# Approximation for the Volatility of an Interest Rate

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.

• Quantitative Finance Stack Exchange might be a better fit for the question. Feb 22, 2023 at 17:55
• Thank you, I'll post it there. Feb 22, 2023 at 18:31

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
• 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? Feb 22, 2023 at 21:06
• 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. Feb 22, 2023 at 22:27