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We study the liquidity of the global currency market by analyzing the price impact of trading volume. We analyze a decade of CLS intraday data representative of global foreign exchange (FX) trading by developing a refinement of the popular Amihud (2002) illiquidity measure that we call realized Amihud, which is the ratio between realized volatility and trading volume. Inversely related to market depth, price impact increases with transaction costs, money market stress, uncertainty, and risk aversion. Furthermore, we analyze whether and how liquidity begets price efficiency by looking at violations of the “triangular” no-arbitrage condition. We find that dollar-based currencies offer a lower trading impact supporting price efficiency.

We study the theoretical and empirical properties of a simple measure of market illiquidity, namely the realized Amihud, which is defined as the ratio between the realized volatility and trading volume and which refines the popular price impact measure proposed by Amihud (2002). In our model, both price volatility and market liquidity are assumed to follow stochastic processes in continuous time. In this setting, characterized by stochastic volatility and liquidity, we prove that the realized Amihud provides a precise measurement of the inverse of integrated liquidity over fixed-length periods (e.g., a day, a week, a month). We consider a number of alternative econometric specifications, hence highlighting the main dynamic and distributional properties of the realized Amihud, including jumps, clustering, and leverage effects.

We study the theoretical properties of the model for fractional cointegration proposed by Granger (1986), namely the FVECM_{d,b}. First, we show that the stability of any discretetime stochastic system of the type Pi(L)Y_t = e_t can be assessed by means of the argument principle under mild regularity condition on Pi(L), where L is the lag operator. Second, we prove that, under stability, the FVECM_{d,b} allows for a representation of the solution that demonstrates the fractional and co-fractional properties and we find a closed-form expressionfor the impulse response functions. Third, we prove that the model is identified for any combination of number of lags and cointegration rank, while still being able to generate polynomial co-fractionality. In light of these properties, we show that the asymptotic properties of the maximum likelihood estimator reconcile with those of the FCVAR_{d,b} model studied in Johansen and Nielsen (2012). Finally, an empirical illustration is provided.

We study the global market liquidity of foreign exchange rates by analysing the price impact of trading. We propose a simple theoretical framework that refines the popular Amihud (2002) illiquidity measure and term our new estimator realized Amihud. We highlight our measure’s consistency and higher accuracy, and these properties are validated numerically and empirically using CLS data representative of global FX trading. The price impact of FX trading increases with transaction costs, money market stress, uncertainty, and risk aversion. Furthermore, currencies with smaller price impact are less prone to deviations from triangular no-arbitrage parity. Our results are corroborated by an instrumental variables approach and a quasi-natural experiment using the shock of the Swiss franc’s uncapping in January 2015.

Traders hedge the risks carried by options and other securities using the so-called Greeks, with the delta and the vega being the most prominent. In this paper, we propose a novel non-structural method for hedging European options, relying on two model-independent results: First, under suitable regularity conditions on the risk-neutral density, an option price can be disentangled into a linear combination of risk-neutral moments. Second, there exists an explicit functional form linking the risk-neutral moments to the price of the underlying asset and the related variance swap contracts. We show that, historically, S&P 500 call prices are mainly explained by two factors that are related to level and volatility of the underlying index. Based on this, we devise and empirically compare the hedging performance of two strategies where the vega exposure is adjusted either by taking a direct position in variance swap contracts or, indirectly, through an ATM call option. While both strategies ensure effective immunization in periods of market turmoil, taking direct exposure on volatility might not be optimal during extended periods of subdued market volatility. We argue that this result is related to the phenomenon known as the "low VIX puzzle".

