The Feller diffusion, filter rules and abnormal stock returns

Abstract

We determine the conditional expected logarithmic (i.e. continuously compounded) return on a stock whose price evolves in terms of the Feller diffusion and then use it to demonstrate how one must know the exact probability density that describes a stock’s return before one can determine the correct way to calculate the abnormal returns that accrue on the stock. We show in particular that misspecification of the stochastic process which generates a stock’s price will lead to systematic biases in the abnormal returns calculated on the stock. We examine the implications this has for the proper conduct of empirical work and for the evaluation of stock and portfolio performance.

Keywords:

• Feller diffusion
• Fokker–Planck equation
• Geometric Brownian Motion
• logarithmic return

JEL Classifications:

• C52
• G14
• G17

View correction statement:

Corrigendum

Acknowledgements

The authors acknowledge the helpful comments and suggestions of the Editor and referee. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We emphasise in particular that we are not suggesting the expected logarithmic returns formula for determining abnormal returns under the Feller diffusion is in any way superior to that of the equivalent formula for the Geometric Brownian Motion.

2 This also makes binomial models particularly attractive because of the straightforward way in which they can be empirically implemented.

3 In order to obtain this result, one must use the recurrence relationships and , where I₀(z) is the modified Bessel function of the first kind of order zero (Abramowitz and Stegun 1964, 376).

4 Feller (1951a, 235) states this probability density without proof, whilst Feller (1951b, 180) uses the Fokker–Planck equation (9) to determine the Laplace transform of the probability density which he then inverts to determine the probability density itself. Unfortunately, the probability density determined in Feller (1951a, 235) differs from that stated in Feller (1951b, 180) by a multiple of 2b – where b = μ in our analysis is the expected buy and hold return on the stock on a per unit time basis. It thus follows that the probability density determined in Feller (1951b, 180) is incorrect – something that is of considerable importance given that the probability density summarised in Feller (1951b, 180) has been widely used in empirical analysis for parameter estimation (Gibbons and Ramaswamy 1993).

5 In the appendix we use this result to show (Feller 1951a, 236): Note how this result shows that at t = 0 the conditional probability density for the Feller diffusion as given by Equation (16) will take the form of a Dirac delta function which is completely concentrated at x(0) (Sneddon 1961, 51–53; Cox and Miller 1965, 209).

6 Feller (1951a, 236) again states this result without proof. Cox and Miller (1965, 236–237) provide an alternative proof based on a series expansion of the moment generating function for the probability density (16).

7 The exponential integral given here arises in a number of areas relating to the Feller diffusion. In the appendix, for example, we demonstrate how the average time it will take for a stock’s price to reach one or other of the upper and lower selling price triggers associated with a filter rule trading strategy (Fama and Blume 1966; Sweeney 1988; Chan, Jegadeesh, and Lakonishok 1996; Hong and Stein 1999; D’Aspremont 2011) is stated in terms of the exponential integral given here.

8 Here it is important to note that a simple application of L’Hôpital’s rule shows: a result which is consistent with the numerical example summarised in Table 1.

9 All integrals associated with the Feller diffusion are evaluated numerically using a combination of 50 point Gauss–Legendre quadrature and 50 point Gauss–Laguerre quadrature. These quadrature rules integrate polynomial expressions of order 99 or less exactly (Carnahan, Luther, and Wilkes 1969, 101–105).

10 One can reinforce this point by using Equation (8) in conjunction with Itô’s formula to determine the distributional properties of instantaneous increments in the logarithmic return, It then follows: This result shows that the instantaneous increment in the logarithmic return (per unit time) will have a mean of and variance Note that when or equivalently, the probability of eventual extinction exceeds then the expected instantaneous change in the stock’s logarithmic return will be negative. This in turn will mean that in expectations there will be a downward spiral in the logarithmic return that will culminate ultimately, in the firm entering bankruptcy (i.e. ultimate extinction).

11 Further insight may be obtained into the biases which arise from mistakenly assuming a stock’s price, x(t), evolves in terms of a Geometric Brownian Motion by applying a Taylor series expansion to log[x(t)] about the point x(0)e^μt. If one then takes expectations across the series expansion and substitutes the first four central moments for the Feller diffusion (Davidson and Tippett 2012, 218 and 232) one will end up with the following series expansion for the expected logarithmic return for the Feller diffusion: Substituting μ = 0.15, σ = 1 and t = 1 shows that the above expansion exactly replicates the results summarised in Table 1 except when x(0) = $5 and where the expansion gives expected logarithmic returns of 0.0461 and −0.0168, respectively. In these latter two cases the series expansion will need to encompass additional terms involving the higher order (fifth, sixth, etc.) moments if it is to give a more satisfactory approximation to the expected logarithmic return. This contrasts with the Geometric Brownian Motion where Equation (23) shows that only the first two moments are necessary to give an exact representation of the expected logarithmic return. Moreover, the above series expansion shows that the opening stock price, x(0), affects the expected logarithmic return for the Feller diffusion in a way that is entirely absent from the expected logarithmic return for the Geometric Brownian Motion.

12 A little reflection will convince the reader that our analysis has important implications for a much wider class of issues – including filter trading rules and momentum in stock returns, the prediction of corporate failure and the pricing of derivative securities (Cox 1996).

13 Here it will be recalled that the expected logarithmic return on the Geometric Brownian Motion is a constant which is independent of the opening stock price, x(0). Whether the stock price is ‘low’ or ‘high’, the expected logarithmic return is the same. This in turn means that the expected logarithmic returns in Table 1 under the false assumption of the Geometric Brownian Motion will be the same irrespective of the opening stock price. One can demonstrate the nature of the biases which arise under the false assumption of the Geometric Brownian Motion by supposing a researcher undertakes parameter estimation when the stock’s price is concentrated around x(0) = £45 per share. Consistent with the results summarised in Table 1 the researcher will find that the average annual logarithmic return over the estimation period will be around 13.96% (per annum). Given this, the researcher will set this return as the ‘benchmark’ against which the stock’s subsequent performance is to be assessed. However, if the researcher applies this benchmark to returns based on opening stock prices that are not in the neighbourhood of x(0) = £45, there will be evidence of (fictitious) abnormal returns. Thus, for example, Table 1 shows that if the stock’s price is in the neighbourhood of x(0) = £30 then applying a benchmark return of 13.96% will lead, on average, to a negative abnormal return of around (0.1343 − 0.1396 =) −0.53% (per annum). Similarly, if the stock’s price is in the vicinity of x(0) = £60 there will, on average, be a positive abnormal return of around (0.1422 − 0.1396 =) 0.26% (per annum).

14 Here it is important to emphasise that the expected abnormal buy and hold return under the filter rule trading strategy considered in this section does not turn out to be zero. This result has important implications for empirical work involving abnormal returns calculated in terms of the buy and hold returns which accrue on stocks (Jegadeesh and Titman 1993, 2001).