Links to all tutorial articles (same as those on the Exam pages)Understanding convexity: first and second derivatives of a price function
First and second derivatives are important in finance – in particular in measuring risk for fixed income and options. In fixed income – the first and second derivatives are modified duration and convexity respectively, and for options, these are delta and gamma. But what do these really mean – and what does one think about them when one sees a number? The rest of this article attempts to provide an intuitive look at how price changes for a bond (or an option) are determined by the first and the second derivative, what they mean, and how they are to be interpreted. Note that this isn’t a repetition of the text book – though some of that is inevitable. The idea is to provide an intuitive understanding of what these variables are, what they really represent, and how to think about them. What this article will cover is as follows:
Based on this illustration, we can say that the bond price moves by 10% for a 1% change in yield. This is the modified duration. It is just the rate of change. Formulaically, we can express the rate of change as
To summarize:
Options: Delta and Gamma
