The Extreme Value Method for Estimating the Variance of the Rate of Return

TLDR

The random walk model applied to stock prices assumes that the logarithm of a common stock’s price follows a random walk, making the diffusion constant—equivalent to the variance of the rate of return—a key quantity to estimate. The paper proposes using the extreme value method to estimate this diffusion constant (variance of the rate of return). The authors first review the traditional estimation of the diffusion constant from closing prices, then introduce the extreme value method based on high and low prices, and apply it to estimate the variance of the rate of return. The extreme value method outperforms the traditional approach, yielding estimates that are 2.5 to 5 times more accurate and demonstrating a far superior precision when using high and low prices.

Abstract

The random walk problem has a long history. In fact, its application to the movement of security prices predates the application to Brownian motion.' And now it is generally accepted that, at least to a good approximation, ln (S), where S is the price of a common stock, follows a random walk.2 The diffusion constant characterizing that walk for each stock thus becomes an important quantity to calculate. In Section II, we describe the general random walk problem and show how the diffusion constant is traditionally estimated. In Section III, we discuss another way to estimate the diffusion constant, the extreme value method. In Section IV, we compare the traditional and extreme value methods and conclude that the extreme value method is about 21/2-5 times better, depending on how you choose to measure the difference. In Section V, we discuss the use of this method for the estimation of the variance of the rate of return of a common stock. If S is the price of a common stock, it is now generally accepted that In (S) follows a random walk, at least to a very good approximation. The diffusion constant characterizing that walk, which is the same as the variance of the rate of return, thus becomes an important quantity to calculate and is traditionally estimated using closing prices only. It is shown that the use of extreme values (the high and low prices) provides a far superior estimate.