If you want to go into pricing, by all means. Learn pde. No phd? Good luck in getting that job. In a nutshell, the candidate asked if knowing pde is helpful prior to joining mfe. Sure it is Is knowing pde helpful in getting a job? Hell no. I have used crank Nicholson to solve finite difference methods in my numerical class. Put that in ur resume and let's see if it gets u past the recruiters trash can.
What is the "PDE"? I'm a newbie. Sorry if my question is stupid. Minor point; it's Crank-Nicolson. ThanhBinh said:. PDE is partial differential equation.
Here is a relevant one for Black Scholes. Thank you very much. This document is useful for me. I'm a computer science engineer. I'm learning about MFE by myself. Le Van. Le Van said:. Perhaps you have overlooked it. Granted, machine learning and big data analytics are gaining popularity but that doesn't undermine the importance of PDE. At the end of the day it all boils down to where you'd like to work; buy sides firm generally want to hire statisticians and data scientists while sell side firms prefer mathematicians.
Good point. I have been focusing mostly on statistics and econometrics the past year. So I think I underestimated the importance of those subjects. Post reply. Then Eq. If Eq. Thus Eq. To compute a solution to Eq. Alternatively, we could now discretize Eqs. For example, if we apply Eq. Equation 48 , the classical Euler's method , can be used to step along the solution of Eq.
Application of Eq. In Eq. Note that Eq. We can now consider using Eq. The finite difference form of Eq. We must also specify two boundary conditions BCs for Eq. Equations 49 , 51 , 53 and 55 constitute the full system of equations for the calculation of the numerical solution to Eq. Note that we have replaced the original PDE, Eq. Also, an analytical solution to Eq. Representative output from this program that compares the numerical solution from Eqs.
Additional parameters follow from the values in Table 2. We can note two additional points about these values:. Finally, some descriptive comments about the details of the program in Appendix 1 are given immediately after the program listing.
We again have an analytical solution to evaluate the numerical solution. Since Eq. An important difference between the parabolic problem of Eq. Equations 66 , 67 , 68 , 69 , 70 , 71 constitute the complete finite difference approximation of Eqs.
A small MATLAB program for this approximation is given in Appendix 2 , including some descriptive comments immediately after the program. The general analytical solution to Eq. The plotted output from the program is given in Figure 1 and includes both the numerical solution of Eqs. We can note the following points about Figure 1 :. The parameters that produced the numerical output in Figure 1 are listed in Table 3.
Additional parameters follow from the values in Table 3. This is a stringent test of the numerical solution since the curvature of the solution is greatest at these peaks.
The results are summarized in Table 4. Of course, the peak analytical values given by Eq. This explicit finite difference numerical solution also has a stability limit like the preceding parabolic problem. In other words, the parameters of Table 3 were chosen primarily for accuracy and not stability. Rather, we will convert Eq. The idea then is to integrate Eq. The analytical solution to Eqs. The analytical solution of Eq.
To develop a numerical solution to Eq. The output from this program is listed in Table 5. The convergence of the solution of Eq. The parameters that produced the numerical output in Table 5 are listed in Table 6. Additional parameters follow from the values in Table 6. The stability constraint for the 2D problem of Eq. The actual path that the parabolic problem takes to the solution of the elliptic problem is not relevant so long as the parabolic solution converges to the elliptic solution.
Some other trial values indicated that this initial value is not critical but it should be as close to the final value, for example 2. This parametrization is an example of continuation in which the solution is continued from the given assumed starting value of Eq. The concept of continuation has been applied in many forms and not just through the addition of a derivative as in Eq. In general, the errors in the numerical solution of PDEs can result from the limited accuracy of all of the approximations used in the calculation.
For example, the 0. In formulating a numerical method or algorithm for the solution of a PDE problem, it is necessary to balance the discretization errors so that one source of error does not dominate, and generally degrade, the numerical solution. Thus, control of approximation errors is central to the calculation of a numerical solution of acceptable accuracy.
In the preceding examples, this control of errors can be accomplished in three ways:. The three preceding numerical solutions were developed using basic finite differences such as in Eqs. However, many approaches to approximating derivatives in PDEs have been developed and used. Among these are finite elements, finite volumes, weighted residuals , e. Each of these methods has advantages and disadvantages, often according to the characteristics of the problem of interest starting with the parabolic, hyperbolic and elliptic geometric classifications.
Thus, an extensive literature for the numerical solution of PDEs is available, and we have only presented here a few basic concepts and examples.
The principal advantage of numerical methods applied to PDEs is that, in principle, PDEs of any number and complexity can be solved which is particularly useful when analytical solutions are not available. As another example, a solution to the Burgers equation could be computed by extending Eq.
While Euler's method is general with respect to the form of the initial value integration, it does have two important limitations:. Thus, the Euler method is limited by both accuracy and stability. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?
Learn more. What exactly are partial differential equations? Ask Question. Asked 9 years, 1 month ago. Active 4 years ago. Viewed 3k times. JohnPhteven JohnPhteven 1, 3 3 gold badges 29 29 silver badges 47 47 bronze badges.
Add a comment. Active Oldest Votes. Intuition aside, the mathematical formulation of a PDE can be stated relatively simply. Further readings Sergiu Klainerman's essay , an abridged version of which appeared in the Princeton Companion to Mathematics. Willie Wong Willie Wong Vrashabh Irde Vrashabh Irde 4 4 silver badges 12 12 bronze badges.
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