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Abstract:
This paper presents the "multifractal model of asset returns" ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second, the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multiscaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.

Abstract:
The Multifractal Model of Asset Returns (See Mandelbrot, Fisher, and Calvet, 1997 ) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the stochastic process.

Abstract:
This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997) (See Mandelbrot, Fisher, and Calvet, 1997 at the following URL: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=78588 ), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework or time-varying volatility in a continuous time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark / US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark / US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.

Abstract:
We present a rational theory of return behavior around seasoned equity offerings, including a pre-issuance price runup, negative announcement effect, and long-run post-issuance underperformance. The main result uses real option principles to relate SEO's to an endogenous decrease in expected returns. Equity issues are associated with firm expansions. When firms invest, they convert growth options to assets in place. Even when the new assets are risky, they will be less risky than the options they replace. Although both size and book-to-market effects are present in our model, standard matching procedures fail to capture the dynamics of risk and expected return. We calibrate the model, and show that it gives a close match to the primary empirical moments.

Event studies, SEOs, Long-Run Performance, dynamic corporate decisions

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favor-ably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly inter-vals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, long memory, Markov regime-switching, maximum likelihood estimation, scaling, stochastic volatility, time deformation, volatility component, Vuong test

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, long memory, Markov regime-switching, maximum likelihood estimation, scaling, stochastic volatility, time deformation, volatility component, Vuong test

Abstract:
We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Abstract:
We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Forecasting, Long Memory, Markov Regime-switching, Maximum Likelihood Estimation, Scaling, Stochastic Volatility, Time Deformation, Volatility Component, Vuong Test

Abstract:
We show that corporate investment decisions can explain conditional dynamics in expected asset returns. Our approach is similar in spirit to Berk, Green, and Naik (1999), but we introduce to the investment problem operating leverage, reversible real options, fixed adjustment costs, and finite growth opportunities. We assume constant revenue betas, but still obtain asset betas that vary through time as a reflection of historical investment decisions and product market demand. Book-to-market effects emerge and relate to operating leverage, while size captures the importance of growth options relative to assets in place. We first develop these results in a simple setting that permits closed-form solutions. Next, we empirically evaluate a more realistic specification that is solved numerically and estimated using simulated method of moments. This provides new quantitative evidence on the importance of
operating leverage and growth options to the cross-section of returns.

Cross-Section of Returns, Size Effect, Book-to-Market Effect, Corporate Investment, Real Options, Simulated Method of Moments

Abstract:
This paper investigates the dynamics of firm level beta and volatility around seasoned equity offerings. Beta increases prior to the SEO and decreases thereafter. This pattern is generally consistent with a real options explanation of SEO underperformance, but existing models predict a sharp risk drop, while we find a gradual decline. To reconcile this difference, we extend the theory to consider investment commitment and internal financing. In the cross-section, we show that firms with high prior return runups experience larger post-issuance underperformance, as well as more substantial post-issuance declines in beta. By contrast, large market-wide runups, which might be taken as a measure of sentiment, do not precede either post-issuance underperformance or post-issuance beta declines. Finally, equity issues coincide with low points in both own firm and market-wide volatility, suggesting the possibility of volatility timing in corporate financing activities.

Seasoned Equity Offering, Real Options, Dynamic Risk, Dynamic Beta

Abstract:
This paper investigates the dynamics of firm level beta and volatility around seasoned equity offerings. Beta increases prior to the SEO and decreases thereafter. This pattern is generally consistent with a real options explanation of SEO underperformance, but existing models predict a sharp risk drop, while we find a gradual decline. To reconcile this difference, we extend the theory to consider investment commitment and internal financing. In the cross-section, we show that firms with high prior return runups experience larger post-issuance underperformance, as well as more substantial post-issuance declines in beta. By contrast, large market-wide runups, which might be taken as a measure of sentiment, do not precede either postissuance underperformance or post-issuance beta declines. Finally, equity issues coincide with low points in both own firm and market-wide volatility, suggesting the possibility of volatility timing in corporate financing activities.

Seasoned Equity Offering, Real Options, Dynamic Risk, Dynamic Beta, Investment Commitment, Time-to-Build, Long Run Event Studies, Abnormal Return

Abstract:
We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (2001, 2004). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by ML for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. We extend the model to multivariate settings with a potentially large number of assets by proposing a parsimonious multifrequency factor structure.

Multivariate MSM, comovement, maximum likelihood, particle filter, Markov-switching, stochastic volatility, multifrequency volatility decomposition, value at risk, quantile forecasts

Abstract:
We examine a model in which two firms strategically compete in a duopolistic product market. Firms produce a homogenous product and face stochastic industry demand. Each firm has a single option either to expand or contract capacity, and hence output. In this setup we analyze the risk characteristics of industries as well as single firms and look at corresponding asset price dynamics. We focus on sequential exercise of options. We find that strategic competition in the product market is risk reducing. Irrespective of expansion or contraction the presence of strategically interacting rivals causes firm's risk to decline. This is the consequence of a simple hedging argument. Moreover, we find that own firm and industry characteristics have opposite risk implications in case of expansion and contraction. Empirical evidence, however, is that a
strong negative relationship exists between firm and rival risk measures.

