ASAC 2005 Toronto, Ontario

Rossitsa Yalamova University of Lethbridge FIXED INCOME MARKET EFFICIENCY

Time series analysis of price fluctuations in the Bank of Canada fixed income securities is presented. The multifractal spectra of bond prices were calculated using the most advanced technique in fractal estimation, the wavelet transform modulus maxima. Correlation structures of bond prices are estimated within each asset. The empirical success of the fractional Brownian motion in multifractal trading time model offers new challenges in risk modeling and derivative pricing. Introduction

Statistical characterization of fixed income market efficiency is important not only in terms of volume liquidity, but also in terms of the pricing of fixed income derivatives. Bernaschi et al (2004) found non-perfect correlation among US T-bonds which implies that empirical data does not fit exactly into the single factor models for the term structure of interest rates. Therefore, research of the price fluctuation of a single bond may lead to better insight into the efficiency of fixed income markets. The research described in the current paper offers wavelet methodology and an empirical test of a bond price volatility model based on a continuous time stochastic process that incorporates outliers and volatility anti-persistence. The Gaussian paradigm of independent normally distributed asset returns has been challenged and numerous attempts have been made to improve it. The behavior of financial markets has been increasingly quantified by the means of statistical physics - scaling, multifractality, multiplicative cascade, etc. Mandelbrot (1997) proposed to use Levy distributions with fat tails decaying as a power law. The fundamental properties of volatility dynamics are volatility clustering (conditional heteroscedasticity) and long memory (slowly decaying autocorrelation). Both properties might be labeled as horizontal dependency when viewing volatility in the time domain. Main stream financial economists prefer ARCH type models that capture only imperfectly the volatility correlation and the fat tails of the probability density function of price variations. These models are also inaccurate in terms of changes of time scales. The asymmetric dependence of volatility across various time scales was described in Gençay et al. (2004). Their findings indicate that a low volatility at a low frequency implies a low volatility at higher frequency, but not vice versa. The conjecture is that volatility decreases at higher frequency earlier than it does at lower frequency. Bernaschi et al (2004) present a time series analysis of price fluctuation in the US Treasury fixed income market and show that price increments do not fulfill the random walk hypothesis. Arneodo et al. (1998) propose a cascade model that accounts for the distribution of

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the volatility of returns across scales but not for the precise fluctuation of returns. The behavior of the autocorrelation function depends on the cascade variables and some scale independent random variable. If that variable is white noise there will be no correlation between the returns while their absolute values (or the associated volatilities) are strongly correlated. The classical assumption of independent returns implies that the degree of fatness of the tails decreases as the holding horizon extends. But Johansen and Sornette (2000) find that asset returns exhibit strong correlations exactly at the time of extreme events. Therefore, using a fixed time scale is not suitable for an analysis of the real dynamics of price moves. Wavelet analysis captures dependencies in the two dimensional time-frequency plane. Wavelet Analysis

Muzy et al. (2001) and Breymann et al. (2000) show that the return volatility displays long-term correlations from large to small time scales. Since fixed time scales are not adequate for capturing the perception of risk and return, a better insight into the dynamics of financial markets can be achieved with a time-adaptive framework that simultaneously takes all time-scales of the statistical distributions into account. Accurately modeling financial price variations is an essential step in defining risk management techniques, portfolio optimization, derivative pricing, fund management and trading. Volatility computation based on Fourier series analysis was proposed by Malliavin and Mamcino (2002). This method is well suited for financial market application as it is fully model free and nonparametric. Since the method is based on the integration of the series rather than the differentiation, it is more robust and well suited to compute volatility for a high frequency time series. Høg and Lunde (2003) propose an estimator of integrated volatility using wavelet methods. The concept of integrated volatility estimated by intra-day quadratic variation, not the squared inter-day returns, has proven to have numerous applications, especially in derivatives. In the sense of the stochastic volatility model, integrated volatility/variance can be normalized to [0,1] over a particular time span

Wavelets are localized both in time and frequency/scale and well suited to approximate discontinuities and sharp spikes. They are derived from a single function (mother wavelet) by dilation and translation

is the scale of the wavelet and is the position (translation). Wavelet coefficients decompose the information from the original time series into pieces associated with

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both time and scale. The wavelet coefficients can be viewed as differences between weighted averages where the weights are determined by a given wavelet filter. Wavelet coefficients are appropriate for the estimation of the total variation of the data at various time scales. The wavelet transform is a well adapted tool for studying scaling processes as the wavelet coefficients at a fixed scale are stationary with short range correlation. The greater the number of the wavelet vanishing moments, the shorter the correlations, e. g. with a large enough number of vanishing moments, the wavelet coefficients can be considered as decorrelated. Therefore, the qorder moment of the wavelet coefficients at scale a reproduces the scaling property of the process and can be estimated by just averaging these coefficients at scale a of a given time series. Data

The data for this research was collected from the Bank of Canada website and consists of 2665 daily observations from March 1, 1994 to August 12, 2004. Selected Government of Canada benchmark bond yields are based on actual mid-market closing yields of selected Canada bond issues that mature approximately in the indicated term areas - 2, 3, 5, 7 and 10 years, Long Term bonds and Real Return on Long Term bonds. Data on Government of Canada Marketable Bond Average Yields of 1-3, 3-5, 5-10 and over 10 years are also analyzed. Measurement Methodology

Recent empirical studies suggest that market price changes exhibit certain properties: price increments are not correlated, but volatilities are power-law-dependent, and the shape of the probability density function (pdf) of the price increments depends on the time scale (from Gaussian at large scales to fat tails at fine scales), as shown in, for example, Bouchaud and Potters (2003). A random process x(t) is said to be self-similar with Hölder exponent H if for any scale a>0 it obeys the scaling relation

A self-similar process is monofractal when its singularity spectrum is reduced to a single point and the Hölder exponent at any point is the same H. Its increments as display long range dependence and their autocovariance function decays as the horizon . Then H is called the Hurst exponent of the series. The Hurst exponent provides us with a means to analyze the dependence characteristics of a financial time series and to determine if they are persistently, neutrally or anti-persistently dependent. When the Hurst exponent is the time series is called anti-persistent. Fractional Brownian Motion (FBM) increments with anti-persistence diffuse more quickly than the neutral Geometric Brownian (GBM) increments with H=0.5. Such an anti-persistent FBM returns continuously to the initial point. The geometric Brownian motion with H=0.5 has independent increments and its autocovariance is a constant γ(τ)=σ², no matter what the time

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horizon. When 0.5