We propose a non-structural pricing method to derive the risk-neutral density (RND) implied by options on the CBOE Volatility Index (VIX). The methodology is based on orthogonal polynomial expansions around a kernel density and yields the RND of the underlying asset without the need for a parametric specification. The classic family of Laguerre expansions is extended to include the GIG and the generalized Weibull kernels, thus relaxing the conditions required on the tail decay rate of the RND to ensure convergence. We show that the proposed methodology yields an accurate approximation of the RND in a large variety of cases, also when the no-arbitrage and efficient option prices are contaminated by measurement errors. Our empirical investigation, based on a panel of traded VIX options, reveals some stylized facts on the RND of VIX. We find that a common stochastic factor drives the dynamic behavior of the risk neutral moments, the probabilities of volatility tail-events are priced in the options as jumps under the risk-neutral measure, and the variance swap term structure depends on two factors, one accounting for the slope and one for the mean-reverting behavior of the VIX.

We explore the interplay between sovereign and bank credit risk in a setting where Danish authorities first let two Danish banks default rather than bail them out and then left the country's largest bank, Danske Bank, to recapitalize privately. We find that the correlation between bank and sovereign credit default swap (CDS) rates changed with these events, indicating that the non-bailout decisions and recapitalization helped to curb the feedback loop between bank and sovereign risk. Following the non-bailout events, the sensitivity to external shocks declined both for Danske Bank and for Danish sovereign debt, as measured by their CDS connection with CDS rates on the European banking sector. After Danske Bank was recapitalized, its exposure to the European banking sector reappeared, while that did not happen for Danish sovereign debt. This decoupling between CDS rates on sovereign and private bank debt indicates that the non-bailout policies succeeded in breaking the vicious circle generated by the risks on the bank and sovereign debts. Our results are reinforced by the use of an indirect testing approach and by focusing on CDS-quantiles.

Short memory models contaminated by level shifts have long-memory features similar to those associated to processes generated under fractional integration. In this paper, we propose a robust testing procedure, based on an encompassing parametric specification, that allows to disentangle the level shift term from the ARFIMA component. The estimation is carried out via a state-space methodology and it leads to a robust estimate of the fractional integration parameter also in presence of level shifts.The Monte Carlo simulations show that this approach produces unbiased estimates of the fractional integration parameter when shifts in the mean, or in other slowly varying trends, are present in the data. Once the fractional integration parameter is estimated, the KPSS test statistic is adopted to assess if the level shift component is statistically significant. The test has correct size and generally the highest power compared to other existing tests for spurious long-memory. Finally, we illustrate the usefulness of the proposed approach on the daily series of bipower variation and share turnover and on the monthly inflation series of G7 countries.

We propose a non-structural pricing method to retrieve the risk-neutral density implied by options contracts on the CBOE VIX. The method is based on orthogonal polynomial expansions around a kernel density and yields the risk-neutral density of the underlying asset without the need for modeling its dynamics. The method imposes only mild regularity conditions on shape of the density. The approach can be thought of as an alternative to Hermite expansions where the kernel has positive support. .e family of Laguerre kernels is extended to include the GIG and the generalized Weibull densities, which, due to their flexible rate of decay, are better suited at modeling the density of the VIX. Based on this technique, we propose a simple and robust way to estimate the expansion coefficients by means of a principal components analysis. We show that the proposed methodology yields an accurate approximation of the risk-neutral density also when the no-arbitrage and efficient option prices are contaminated by measurement errors. A number of numerical illustrations support the adequacy and the flexibility of the proposed expansions in a large variety of cases.

Short memory models contaminated by level shifts have similar long-memory features as fractionally integrated processes. This makes hard to verify whether the true data generating process is a pure fractionally integrated process when employing standard estimation methods based on the autocorrelation function or the periodogram. In this paper, we propose a robust testing procedure, based on an encompassing parametric specification that allows to disentangle the level shifts from the fractionally integrated component. The estimation is carried out on the basis of a state-space methodology and it leads to a robust estimate of the fractional integration parameter also in presence of level shifts. Once the memory parameter is correctly estimated, we use the KPSS test for presence of level shift. The Monte Carlo simulations show how this approach produces unbiased estimates of the memory parameter when shifts in the mean, or other slowly varying trends, are present in the data. Therefore, the subsequent robust version of the KPSS test for the presence of level shifts has proper size and by far the highest power compared to other existing tests. Finally, we illustrate the usefulness of the proposed approach on financial data, such as daily bipower variation and turnover.