Growth options and industry risk, risk dynamics in oligopolistic industries

Abstract:
Recent research documents that aggregate stock prices are driven by shocks with persistence levels ranging from daily intervals to several decades. Building on these insights, we introduce a parsimonious equilibrium model in which regime-shifts of heterogeneous durations affect the volatility of dividend news. We estimate tightly parameterized specifications with up to 256 discrete states on daily U.S. equity returns. The multifrequency equilibrium has significantly higher likelihood than the classic Campbell and Hentschel (1992) specification, while generating volatility feedback effects 6 to 12 times larger. We show in an extension that Bayesian learning about stochastic volatility is faster for bad states than good states, providing a novel source of endogenous skewness that complements the "uncertainty" channel considered in previous literature (e.g., Veronesi, 1999). Furthermore, signal precision induces a tradeoff between skewness and kurtosis, and economies with intermediate investor information best match the data.

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Abstract:
Recent research documents that aggregate stock prices are driven by shocks with persistence levels ranging from daily intervals to several decades. Building on these insights, we introduce a parsimonious equilibrium model in which regime-shifts of heterogeneous durations affect the volatility of dividend news. We estimate tightly parameterized specifications with up to 256 discrete states on daily U.S. equity returns. The multifrequency equilibrium has significantly higher likelihood than the classic Campbell and Hentschel (1992) specification, while generating volatility feedback effects 6 to 12 times larger. We show in an extension that Bayesian learning about stochastic volatility is faster for bad states than good states, providing a novel source of endogenous skewness that complements the uncertainty channel considered in previous literature (e.g., Veronesi, 1999). Furthermore, signal precision induces a tradeoff between skewness and kurtosis, and economies with intermediate investor information best match the data.

Multifrequency news, volatility feedback, learning, information quality, endogenous skewness and kurtosis, Epstein-Zin utility

Abstract:
We investigate truth-telling by an informed insider, or manager, who repeatedly forecasts cash flows to competitive investors in a standard message game. The insider cannot trade on or sell private information, but faces imperfectly hedgeable nonwage income shocks. When compensation depends on the current stock price, a partially revealing equilibrium may exist in which the manager manipulates his reports, and hence the stock price, to reduce consumption variance. Intuitively, the manager builds reputation in good times when honesty is affordable, and exploits reputation in times of need. Endogenous reputation for honesty thus follows from a self-insurance motive.

Abstract:
We investigate truth-telling by an informed insider, or manager, who repeatedly forecasts cash flows to competitive investors in a standard message game. The insider may not trade on or sell private information, but receives wages that can depend on the current stock price or forecast accuracy. The manager faces imperfectly hedgeable shocks to non-wage income. When compensation is value-based, this induces a partially revealing equilibrium in which the insider manipulates the stock price to minimize consumption variance. Intuitively, the manager builds reputation when times are good and honesty is affordable, and exploits reputation in times of need. The paper thus formalizes a self-insurance motivation for endogenous reputation for honesty.

Reputation, Information, Truth-telling, Self-insurance, Managerial Compensation

Abstract:
Unconditional alpha estimates are biased when conditional beta covaries with market risk premia ("market-timing'') or volatility ("volatility-timing''). We demonstrate an additional bias ("overconditioning'') that can occur any time an empiricist uses a risk proxy not in the investor information set -- for example when asset payoffs are nonlinear and the conditional loading is proxied by contemporaneous realized beta. Calibrating to U.S. equity returns, volatility-timing and overconditioning plausibly impact alphas much more than market-timing, which has been the focus of prior literature. A variety of instrumental variables estimators using realized betas can substantially correct market- and volatility-timing biases, while eliminating overconditioning. Empirically, appropriate instrumentation reduces momentum alphas by 20-40% relative to unconditional, whereas overconditioned alphas overstate performance by up to 2.5 times. Volatility-timing inflates unconditionally estimated momentum alpha because the formation-period market return (i) positively predicts holding-period beta (Grundy and Martin, 2001) and (ii) negatively predicts holding-period market volatility (French, Schwert, and Stambaugh, 1987), inducing negative covariation between conditional beta and market volatility.

Overconditioning, Conditional CAPM, Performance Evaluation, Momentum

Abstract:
This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock`s persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Abstract:
This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Endogenous jumps, general equilibrium, Markov regime-switching, multifrequency, fat tails, stochastic volatility, time deformation, volatility component

Abstract:
This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. Out model captures the thick tails and volatility persistence exhibited by many financial time series. We assume that the forecaster knows the true generating process with certainty, but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We show that a discretized version of the model has a finite state space, which allows an analytical solution to the conditioning problem. Further, the discrete model converges to the continuous-time model as time scale goes to zero, so that forecasts are consistent. The methodology is implemented on simulated data calibrated to the Deutschemark/US Dollar exchange rate. Applying these results to option pricing, we find that the model captures both volatility smiles and long-memory in the term structure of implied volatilities.

