纯跳Lévy波动下百慕大期权估值的总价值调整
Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations
AI总结：
针对交易对手违约风险下的期权定价问题，采用CGMY纯跳Lévy过程研究百慕大期权的总价值调整，提出MC-FF数值方法并通过离散能量估计证明收敛性，与基准方法对比验证有效性。
摘要：
新冠疫情期间，多家机构宣布其交易对手难以履行合约，因此在期权定价时有必要考虑交易对手违约风险。2008年金融危机后，各类价值调整在金融业受到重视。总价值调整（XVA）是多项价值调整的总和，已在Heston模型、Bates模型等多种随机模型中得到研究。本文采用一种应用广泛的纯跳Lévy过程——CGMY过程，对含多种价值调整的百慕大期权进行定价。在纯跳Lévy过程下，衍生品价值满足分数阶偏微分方程（FPDE）。为此，我们构建了蒙特卡洛与FPDE有限差分相结合的方法（MC-FF）来求风险敞口的数值近似，并将其与基准方法蒙特卡洛-COS（MC-COS）进行对比。我们采用与现有研究不同的离散能量估计方法推导数值格式的收敛性。基于数值结果，通过金融计算得到XVA。
英文摘要：
During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts.Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjustment (XVA) is the sum of multiple value adjustments, which is also investigated in many stochastic models such as Heston and Bates models. In this work, a widely used pure jump Lévy process, the CGMY process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump Lévy process, the value of derivatives satisfies a fractional partial differential equation(FPDE). Therefore, we construct a method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compare it with the benchmark Monte Carlo-COS (MC-COS) method. We use the discrete energy estimate method, which is different with the existing works, to derive the convergence of the numerical scheme.Based on the numerical results, the XVA is computed by the financial
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