Short memory models contaminated by level shifts have similar long-memory features as fractionally integrated processes. This makes it hard to verify whether the true data generating process is a pure fractionally integrated process when employing standard estimation methods based on the autocorrelation function or the periodogram. In this paper, we propose a robust testing procedure, based on an encompassing parametric specification that allows us to disentangle the level shifts from the fractionally integrated component. The estimation is carried out on the basis of a state-space methodology and it leads to a robust estimate of the fractional integration parameter also in presence of level shifts. Once the memory parameter is correctly estimated, we use the KPSS test for presence of level shift. The Monte Carlo simulations show how this approach produces unbiased estimates of the memory parameter when shifts in the mean, or other slowly varying trends, are present in the data. Therefore, this subsequent robust version of the KPSS test for the presence of level shifts has proper size and by far the highest power compared to other existing tests. Finally, we illustrate the usefulness of the proposed approach on financial data, such as daily bipower variation and turnover.

We analyze the properties of the indirect inference estimator when the observed series are contaminated by measurement error. We show that the indirect inference estimates are asymptotically biased when the nuisance parameters of the measurement error distribution are neglected in the indirect estimation. We propose to solve this inconsistency by jointly estimating the nuisance and the structural parameters. Under standard assumptions, this estimator is consistent and asymptotically normal. A condition for the identification of ARMA plus noise is obtained. The proposed methodology is used to estimate the parameters of continuous-time stochastic volatility models with auxiliary specifications based on realized volatility measures. Monte Carlo simulations shows the bias reduction of the indirect estimates obtained when the microstructure noise is explicitly modeled. Finally, an empirical application illustrates the relevance of a realistic specification of the microstructure noise distribution to match the features of the observed log-returns at high frequencies. We analyze the properties of the indirect inference estimator when the observed series are contaminated by measurement error. We show that the indirect inference estimates are asymptotically biased when the nuisance parameters of the measurement error distribution are neglected in the indirect estimation. We propose to solve this inconsistency by jointly estimating the nuisance and the structural parameters. Under standard assumptions, this estimator is consistentand asymptotically normal. A condition for the identification of ARMA plus noise is obtained. The proposed methodology is used to estimate the parameters of continuous-time stochastic volatility models with auxiliary specifications based on realized volatility measures. Monte Carlo simulations shows the bias reduction of the indirect estimates obtained when the microstructure noise is explicitly modeled. Finally, an empirical application illustrates the relevanceof a realistic specification of the microstructure noise distribution to match the features of the observed log-returns at high frequencies.

This paper discusses identification problems in the fractionally cointegrated system of Johansen (2008) and Johansen and Nielsen (2012). It is shown that several equivalent re-parameterizations of the model associated with different fractional integration and cointegration parameters may exist for any choice of the lag length, also when the true cointegration rank is known. The properties of these multiple non-identified models are studied and a necessary and sufficient condition for the identification of the fractional parameters of the system is provided. The condition is named F(d) and it is a generalization to the fractional case of the I(1) condition in the VECM model. The assessment of the F(d) condition in the empirical analysis is relevant for the determination of the fractional parameters as well as the number of lags. The paper also illustrates the indeterminacy between the cointegration rank and lag length. It is proved that, under certain restrictions on the fractional parameters, the model with rank zero and k lags is equivalent to the model with full rank and k − 1 lags. This precludes the possibility to test for the nullity of the cointegration rank. This paper discusses identification problems in the fractionally cointegrated system of Johansen (2008) and Johansen and Nielsen (2012). It is shown that several equivalent re-parameterizations of the model associated with different fractional integration and cointegration parameters may exist for any choice of the lag length, also when the true cointegration rank is known. The properties of these multiple non-identified models are studied and a necessary and sufficient condition for the identification of the fractional parameters of the system is provided. The condition is named F(d) and it is a generalization to the fractional case of the I(1) condition in the VECM model. The assessment of the F(d) condition in the empirical analysis is relevant for the determination of the fractional parameters as well as the number of lags. The paper also illustrates the indeterminacy between the cointegration rank and lag length. It is proved that, under certain restrictions on the fractional parameters, the model with rank zero and k lags is equivalent to the model with full rank and k - 1 lags. This precludes the possibility to test for the nullity of the cointegration rank.

