American Put Front-Fixing Newton Jacobian Structure

Summary

The document asks whether a displayed matrix is the Jacobian for applying Newton’s method to a front-fixing numerical method for an American put option. Its entries show a mostly banded structure, with coefficients on neighboring option-value variables and a final column containing terms tied to the moving exercise boundary and derivatives of the coefficients.

The excerpt provides no derivation, underlying discretized equations, boundary conditions, or confirmation of the proposed matrix. It therefore serves mainly as a narrowly focused implementation question rather than a worked method. To check the Jacobian, a reader would need the source paper’s residual equations and would need to differentiate each residual with respect to every unknown, including the front-fixing boundary variable.

Key ideas

• The question concerns Newton iteration for an American put solved with a front-fixing method.
• The proposed Jacobian has a banded structure for option-value variables and a separate column for the moving boundary.
• Several entries include derivatives of discretization coefficients with respect to the boundary-related unknown.
• The excerpt does not supply equations or a derivation that would establish whether the matrix is correct.

Tags

# Jacobian for Newton method for American options by front fixing


# Jacobian for Newton method for American options by front fixing












In this paper Penalty and front-fixing methods for the numerical solution of American option problems a front fixing method based on Newton is described for an American put option is described. I am trying to implement this method. Can someone confirm that the following is the Jacobian matrix $J$. $$\left( \begin{array}{ccccccc} \gamma^n_1 & & &&&&-\beta^n_1-\alpha^n_1(1+\Delta x)+(\gamma^n_1)'p^n_2+(\beta^n_1)'(E-\bar{S}^n)\\ \alpha^n_2 &\gamma^n_2 & &&&&-\beta^n_2(1+\Delta x)+(\gamma^n_2)'p^n_3+(\beta^n_2)'(E-(1+\Delta x)\bar{S}^n)\\ \beta^n_3 & \alpha^n_3 & \gamma^n_3& &&&(\beta^n_3)'p^n_2+(\gamma^n_3)'p^n_4\\ &\ddots&\ddots&\ddots&&&\vdots\\ &&\ddots&\ddots&\ddots&&\vdots\\ &&& \beta^n_{M-1} & \alpha^n_{M-1} & \gamma^n_{M-1}&(\beta^n_{M-1})'p^n_{M-2}+(\gamma^n_{M-1})'p^n_{M} \\ &&&&\beta^n_{M} &\alpha^n_{M}&(\beta^n_{M})'p^n_{M-1}\\ \end{array} \right)$$

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