An ETD Method for Vulnerable American Options

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the authors propose the use of exponential time differencing (ETD) technique as a numerical method to solve the pricing problem of vulnerable American options. They address three key challenges, including eliminating cross-derivative terms, handling early-exercise opportunities, and substituting fixed boundary conditions. Subsequently, the accuracy and efficiency of the proposed method are verified through numerical experiments and comparison with existing methods. Finally, an analysis of numerical stability and convergence speed is conducted.

I find the topic and model to be novel and interesting, the methods to be correct, and the results to be discussed in an appropriate context. Therefore, the paper should be accepted. 

Author Response

We are grateful for the positive feedback and comments of the reviewer and support for the publication of our paper.

Reviewer 2 Report

Comments and Suggestions for Authors

  In this article, an exponential time differencing (ETD) numerical approximation is applied to solve for efficiently American vulnerable options pricing. 

 

  The vulnerable American put option price turns out to be the solution of the nonlinear PDE (3), subject to the initial condition (4). 

 

  The numerical scheme is constructed in Section 3. The transformed PDE takes the form of (22), with a transformed initial data (23). The centered difference discretization is given by (26)-(27), combined with the boundary extrapolation approximations (28)-(31). The second order ETD temporal discretization is formulated as (37). Moreover, the default case solution is derived in section 3.3. 

 

  Finally, some numerical results are presented in Section 4, in which the numerical stability and convergence are validated by the numerical experiments.  

 

  The mathematical is of certain interests, with optical application in financial markets. The numerical idea is interesting, and the numerical results are convincing.  

 

  The reviewer strongly supports its publication at "Mathematics" after the following improvements are made. 

 

 

(1) For the proposed ETD scheme (37), the numerical stability and convergence has been demonstrated by the numerical experiments. Meanwhile, the reviewer wonders whether a theoretical analysis could be established for the numerical stability and convergence or not. 

 

  In fact, there have been quite a few existing works of stability and convergence analysis for the multistep ETD schemes for various nonlinear PDEs, such as a stabilized second order exponential time differencing multistep method for thin film growth model without slope selection, a third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, energy stable higher order linear ETD multi-step methods for gradient flows and application to thin film epitaxy, etc. 

 

  The reviewer believes some theoretical techniques of stability and convergence analysis for the ETD schemes may be applicable to the nonlinear PDE studied in this article. 

 

  The authors should address this issue in the revision. AT least a remark should be made. 

 

 

(2) In addition to the second order accurate ETD scheme, the reviewer wonders whether a third order and even higher order ETD schemes could be similarly designed or not. 

 

  The authors should explain this issue in the revision. 

 

Author Response

We are grateful for the constructive feedback from the reviewer and the chance to improve our manuscript. Here we answer to the comments of the reviewer:

1. Theoretical Analysis for Stability and Convergence: We have added a remark to the manuscript in response to the inquiry about theoretical analysis. This note discusses the relevance and potential applicability of existing theoretical techniques for analyzing stability and convergence in our ETD scheme and explains our study's empirical approach. We have also reviewed and incorporated additional relevant references.
2. Higher Order ETD Schemes: The manuscript now includes a more detailed justification for choosing a second-order scheme over higher-order alternatives, considering computational resource constraints and the complexity of implementation. 

   We would like to highlight that our primary interest in assessing these methods' effectiveness in practical scenarios, rather than seeking potentially higher-order solutions that may not significantly benefit the problems addressed.

We believe these revisions have significantly enhanced our manuscript, making it a substantial contribution to the field. Moreover, we acknowledge the importance of the issues raised by Reviewer 2 for future research, viewing them as potential subjects for a further study.