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1 Example code in MATLAB. 2 See also. 3 References. ... Download QR code; Print/export Download as PDF; Printable version; In other projects
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example the Black–Scholes PDE. Once in this form, a finite difference model can be derived, and the valuation obtained. [2]
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The source code for a function is replaced by an automatically generated source code that includes statements for calculating the derivatives interleaved with the original instructions. Source code transformation can be implemented for all programming languages, and it is also easier for the compiler to do compile time optimizations.
Figure 1.Comparison of different schemes. In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and provides numerical solutions to differential equations. [1]
For example, for bond options [3] the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff.
The following procedure can be used to easily test if any source code is derivative code or not. Delete the code in question; Build (or compile) the project; If the build process simply replaces the source code which has been deleted, it is (obviously) code which has been derived from something else and is therefore, by definition, derivative code.