Ulrich A. Müller

A geographical model for the daily and weekly seasonal volatility in the foreign exchange market

Abstract The daily and weekly seasonality of foreign exchange volatility is modelled by introducing an activity variable. This activity is explained by a simple model of the changing and sometimes overlapping market presence of geographical components (East Asia, Europe, and America). Integrating this activity over time results in the new time scale, characterized by non-seasonal volatility. This scale, applied to dense datastreams of absolute price changes, suceeds in removing most of the seasonal heteroscedasticity in an autocorrelation study. Unexpectedly, the positive autocorrelation is found to decline hyperbolically rather than exponentially as a function of the lag.

Volatilities of different time resolutions — Analyzing the dynamics of market components

The heterogeneous market states that the diversity of actors causes different behaviors of volatilities of different time resolutions. A lagged correlation study reveals that statistical volatility defined over a coarse time grid significantly predicts volatility defined over a fine grid. This empirical fact is not explained by conventional theories and models.

Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis☆

Abstract In this paper we present a statistical analysis of four foreign exchange spot rates against the U.S. Dollar with several million intra-day prices over 3 years. The analysis also includes gold prices and samples of daily foreign exchange prices over 15 years. The mean absolute changes of logarithmic prices are found to follow a scaling law against the time interval on which they are measured. This empirical law holds although the distributions of the price changes strongly differ for different interval sizes. Systematic variations of the volatility are found even during business hours by an intra-day analysis of price changes. Seasonal heteroskedasticity is observed with a period of one day as well as one week as the result of an analogous intra-week analysis; taking this into account is necessary for any future study of intra-day price change distributions and their generating process. The same type of analysis is also made for the bid-ask spreads.

From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets

Abstract.This paper presents stylized facts concerning the spot intra-daily foreign exchange markets. It first describes intra-daily data and proposes a set of definitions for the variables of interest. Empirical regularities of the foreign exchange intra-daily data are then grouped under three major topics: the distribution of price changes, the process of price formation and the heterogeneous structure of the market. The stylized facts surveyed in this paper shed new light on the market structure that appears composed of heterogeneous agents. It also poses several challenges such as the definition of price and of the time-scale, the concepts of risk and efficiency, the modeling of the markets and the learning process.

Modelling Short-Term Volatility with GARCH and HARCH Models

In this paper we present both a new formulation of the HARCH process and a study of the forecasting accuracy of ARCH-type models for predicting short-term volatility. Using high frequency data, the market volatility is expressed in terms of partial volatilities which are formally exponential moving averages of squared returns measured at different frequencies. This new formulation is shown to produce more accurate fits to the data and, at the same time, to be easier to compute than the earlier version of the HARCH process. This is obtained without losing the nice property of the HARCH process to identify different market components. In a second part, some performance measures of forecasting accuracy are discussed and the ARCH-type models are shown to be good predictors of the short-term hourly historical volatility with the new formulation of the HARCH process being the best predictor.

Extremal forex returns in extremely large data sets

Exciting information for risk and investment analysis is obtained from an exceptionally large and automatically filtered high frequency data set containing all the forex quote prices on Reuters during a ten-year period. It is shown how the high frequency data improve the efficiency of the tail risk cum loss estimates. We demonstrate theoretically and empirically that the heavy tail feature of foreign exchange rate returns implies that position limits for traders calculated under the industry standard normal model are either not prudent enough, or are overly conservative depending on the time horizon.

Effective Return, Risk Aversion and Drawdowns

We derive two risk adjusted performance measures for investors with risk averse preferences. Maximizing these measures is equivalent to maximizing the expected utility of an investor. The first measure, X(eff), is derivedassuming a constant risk aversion while the second measure, R(eff),is based on a stronger risk aversion to clustering of losses than of gains. The clustering of returns is captured through a multi-horizon framework. The empirical properties of X(eff), R(eff) are studied within the context of real-time trading models for foreign exchange rates and their properties are compared to those of more traditional measures like the annualized return, the Sharpe Ratio and the maximum drawdown. Our measures are shown to be more robust against clustering of losses and has the ability to fully characterize the dynamic behavior of investment strategies.

Changing time scale for short‐term forecasting in financial markets

A forecasting model based on high-frequency market makers quotes of financial instruments is presented. The statistical behaviour of these time series leads to discussion of the appropriate time scale for forecasting. We introduce variable time scales in a general way and define the new concept of intrinsic time. The latter reflects better the actual trading activity. Changing time scale means forecasting in two steps, first an intrinsic time forecast against physical time, then a price forecast against intrinsic time. The forecasting model consists, for both steps, of a linear combination of non-linear price-based indicators. The indicator weights are continuously re-optimized through a modified linear regression on a moving sample of past prices. The out-of-sample performance of this algorithm is reported on a set of important FX rates and interest rates over many years. It is remarkably consistent. Results for short horizons as well as techniques to measure this performance are discussed.

The Intraday Multivariate Structure of the Eurofutures Markets

We investigate the multivariate intraday structure in interest rates, focusing on implied forward rates from Eurofutures contracts. Since futures markets are the most liquid for interest rate instruments and they yield high-quality intraday data, it is somehow surprising that their intraday behavior has not been thoroughly studied in the literature. We find interesting similarities with the foreign exchange market in terms of scaling law, intraday patterns, all of which point to the heterogeneity of market participants. Other properties like asymmetric causal information flow between fine and coarse volatilities for the same time series are present in our data. There are also lead/lag correlation across maturities and currencies, but they that tend to disappear as markets mature. A principal component analysis of the short end of the yield curve allows us to determine the most important components and to reduce the number of time series needed to describe the term structure. We find the decomposition rather stable over time. The first component which describes the curve level presents a HARCH effect while the remaining ones do not, having instead significant negative autocorrelations for the time series themselves. A HARCH model is applied to the first component and the impact of different market agents is discussed.

From default probabilities to credit spreads: Credit risk models do explain market prices

Abstract Credit risk models like Moodys KMV are now well established in the market and give bond managers reliable estimates of default probabilities for individual firms. Until now it has been hard to relate those probabilities to the actual credit spreads observed on the market for corporate bonds. Inspired by the existence of scaling laws in financial markets by Dacorogna et al. [2001. An Introduction to High Frequency Finance. Academic Press, San Diego, CA] and Di Matteo et al. [2005. Journal of Banking and Finance 29, 827–851] deviating from the Gaussian behavior, we develop a model that quantitatively links those default probabilities to credit spreads (market prices). The main input quantities to this study are merely industry yield data of different times to maturity and expected default frequencies (EDFs) of Moodys KMV. The empirical results of this paper clearly indicate that the model can be used to calculate approximate credit spreads (market prices) from EDFs, independent of the time to maturity and the industry sector under consideration. Moreover, the model is effective in an out-of-sample setting, it produces consistent results on the European bond market where data are scarce and can be adequately used to approximate credit spreads on the corporate level.