Forecasting, Implied Volatility, Long Memory, Multifractal Model of Asset Returns, Option Pricing, Poisson Multifractal, Trading Time, Volatility Smile

Abstract:
We study own and rival risk in a dynamic duopoly with a homogeneous output good, stochastic industry demand, real options to expand or contract capacity, and potentially different adjustment costs across firms. In general, a competitor's options to adjust capacity reduce own-firm risk through a simple hedging channel. When a rival possesses a growth option, an increase in industry demand directly enhances current profits and value, but also increases the prospect of value-reducing competitor expansion. As the rival moves closer to its expansion or contraction boundaries, such hedging effects become more important and generally differ from the own-firm effects of real options. As a consequence, when a leader and a follower emerge in equilibrium, risk dynamics depart substantially from a simultaneous move benchmark. In leader-follower equilibria own-firm and competitor required returns tend to move together through contractions and oppositely during expansions. Thus, the common practice of using industry peer betas to proxy for own-firm risk should work well in certain environments, but not in others, providing testable new empirical predictions.

Growth options and industry risk, asset pricing and investmentdecisions, risk dynamics in oligopolistic industries

Abstract:
This paper investigates the Multifractal Model of Asset Returns, a continuous-time process that incorporates the thick tails and volatility persistence exhibited by many financial time series. The model is constructed by compounding a Brownian Motion with a multifractal time-deformation process. Return moments scale as a power law of the time horizon, a property confirmed for Deutschemark / U.S. Dollar exchange rates and several equity series. The model implies semi-martingale prices and thus precludes arbitrage in a standard two-asset economy. Volatility has long-memory, and the highest finite moment of returns can have any value greater than two. The local variability of the process is characterized by a renormalized probability density of local Hölder exponents. Unlike standard models, multifractal paths contain a multiplicity of these exponents within any time interval. We develop an estimation method, and infer a parsimonious generating mechanism for the exchange rate series. Simulated samples replicate the moment-scaling found in the data.

Multifractal Model of Asset Returns, Compound Stochastic Process, Time Deformation, Scaling, Self-Similarity, Multifractal Spectrum, Stochastic Volatility

Abstract:
This paper reassesses the role of economic fundamentals in the 1987 stock market crash using a two factor common-component model of returns. The model decomposes returns into idiosyncratic components, a common white noise component, and a common source of Poisson jumps. Among three two-year sample periods for Major Market Index stocks, only a 1987-88 sample results in an estimated jump component with low frequency and large size. Using Bayes' rule, we infer ex post jump probabilities for each sample day. In contrast to an analogous univariate model for an index return, the multivariate model captures information in the cross-section of returns. Leading financial news on the most likely jump days from the multivariate model is compared with news on a control group of high index return days. Days with high jump probabilities under the multivariate model contain systematically more news related to the dollar, trade deficits, and financing of the U. S. budget deficit. This suggest that the common jump component proxies for economic fundaments related to this cluster of news events, and that the unexpectedly large U.S. trade deficit news released on the Wednesday prior to the crash provided an economic catalyst for the event.

Abstract:
This paper reassesses the role of economic fundamentals in the 1987 stock market crash using a two factor common-component model of returns. The model decomposes returns into idiosyncratic components, a common white noise component, and a common source of Poisson jumps. Among three two-year sample periods for Major Market Index stocks, only a 1987-88 sample results in an estimated jump component with low frequency and large size. Using Bayes' rule, we infer ex post jump probabilities for each sample day. In contrast to an analogous univariate model for an index return, the multivariate model captures information in the cross-section of returns. Leading financial news on the most likely jump days from the multivariate model is compared with news on a control group of high index return days. Days with high jump probabilities under the multivariate model contain systematically more news related to the dollar, trade deficits, and financing of the U. S. budget deficit. This suggest that the common jump component proxies for economic fundaments related to this cluster of news events, and that the unexpectedly large U.S. trade deficit news released on the Wednesday prior to the crash provided an economic catalyst for the event.

Abstract:
We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low-frequency variations. Second, they specify intermediate-frequency dynamics usually assigned to smooth autoregressive transitions. Finally, high-frequency switches generate substantial outliers. Thus a single mechanism captures three features that are typically viewed as distinct in the literature. Maximum-likelihood estimation is developed and performs well in finite samples. Using exchange rates, we estimate a version of the process with four parameters and more than a thousand states. The multifractal outperforms GARCH, MS-GARCH, and FIGARCH in- and out-of-sample. Considerable gains in forecasting accuracy are obtained at horizons of 10 to 50 days.

forecasting, long memory, Markov-switching multifractal (MSM), closed-form likelihood, scaling, stochastic volatility, volatility component, Vuong test

Abstract:
We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (2001, 2004). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by ML for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. We extend the model to multivariate settings with a potentially large number of assets by proposing a parsimonious multifrequency factor structure.