The realized volatility of financial returns is characterized by persistence and occurrence of unpredictable large increments. To capture those features, we introduce the Multiplicative Error Model with jumps (MEM-J). When a jump component is included in the multiplicative specification, the conditional density of the realized measure is shown to be a countably infinite mixture of Gamma and K distributions. Strict stationarity conditions are derived. A Monte Carlo simulation experiment shows that maximum likelihood estimates of the model parameters are reliable even when jumps are rare events. We estimate alternative specifications of the model using a set of daily bipower measures for 7 stock indexes and 16 individual NYSE stocks. The estimates of the jump component confirm that the probability of jumps dramatically increases during the financial crises. Compared to other realized volatility models, the introduction of the jump component provides a sensible improvement in the fit, as well as for in-sample and out-of-sample volatility tail forecasts. The realized volatility of financial returns is characterized by persistence and occurrence of unpreditable large increments. To capture those features, we introduce the Multiplicative Error Model with jumps (MEM-J). When a jump component is included in the multiplicative specification, the conditional density of the realized measure is shown to be a countable infinite mixture of Gamma and K distributions. Strict stationarity conditions are derived. A Monte Carlo simulation experiment shows that maximum likelihood estimates of the model parameters are reliable even when jumps are rare events. We estimate alternative specifications of the model using a set of daily bipower measures for 7 stock indexes and 16 individual NYSE stocks. The estimates of the jump component confirm that the probability of jumps dramatically increases during the financial crisis. Compared to other realized volatility models, the introduction of the jump component provides a sensible improvement in the fit, as well as for in-sample and out-of-sample volatility tail forecasts.

The volatility of financial returns is characterized by rapid and large increments. We propose an extension of the Heterogeneous Autoregressive model to incorporate jumps into the dynamics of the ex-post volatility measures. Using the realized-range measures of 36 NYSE stocks, we show that there is a positive probability of jumps in volatility. A common factor in the volatility jumps is shown to be related to a set of financial covariates (such as variance risk premium, S&P500 volume, credit-default swap, and federal fund rates). The credit-default swap on US banks and variance risk premium have predictive power on expected jump moves, thus confirming the common interpretation that sudden and large increases in equity volatility can be anticipated by credit deterioration of the US bank sector as well as changes in the market expectations of future risks. Finally, the model is extended to incorporate the credit-default swap and the variance risk premium in the dynamics of the jump size and intensity. The volatility of financial returns is characterized by rapid and large increments. We propose an extension of the Heterogenous Autoregressive model to incorporate jumps into the dynamics of the ex-post volatility measures. Using the realized-range measures of 36 BNYSE stocks, we show that there is a positive probability of jumps in volatility. A common factor in the volatility jumps is shown to be related to a set of financial covariates (such as variance risk premium, S&P500 volume, credit-default swap, and federal fund rates). The credit-default swap on US banks and variance risk premium have predictive power on expected jump moves, thus confirming the common interpretation that sudden and large increases in equity volatility can be anticipated by credit deterioration of the US bank sector as well as changes in the market expectations of future risks. Finally, the model is extended to incorporate the credit-default swap and the variance risk premium in the dynamics of the jump size and intensity.