آنالیز روشن موجک نرخ FX آسیایی با فرکانس بالا، تابستان 1997

Foreign exchange (FX) pricing processes are nonstationary: Their frequency characteristics are time dependent. Most do not conform to Geometric Brownian Motion (GBM), because they exhibit a scaling law with Hurst exponents between zero and 0.5 and fractal dimensions between 1.5 and 2. Wavelet multiresolution analysis (MRA), with Haar wavelets, is used to analyze these time and scale dependencies (self-similarity) of intraday Asian currency spot exchange rates. We use the ask and bid quotes of the currencies of eight Asian countries (Japan, Hong Kong, Indonesia, Malaysia, Philippines, Singapore, Taiwan, and Thailand) and, for comparison, of Germany for the crisis period May 1, 1998–August 31, 1997, provided by Telerate (U.S. dollar is the numéraire). Their time-scale-dependent spectra, which are localized in time, are observed in wavelet scalograms. The FX increments are characterized by the irregularity of their singularities. Their degrees of irregularity are measured by homogeneous Hurst exponents. These critical exponents are used to identify the global fractal dimension, relative stability, and long-term dependence, or long-term memory, of each Asian FX series. The invariance of each identified Hurst exponent is tested by comparing it at varying time and scale (frequency) resolutions. It appears that almost all investigated FX markets show antipersistent pricing behavior. The anchor currencies of the D-mark and Japanese Yen (JPY) are ultraefficient in the sense of being most antipersistent or “fast mean-reversing.” This is a surprising result because most financial analyst either assume neutral or persistent behavior in the financial markets, based on earlier research by Granger in the 1960s. This is a pedagogical paper explaining the most rational methodology for the identification of long-term memory in financial time series.

There is still a remarkable lack of scientific analysis of the crucial financial market phenomena that characterized the Asian Financial Crisis from a risk management perspective. The phenomena of financial crisis, turbulence, friction, and persistence are clearly inadequately described by the affine Markow (ARIMA) and Geometric Brownian Motion (GBM) models (Bouchaud & Potters, 2000). The main objective of this paper is to present more representative models and analytic procedures, adapted from hydrology, biometrics, and signal processing, to empirically identify foreign exchange (FX) rate models. The measurement of the empirical efficiency of FX markets for risk management purposes dates back to the early 1970s, when the 1944 Breton Woods Agreement of fixed exchange rates was discarded in 1971 and replaced by the current system of flexible exchange rates in 1973 (Cornell & Dietrich, 1978, Friedman & Vandersteel, 1982 and McFarland et al., 1982). But the then existing research methodology was inadequate to identify the model of the data. Recently, more discerning signal processing techniques to the same model identification problem have been proposed and appear to have more success (Bollerslev & Domowitz, 1993, Corazza & Malliaris, 2002, Gençay et al., 2001a, Gençay et al., 2001b, Van De Gucht et al., 1996, Los, 2000b and Los, 2003). In an efficient market, the arrival of new information produces instantaneous price correction, leaving no prospect for price prediction and therefore minimal opportunity for reaping abnormal profits. The efficient market hypothesis (EMH) states that a market where the best prediction of a price τ periods into the future (=the prediction horizon), based on current and past information, Et{x(t+τ)}, is its current, exact, and known price, xˆ(t)=x(t), is martingale efficient ( Fama, 1970 and Fama, 1991): 1 equation(1) View the MathML sourceEt{x(t+τ)}=xˆ(t)for anyτ>0 Turn MathJax on This implies that the martingale increments equation(2) View the MathML sourcex˜−τ(t)=x(t+τ)−xˆ(t) Turn MathJax on have zero mean (per definition): equation(3) View the MathML sourceEt{x˜−τ(t)}=0for anyτ>0 Turn MathJax on and that these increments are independent from each other. They may also be identically distributed=stationary, up to a Fickian Hurst exponent, or time-scaling parameter, of 0.5, so that their volatility equals that of the Geometric Brownian Motion (GBM): equation(4) View the MathML sourceσx˜−τ=στH=στ0.5 Turn MathJax on Earlier, we investigated the martingale efficiency of seven Asian FX markets by using nonparametric tests of the historical increments of high-frequency, minute-by-minute data (Bollerslev & Wright, 2000, Los, 1999 and Los, 2000a):2 equation(5) View the MathML sourcex˜τ(t)=x(t)−xˆ(t−τ)=x(t)−xˆτ(t) Turn MathJax on From that research, we concluded that: (1) All nine investigated currencies [Hong Kong dollar (HKD), Indonesian rupiah (IDR), Japanese Yen (JPY), Malaysian ringgit (MYR), Philippines peso (PHP), Singapore dollar (SGD), Thai baht (THB), Taiwan dollar (TWD), and the benchmark currency of the Deutschemark] exhibit wide-sense stationarity in their increments, before the currency turmoil started on July 2, 1997, the day after Hong