程序来进行公平的跨链交换，并提出一种算法来估计保证金交易下的公平溢价。数值实验证明了算法的有效性。