FRM questions by bionic turtle

• The partial first derivative is very common in risk measurement. It appears in various asset classes and metrics; e.g., option delta, bond duration, risk contribution, marginal value at risk (marginal VaR)
• The first derivative is an instantaneous rate of change; i.e., the limiting ratio illustrated by the convergence of a secant line to a tangent line.
• We looked at some basic differentiation rules (e.g., power rule).
• I highly recommend the excellent, free calculus resources at www.analyzemath.com. My favorite calculus texts are The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) and Calculus Know-It-ALL: Beginner to Advanced, and Everything in Between by Stan Gibilisco. I also like the affordable Schaum's Outline of Calculus.
• We looked at a handy idea: The first derivative of the natural log of a function equals the function’s growth rate or relative rate of change: if f(x) = ln g(x), f’(x) = g’(x)/g(x).
• Given the price function of a 30-year zero-coupon bond under continuous discounting, p(y) = 100*EXP(-y*30), we showed that the first derivative is the dollar duration: p’(y) = –3000*EXP(-y*30). Further, while duration has various specific "flavors," all are variations on (functions of) this first derivative dollar duration. For example, by dividing by price [i.e., p(y)], we produce the modified/Macaulay duration of 30.
• It is helpful to be mindful of the axis units. In the case of dollar duration, our first derivative at 5% yield was about –670. What are the units? In this case (i.e., dollar duration), this refers to –670 $/%. Specifically, 670 dollars in price change for a one-unit (100 basis point) change in yield. Still confused about duration units? See the comments to this post.
• The option delta is a first derivative; i.e., the change in call price with respect to a change in stock price. What are the units in this case? Since we have dollars (option price on y axis) divided by dollars (stock price on x axis), they cancel and option delta is unitless.
• We applied the first derivative rule to confirm that (i) futures contract delta is EXP(r*t) and (ii) the Eurodollar futures contract implies a $25 dollar change for each one basis point move.
• Finally, I hope I succeeding in conveying the one big idea underlying this webinar. We may characterize portfolios as responding to (mapped to) a set of underlying risk factors (the call option analogy: the value of the call option reacts to risk factors such as volatility and interest rates). The partial first derivative returns a linear approximation of the portfolio’s sensitivity to an underlying risk factor.
• Phillip made an excellent point about the key weakness of the first derivative linear approximation: is it is only locally accurate; the larger the change in the underlying factor, the less accurate it becomes.