Kong was handed over to the People's Republic of China. When all 1997 FX increments are tested together, only five (HKD, JPY, PHP, TWD, and Deutschemark) show wide-sense stationarity. This suggests that FX models must allow for both stationarity and nonstationarity. In particular, they must allow for sharp discontinuities. (2) All nine currencies exhibit significant time dependencies of various lengths τ in their increments, in both the whole 1997 data set and in each of the half year data sets, from before and after the midyear currency break. This suggests that FX models must allow for both serial and long-term time dependence. (3) Significant trading windows of up to 20 min are identifiable in the FX increments throughout 1997, and more complex persistence behavior than the affine Markov (ARIMA) processes are identifiable. This suggests that FX models must allow for nonlinearity.3 This paper examines long-term dependence (long memory) observed in FX markets, that is not identifiable by serial, linear dependent (correlation) models of the affine Markov type. Analysts, who use serial correlation models, often observe that serial FX price residuals are uncorrelated (=linearly independent) and then, erroneously, conclude that these markets are efficient because they ignore the more difficult to detect nonlinear long-term dependence. Already in the mid-1960s, Mandelbrot observed that market pricing processes are nonstationary and possibly nonlinear, and that martingale models are inadequate to describe them (Mandelbrot, 1966, Mandelbrot, 1971 and Mandelbrot, 2002). His warnings were ignored: In the early 1970s, the affine GBM models became popular because of the usefulness of the Black–Scholes risk-neutral valuation method, in particular, for the valuation of derivatives. Now, recently, considerable evidence has been accumulated that empirical market pricing processes do not conform to such affine GBMs because they exhibit empirical scaling laws with scale exponents that do not conform to the Fickian Hurst exponent of H=0.5 of the GBM. Fortunately, the valuation of derivatives can be performed with non-Fickian exponents, H≠0.5 Several authors have found that stock market price increments, or rates of return, exhibit empirical Hurst exponents in the range of H≈0.6–0.7 (Cf. Peters, 1994, for an extensive bibliography). Thus, stock market rates of return on such risky assets are now known to be persistent. In contrast, this paper finds that most FX rates exhibit empirical Hurst exponents in the range of 0.3–0.5. In particular, the anchor currencies of the Deutschemark (now replaced by the Euro) and the JPY exhibit Hurst exponents of about H≈0.2–0.3. In other words, most FX rates are found to be antipersistent. From what we have been able to determine from the financial literature, ours are the first empirical measurements of antipersistent Hurst exponents of FX rates (Elliott & van der Hoek, 2003).4

The fractal or self-similar nature of the FX increments, visualized in the scalograms by an even distribution of the pricing energy over the various scales and frequencies, is measured by the Hurst exponent. The empirical findings of this paper support the FMH of Mandelbrot, 1966 and Mandelbrot, 1971 and Peters (1994), which describes the FX market pricing processes more accurately (but still not completely) than does the EMH of Fama, 1970 and Fama, 1991 based on martingale theory. This paper uses various time-series processing techniques on high-frequency, intraday, minute-by-minute FX rates of the German Deutschemark and eight Asian currencies to characterize the FX market pricing. Spectral analysis shows that the spectral power of the FX increments resides mostly in the smallest frequencies, that is, in the fast trading with small price steps, which is not the same as “noise” trading but more like “scalping”. The spectrograms of the increments offer a clear visualization of the persistent differences between the various markets. Most Asian FX rates—the MYR, the PHP, the THB, and the TWD—display changes in frequency (i.e., nonstationarity) in July 1997, when the Asian Financial Crisis began. The Deutschemark, JPY, and SGD are stationary, proving that these currencies were not greatly affected by the turbulence in the THB, but continued to trade as before. The spectrograms of the FX increments verify that they are not white noise or Wiener processes, a conclusion that is corroborated by the wavelet-based scalograms. Hence, the FX rates of the nine currencies do not follow GBMs. The scalograms, which provide both scale and time information, reveal self-similarity of the FX increments at various scales. A closer look at the scalograms of the last week in June and the first week in July suggests that the fractal nature of FX pricing is more induced by the timing of trading activity than by the actual valuation processes. This particular aspect of the microstructure of FX markets can reveal more about the FX-data-generating process. Hence, further research is required in that direction, but needs to use unregularized tick-by-tick FX data instead of our regularized FX