بررسی بازده نرخ ارز در افق های زمانی مختلف

This paper explores and compares the empirical distribution of the US dollar–deutsche mark exchange rate returns with well-known continuous-times processes at different frequencies. We use a variety of parametric models to simulate the unconditional density of the exchange rate returns at different frequencies, and show that the studied models do not fit the empirical distribution of exchange rate returns at both the high and low frequencies.

Many models in continuous-time finance rely on the assumption of a specific stochastic process, while little attention is paid to the empirical fit of an assumed process to the actual data across different time scales. Consequently, the application of such statistical models to financial data would result in specification errors when the underlying data-generating process scales differently across time. This assumption can also lead to mispricing of financial assets, and can have serious implications on portfolio selection and risk management. There is now evidence that investors do have heterogeneous expectations differentiated according to their time dimensions (see Ref. [4]). The co-existence of short-term as well as long-term traders indicates that there are different time scales for different traders in the market. Therefore, different time scales can lead to different price formation processes, which have other effects such as volatility clustering and foreign exchange adjustment. However, investigating the scaling properties of foreign exchange returns and modelling its dynamics is far away from being trivial, and one recent promising attempt is the wavelet multi-scaling approach (see Ref. [7]). The aim of this study is to investigate the performance of the well-known stochastic processes in fitting the empirical distribution of the exchange rate returns at different time scales. We start with estimating the parameters of the candidate processes at different time scales and proceed with simulating the empirical distributions of exchange rate returns from selected candidate processes. The theoretical distributions are then compared with the empirical distribution via a Kolmogorov–Smirnov goodness-of-fit test.

This paper presents evidence that the empirical distribution of returns behaves differently at different frequencies. In fact, different traders who exist in the market do not react the same way to different flows of information. Furthermore, we see that the simulated return distributions do not replicate the empirical distribution according to the Kolmogorov–Smirnov goodness-of-fit test at the 5% significance level, with an exception for the random walk-GARCH(1,1) with student-t errors and the Jump–Diffusion model at weekly frequency. Finally, none of the studied models fit the empirical distribution of exchange rate returns at both the high and low frequencies. Broadening the analysis to several different exchange rates would determine whether the results of this paper could be generalized to the currency market in its entirety.