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Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Gangnan Yuan, Ding Deng, Jinqiao Duan, Weiguo Lu, Fengyan Wu

TL;DR

A method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compares it with the benchmark Monte Carlo-COS (MC -COS) method to derive the convergence of the numerical scheme.

Abstract

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

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

TL;DR

A method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compares it with the benchmark Monte Carlo-COS (MC -COS) method to derive the convergence of the numerical scheme.

Abstract

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

Paper Structure

This paper contains 16 sections, 5 theorems, 74 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Models For XVA and Exposure
  3. Numerical Methods
  4. Numerical Results
  5. Conclusion
  6. Acknowledgements

Key Result

Lemma 1

lican2015 The Y-th order left and right RL tempered fractional derivatives of $V(x)$ at point $x_n$ can be approximated by the tempered-WSGD operators: here and the weights are given by and the parameters $\gamma_1,~\gamma_2$ and $\gamma_3$ admit the following linear system here $\gamma_3$ is the free variable.

Figures (5)

  • Figure 1: Distributions of $S_{T}$ under the CGMY process with default parameters when $S_{0}=40$, generated by Monte Carlo Method.
  • Figure 2: $EE$ of Examples 1 and 2 for Bermudan call option, comparison of MC-FF method and MC-COS method, for 50 exercise times.
  • Figure 3: $EE$, $PFE_{97.5\%}$ and $PFE_{2.5\%}$ of Examples 3 and 4 for Bermudan call option, comparison of MC-FF method and MC-COS method, for 12 exercise times.
  • Figure 4: The effects of different parameters in the CGMY model on $EE$ for Bermudan call option, for 12 exercise times. Except for the changing parameters, the other reference parameters are $C=1$, $G=25$, $M=26$, $Y=1.5,$ and strike price $K=50$.
  • Figure 5: Bermudan call option values with CVA and XVA against the strike price $K$ with default CGMY parameters.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 1
  • proof