rates to avoid aliasing of the results. This paper uses wavelet multiresolution to compute homogeneous Hurst exponents. The Kaplan and Kuo (1993) method, which is a modification of the Wornell and Oppenheim (1992) method, is applied to the FX increments. The graphic fractal dimension D of all the FX series, with the exception of the new TWD, lies between 1.5 and 2, indicating that the FX increments are antipersistent. The new TWD is persistent and thus exceptionally prone to sharp and completely unpredictable discontinuities, induced by administrative control of the currency. The Hurst exponent values of the Deutschemark and the JPY reveal strong antipersistence in the H=0.2–0.3 range. This should warn speculators against taking long positions in these anchor currencies. Dynamic valuation models, such as the Black-Scholes equation, which is on the GBM, is likely to result in inaccurate pricing of financial instruments in these markets. However, most Asian FX markets, except the JPY, are less antipersistent and their Hurst exponents values are closer to 0.5, suggesting that the GBM could provide an accurate law of price motion. Unfortunately, the FX increments show a much wider dispersion and, thus, more uncertainty about the actual value of their Hurst exponents and, thus, about their degrees of persistence. That makes valuation in these very unpredictable markets extraordinarily hazardous. A move towards a currency “snake” and, ultimately, to a currency union could be very desirable for the ASEAN countries, although bandwith limits may cause “intrasnake” currency turbulence. Varying Hurst exponent values across scales and months are an indication of multifractality, that is, the occurrence of different Lipschitz irregularity at different scales. This added complexity certainly poses a problem in modeling the FX pricing processes. We suspect that it is necessary to gain a better understanding of the nonsynchronous timing of FX trading activity to improve the risk-neutral valuation modeling in these markets. According to the FMH, the existence of investors with different investment horizons τ ensures the continuity of the FX markets. Any instability or discontinuity will be absorbed once the investors assess the value of the information and its impact on their investment horizon and act accordingly in feedback fashion—hence, the observed antipersistence—to bring about stability. The driving force behind the EMH is that there are many investors with similar objectives and risks. It assumes that all investors are rational and everyone acts on the same information set and time horizon τ. Thus, the EMH is neither able to explain antipersistence or mean-reverting, nor instabilities or discontinuities. The EMH ignores the scarcity of trading liquidity and the impact of capital controls, which could actually lead to investors transacting at a price that is different from their assessed fair value. The EMH is limited, especially so for the FX markets, as the movements in the FX rates are not directly tied to economic activity. The FX markets are not used to raise capital, unlike other security markets. FX markets are trading market dominated by arbitragers with different horizons τ, actively transacting to take advantage of price discrepancies. For these peculiar reasons, in our opinion, the FMH offers a better explanation of the laws of motion of FX rates than the EMH does. FX markets exhibit nonlinear dynamic structures, high degrees of small amplitude, and fast, noninformational trading and nonperiodic cyclicities. This behavior is possibly induced by frequent international capital flows induced by nonsynchronized, but cyclically occurring, national business cycles, rapidly changing political regimes, and country risk perceptions, unexpected informational shocks to investment opportunities, and, in particular, investment strategies to synthesize and diversify risk claims, using cash swaps between the various national asset markets. Because information trading occurs in lower frequencies, in about the two hour periods according to Ramsey and Zhang (1997), this is, in principle, encompassed by our largest 4.3-h resolution scale. Still, high-frequency data might not reveal much about this kind of information trading. This shortcoming can easily be overcome by increasing the levels of resolution of the scalograms (but one needs very high resolution monitors, and subsampling windows, to continue to discern any details). By making the FX data regular, their distribution is slightly altered or aliased and, therefore, may not be a complete representation of the actual FX processes. Although regular intervals facilitate the wavelet multiresolution, irregular intervals can be used by dictionaries of nonuniform wavelets. Therefore, future research should steer in the directional using the actual irregular ask and bid data to model the actual data-generating processes. The use of midprices may not reflect the different structure in the demand and supply side of each market. For example, there are noticeable differences between the scalograms of the bid and of the ask FX rates.