Monte Carlo Method for Pricing Lookback Type Options in Lévy Models

IF 0.5 4区 数学 Q4 STATISTICS & PROBABILITY
O. E. Kudryavtsev, A. S. Grechko, I. E. Mamedov
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引用次数: 0

Abstract

Theory of Probability &Its Applications, Volume 69, Issue 2, Page 243-264, August 2024.
We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener--Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.
莱维模型中回溯型期权定价的蒙特卡罗方法
概率论及其应用》第 69 卷第 2 期第 243-264 页,2024 年 8 月。 我们构建了一种通用蒙特卡洛方法,用于为支付函数取决于莱维过程极值最终位置的期权定价。所提出的方法不仅能评估浮动和固定执行回看期权在初始时的价格,还能评估莱维过程当前位置可能不同于其极值的整个期间的价格。我们的算法包括三个阶段:极值过程累积分布函数(c.d.f.)的近似、反转评估以及莱维过程极值最终位置的模拟。我们通过维纳--霍普夫因式分解,得到了莱维模型上极值和下极值过程的 c.d.f. 的新近似公式。我们还描述了混合蒙特卡洛方法的开发原理,该方法结合了用于构建极值过程最终位置的c.d.f.的经典数值方法和借助张量神经网络反演c.d.f.的机器学习方法。数值实验证明了通用蒙特卡洛法在回溯期权定价方面的效率。
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来源期刊
Theory of Probability and its Applications
Theory of Probability and its Applications 数学-统计学与概率论
CiteScore
1.00
自引率
16.70%
发文量
54
审稿时长
6 months
期刊介绍: Theory of Probability and Its Applications (TVP) accepts original articles and communications on the theory of probability, general problems of mathematical statistics, and applications of the theory of probability to natural science and technology. Articles of the latter type will be accepted only if the mathematical methods applied are essentially new.
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