Return Smoothing in Discrete and Continuous Time

Abstract

In this thesis we propose four novel continuous-time return smoothing models. Borrowing from the private commercial real estate literature, we start with a benchmark model for return smoothing in discrete time. The benchmark model is translated into continuous time, taking us from an autoregressive moving average (ARMA) specification for the smoothed holding period return to an Ornstein-Uhlenbeck (OU) specification for the instantaneous continuously compounding rate of return on the smoothed asset price. The model is then extended to allow for the possibility of predictability in the underlying “true” return.

In a second line of investigation, we propose an alternative continuous-time return smoothing model in which we keep the OU smoothing mechanism, but replace the instantaneous smoothed return with the smoothed detrended log price. This model leads to unrealistic autocorrelations in the smoothed return, and we address this with an extension that introduces a higher-order smoothing equation described by a continuous-time autoregressive moving average (CARMA) process—the continuous-time analogue of the ARMA process.

We show that each of our four models belongs to a general framework for linear return smoothing in continuous time in which a CARMA process governing the underlying “true” asset price is overlaid with a CARMA-type smoothing equation that summarises the market mechanism whereby the “true” price is transformed into a market-observed, smoothed price. In each model, as with the general framework, the noise in the “true” price is represented by a Lévy process, allowing for non-normality and sample path discontinuities.

To quantify the effect the smoothing models have on holding period returns, we develop a common set of smoothing metrics. These metrics are then computed for each of the models (including the discrete-time benchmark model), and form a basis on which the models can be compared. We also rely on the autocorrelation function in appraising the impact of return smoothing. We comment on the ability of the continuous-time smoothing models to reproduce stylised statistical properties commonly associated with smoothed returns, such as a reduction in return variance and an increase in return autocorrelation. Additionally, we develop the theory needed to operationalise the smoothing metrics for the smoothing framework in general.