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t. e. In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, and is defined as the second derivative of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the duration, the more sensitive the bond price is to the change in ...
In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively straightforward to translate the mathematical description of the evolution of future interest rates ...
Essentially, the Monte Carlo method solves a problem by directly simulating the underlying (physical) process and then calculating the (average) result of the process. [ 1 ] This very general approach is valid in areas such as physics, chemistry, computer science etc. In finance, the Monte Carlo method is used to simulate the various sources of ...
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
It does not take into effect other risk factors, such as non-parallel yield curve shifts, convexity, option-adjusted spreads, and others. However, effective duration may suffice for many managers as a basic risk measure. Virtually no research has been published on the attribution of other sources of risk for MBS.
The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations: [7]: chpt.4 [10] A hierarchy of convex optimization problems. (LP: linear programming, QP: quadratic programming, SOCP second-order cone program, SDP: semidefinite programming, CP: conic optimization.)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point lies in the convex hull of a set , then lies in some -dimensional simplex with vertices in . Equivalently, can be written as the convex combination of at most points in . Additionally, can be written as the convex combination of at most extremal points in , as ...
Newton's method in optimization. A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding ...