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Convexity
Convexity and describe the curvature of a function. While they can easily be identified visually on a graph (convexity graphed below), they can also be determined mathematically by taking the second derivative of the curve function. A negative return indicates concavity, and a positive return indicates convexity.
For example, notice how the slope of the curve in Figure 1 increases as it progresses down the X-axis. This represents an accelerating positive rate of change (convex up). Figure 2, on the other hand, shows a curve with a slope that becomes less negative as it progresses. This represents a slowing negative rate of change (convex down). Both of these figures depict a convex relationship between the X and Y parameters.
How does convexity apply to investing?
An example of a convex up relationship as shown in Figure 1 is the relationship between and The yield of an investment that collects interest at a fixed rate (such as a will grow exponentially due to the This exponential growth is the very reason we invest.
An example of a convex down relationship as shown in Figure 2 is the relationship between and As the puts move from deep to far their delta values become less negative until they reach 0.