Risk Latte - What is Convexity & Concavity?

What is Convexity & Concavity?

Team Latte
March 10, 2005

We all know what a straight line is. It is the shortest distance between any two points on a map or a graph and is represented by the general form y = mx + c where y is the dependent variable, x is the independent variable, m is the slope of the line and c is an intercept. In the above equation m, the slope of the line is constant. All straight lines have constant slopes.

In contrast to a straight line, a curve has varying slope. What is a curve? A curve is wiggly line or bent line that wiggles or bends to join any two points on a graph or a map. A curve could be a quadratic function given by the equation y = ax2 + bx + c. A characteristic feature of a curve is that the slope of the curve always changes. In this quadratic equation x is the dependent variable and y is the independent variable and a, b and c are constants. The slope of the curve is 2ax + b and it changes every time the independent variable x changes. Look at the figure below where a simple (x,y) plot is done to show the dependence of y with x in the above quadratic equation.

Quadratic Function Convex Curve

The above equation may seem familiar to option traders and option theorists as it resembles the payoff from a bought (long) vanilla option. In fact the form of the equation y = ax2 + bx + c is at the heart of option pricing theory, though the expression there is in the form of a partial differential equation.

The payoff - or the (x,y) plot - in the above graph is a convex payoff and the curve is known as a convex curve. Any financial instrument that displays a payoff which has a shape like the above graph is said to display convexity.

What is convexity then in simple terms? If a curve is convex, a straight line connecting any two points (on that curve) lies totally above the curve. You can see that this is true by looking at the above curve. Take any two points on the curve and draw a straight line between them; you will see that the straight line will lie entirely above the curve.

Is there a negative convexity? Yes, there is and it is called a concavity

If we turn the above graph upside down we will get a concave curve. In other words if we have a quadratic function like y = - ax2 + bx + c then we will have a (x,y) plot that would look like:

Quadratic Function Concave Curve

Option traders are once again familiar with the above type of payoffs. It represents the payoff from a sold (short) option position. This curve is a concave curve and any (financial or derivative) instrument or structure that displays a payoff which has a shape resembling the above graph is said to have concavity.

A concave curve is one where a straight line connecting any two points on the curve lies entirely under the curve.

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