If is a constant and is a variable, then what is common between the functions and ?

Most of us would immediately recognize the second function as the payoff from a call option. But what about the first function? That doesn’t seem like the payoff of a financial derivative. Actually, both the above functions share the property of "convexity". In other words, both these functions are convex. And the function, is as much a payoff of a financial derivative as the function . One of the chief reasons why every financial derivative has inherent value (and which sometimes is also known as the "time value") is because all financial derivatives have a convex payoff. Convexity is also known as "gamma" in option parlance. It is the same as what mathematicians call "curvature" of function (or a graph).

Actually, the reason why financial derivatives have value is due to something called Jensen’s inequality. If is a random variable and is a convex function of that random variable then Jensen’s inequality holds for such a function. We’ll talk about Jensen’s inequality in the next section.

Therefore, for financial derivatives to have value they must have a convex payoff. Every option, vanilla or exotic, every structured product has to have a convex payoff for it to have inherent value based on which it can be traded. Without convexity in its payoff, the financial derivative would just become the underlying asset (equity, FX, interest rate, etc.)

How do we define convexity?

If, , is a small parameter (constant) between 0 and 1 (i.e. 0 << 1 ) then, for a given interval , a convex function has the following property:

Let’s take some numbers to see if the two functions above share the above property.

Let’s take and let’s take the interval [100, 110]. Applying the above inequality on the first function, , we get :

Therefore, the left hand side is less than (or equal to) the right hand side. So, the function is convex.

Now, let’s take the second function, which is in fact, the payoff function for a call option. Let’s keep the interval the same, i.e. [100, 110] and . We choose the constant, (strike) to be 102. In fact, the choice of doesn’t matter at all and we can choose any value for . Now, applying the property of convexity to this function we get:

Once again we see that the left hand side is less than (or equal to) the right hand side. This shows that the function for the call option payoff is indeed convex.

In fact, we can take any values for between 0 and 1 and choose any interval , the above property (inequality) for convexity will always hold. We leave it as an exercise for the readers to choose different values of in (0 << 1) and any arbitrary interval to test whether the inequality for convexity holds for both the above functions.

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