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I'm a pure math student teaching myself the basics of quantitative finance, and I'm having a hard time understanding some of the approximations/nonrigorous claims that are common in applied statistics. Here's a representative example from Stephen Blyth's An Introduction to Quantitative Finance. One of the first exercises in the book begins:

The volatility of a[n interest] rate $y$ is usually defined as the standard deviation of its logarithm, and can be well approximated by $\frac{dy}{y}$.

This confuses me in two ways:

  1. Treating $y$ as a random variable, the standard deviation of $\log{y}$ is a single number, not an expression involving $y$. So how could $dy/y$ possibly be an approximation for it? Moreover, the exercise then asks the reader to compute the ratio of the volatilities of two rates $y_{SB}$ and $y_{AM}$ (their definitions are not important, just that $y_{AM}$ is a function of $y_{SB}$), "assuming that the current level of rates is $y_{SB} = 6\%$." But again, why should the volatilities of the rates depend at all on their current levels, given the definition of volatility as a standard deviation of an r.v.?

  2. How am I meant to interpret the symbol $dy$ in this context? There's clearly some level of hand-waving here, but I'm not sure how much or in what way.

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  1. $d \ln y=\frac{dy}y$. there's a little issue here, of course, with $y$ being a stochastic process, so the differential is a bit tricker (see Ito calculus), thus $d \ln y\approx\frac{dy}y$. Why would volatilities change with level? This happens all the time in statistics. In fact the question is why should not gtehy change at all? But here's one reason: the interest rates are often assumed to be strictly positive or non-negative. Hence, the volatility of a very low interest rate should be be different from when it's very high.
  2. $dy$ is simply the change in y, say, the daily absolute change in interest rate
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  • $\begingroup$ I see, I think my problem was in assuming that $y$ is a single random variable. Would it be more accurate to think of $y$ as a family of random variables $y_t$, for $t$ discrete or continuous, each of which have their own volatility? $\endgroup$
    – Nick A.
    Feb 22, 2023 at 21:06
  • $\begingroup$ it's best to think of $y_t$ as a stochastic process or a time series. you can model it in many different ways, e.g. stochastic vol process where volatility is proportional to the level of $y_t$ etc. $\endgroup$
    – Aksakal
    Feb 22, 2023 at 22:27

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