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A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
A strictly convex function is a function that the straight line between any pair of points on the curve is above the curve except for the intersection points between the straight line and the curve. An example of a function which is convex but not strictly convex is f ( x , y ) = x 2 + y {\displaystyle f(x,y)=x^{2}+y} .
If the combined convexity and duration of a trading book is high, so is the risk. [16] However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur. (Parallel in the yield curve) [17]
If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets). For simplicity, we shall assume that the preferences of a consumer can be described by a utility function that is a continuous function , which implies that the ...
The boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A. A convex function is a real-valued function defined on an interval with the property that its epigraph (the set ...
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space , or equivalently as the set of all convex combinations of points in the subset.
Convex functions are related to convex sets. Specifically, the function is convex if and only if its epigraph. A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set. A graph of the bivariate convex function + +.
Geometrically, if the model price curves up on both sides of the present value (the payoff function is convex up, and is above a tangent line at that point), then if the price of the underlying changes, the price of the output is greater than is modeled using only the first derivative.