In general, no analytical formulas exist for pricing discretely monitored exotic options, even when a geometric Brownian motion governs the risk-neutral underlying. While specialized numerical algorithms exist for pricing particular contracts, few can be applied universally with consistent success and with general Lévy dynamics. This paper develops a general methodology for pricing early exercise and exotic financial options by extending the recently developed PROJ method. We are able to efficiently obtain accurate values for complex products including Bermudan/American options, Bermudan barrier options, survival probabilities and credit default swaps by value recursion, European barrier and lookback/hindsight options by density recursion, and arithmetic Asian options by characteristic function recursion. This paper presents a unified approach to tackling these and related problems. Algorithms are provided for each option type, along with a demonstration of convergence. We also provide a large set of reference prices for exotic, American and European options under Black-Scholes-Merton, Normal Inverse Gaussian, Kou’s double exponential jump diffusion, Variance Gamma, KoBoL/CGMY and Merton’s jump diffusion models.

中文翻译：

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带有跳跃扩散和其他 Levy 过程的美国和奇异期权定价

一般而言，不存在用于对离散监控的奇异期权定价的分析公式，即使几何布朗运动控制风险中性标的。虽然存在用于定价特定合同的专门数值算法，但很少有人能够普遍应用并取得一致的成功和一般的 Lévy 动力学。本文通过扩展最近开发的 PROJ 方法，开发了一种为早期行使和奇异金融期权定价的通用方法。我们能够有效地获得复杂产品的准确值，包括百慕大/美式期权、百慕大障碍期权、通过价值递归的生存概率和信用违约掉期、通过密度递归的欧洲障碍和回顾/事后期权，以及通过特征函数递归的算术亚洲期权. 本文提出了一种统一的方法来解决这些问题和相关问题。为每种选项类型提供了算法，以及收敛性演示。我们还在 Black-Scholes-Merton、Normal Inverse Gaussian、Kou 的双指数跳跃扩散、方差 Gamma、KoBoL/CGMY 和 Merton 的跳跃扩散模型下提供了大量的奇异期权、美式和欧式期权的参考价格。