Adlai J. Fisher

Multifractality In Asset Returns: Theory And Evidence

This paper investigates the multifractal model of asset returns (MMAR), a class of continuous-time processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the MMAR compounds a Brownian motion with a multifractal time-deformation. Prices follow a semi-martingale, which precludes arbitrage in a standard two-asset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than 2. The local variability of a sample path is highly heterogeneous and is usefully characterized by the local Hlder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The MMAR predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche mark/U.S. dollar exchange rates and several equity series. We develop an estimation procedure and infer a parsimonious generating mechanism for the exchange rate. In Monte Carlo simulations, the estimated multifractal process replicates the scaling properties of the data and compares favorably with some alternative specifications.

Forecasting Multifractal Volatility

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.

Leaders, Followers, and Risk Dynamics in Industry Equilibrium

We study the distinct impacts of own and rival actions on risk and return when firms strategically compete in the product market. Contrary to simple intuition, a competitor’s options to adjust capacity reduce own-firm risk. For example, if a rival possesses a growth option, an increase in industry demand directly enhances profits but also encourages value-reducing competitor expansion. The rival option thus acts as a natural hedge. Within the industry, we obtain endogenous differences in expected returns. In a leader-follower equilibrium, own-firm and competitor risks and required returns move together through contractions and oppositely during expansions, providing testable new predictions.

Leverage and the Limits of Arbitrage Pricing: Implications for Dividend Strips and the Term Structure of Equity Risk Premia

Negligible pricing frictions in underlying asset markets can become greatly magnified when using no-arbitrage arguments to price derivative claims. Amplification occurs when a replicating portfolio contains partially offsetting positions that lever up exposures to primary market frictions, and can cause arbitrarily large biases in synthetic return moments. We show theoretically and empirically how synthetic dividend strips, which shed light on the pricing of risks at different horizons, are impacted by this phenomenon. Dividend strips are claims to dividends paid over future time intervals, and can be replicated by highly levered long-short positions in futures contracts written on the same underlying index, but with different maturities. We show that tiny pricing frictions can help to reproduce a downward-sloping term structure of equity risk premia, excess volatility, return predictability, and a market beta substantially below one, consistent with empirical evidence. Using more robust return measures we find smaller point estimates of the returns to short-term dividend claims, and little support for a statistical or economic difference between the returns to short- versus long-term dividend claims.

What's Beneath the Surface? Option Pricing with Multifrequency Latent States

We introduce a tractable class of multi-factor price processes with regime-switching stochastic volatility and jumps, which flexibly adapt to changing market conditions and permit fast option pricing. A small set of structural parameters, whose dimension is invariant to the number of factors, fully specifies the joint dynamics of the underlying asset and options implied volatility surface. We develop a novel particle filter for efficiently extracting the latent state from joint S&P 500 returns and options data. The model outperforms standard benchmarks in- and out-of-sample, and remains robust even in the wake of seemingly large discontinuities such as the recent financial crisis.

Extreme Risk and Fractal Regularity in Finance

As the Great Financial Crisis reminds us, extreme movements in the level and volatility of asset prices are key features of financial markets. These phenomena are difficult to quantify using traditional models that specify extreme risk as a rare event. Multifractal analysis, whose use in finance has considerably expanded over the past fifteen years, reveals that price series observed at different time horizons exhibit several forms of scale invariance. Building on these regularities, researchers have developed a new class of multifractal processes that permit the extrapolation from high-frequency to low-frequency events and generate accurate forecasts of asset volatility. The new models provide a structured framework for studying the likely size and price impact of events that are more extreme than the ones historically observed.

Staying on Top of the Curve: A Cascade Model of Term Structure Dynamics

We develop a class of dynamic term structure models that accommodates arbitrarily many interest-rate factors with very few parameters. The model builds on a short-rate cascade, a parsimonious recursive structure that naturally ranks the latent state variables b y their rates of mean reversion, each revolving around the next lowest-frequency factor. With appropriate assumptions on volatilities and risk premia, the model overcomes the curse of dimensionality associated with general affine models. Using a panel of 15 LIBOR and swap rates, we estimate models using from one to 15 factors and only five parameters. The in-sample fit of high-dimensional specifications is near exact, with absolute pricing errors averaging less than one basis point, permitting yield-curve strippin g in an arbitrage-free, dynamically consistent environment. Cross-maturity correlations accurately reflec t empirical evidence, and out-of-sample interest rate forecasts significantly improve on prior benchmarks.

Macroeconomic Attention and the Stock Market

We construct indices of attention to macroeconomic risks including employment, output growth, and monetary policy. Attention rises around macroeconomic announcements and following changes in fundamentals over quarterly, annual, and business cycle horizons. The effect is asymmetric: Bad news raises attention more than good news. Moreover, attention to macroeconomic news provides a useful instrument for the level of the risk premium associated with unemployment and FOMC announcements. The findings support the central predictions of theories of endogenous attention and announcement risk premia, and establish the validity of the proposed empirical measures of macroeconomic attention.We construct indices of media attention to macroeconomic risks including employment, growth, inflation, monetary policy, and oil prices. Attention rises around macroeconomic announcements and following changes in fundamentals over quarterly, annual, and business cycle horizons. The effect is asymmetric, with bad news raising attention more than good news. Attention relates to the stock market in two ways. First, increases in aggregate trade volume and volatility coincide with rising attention, controlling for announcements. Second, changes in attention prior to the unemployment announcement predict both the announcement surprise and stock returns on the announcement day. We conclude that media attention to macroeconomic fundamentals provides useful information beyond the dates and contents of macroeconomic announcements.

Multifractal Diffusions Through Time Deformation and the MMAR

Time deformation provides a powerful tool to construct multifractal processes out of general multifractal measures. The first example of this technique in the literature is the Multifractal Model of Asset Returns (MMAR), which incorporates the outliers and volatility persistence exhibited by many financial time series, as well as a rich pattern of local variations and moment-scaling properties. This chapter illustrates that multifractal diffusions can be created by compounding a standard Brownian motion. The resulting price process is a semimartingale with a finite variance, which precludes investors from making arbitrage profits. The time deformation implies that the moments of returns scale is a power function of the frequency of observation. The time deformation approach is used to define the first multifractal diffusion with uncorrelated increments, the MMAR, which is also reviewed. In the MMAR, the multifractal time deformation is the cumulative distribution function of a random multiplicative cascade. The construction produces the moment-scaling, thick tails, and long-memory volatility persistence exhibited by many financial time series. It substantially improves on traditional fractal specifications and accommodates flexible tail behaviors with the highest finite moment taking any value greater than two. The model captures the nonlinear changes in the unconditional distribution of returns at various sampling frequencies, while retaining the parsimony and tractability of fractal approaches. The MMAR provides a fundamentally new class of stochastic processes for financial applications.

The Markov-Switching Multifractal (MSM) in Discrete Time

This chapter presents the discrete-time version of Markov-Switching Multifractal (MSM) model. MSM closely matches the intuition that a range of economic uncertainties with varying degrees of persistence impact financial markets. Using a tight set of restrictions inspired by the multifractal literature, a pure regime-switching specification is defined with multiple frequencies, arbitrarily many states, and a dense transition matrix. MSM volatility is derived by multiplying together a finite number of random first-order Markov components. It is assumed that the volatility components are identical except for differences in their switching probabilities, which follow an approximately geometric progression. The construction delivers multifrequency stochastic volatility model with a closed-form likelihood, enabling us for the first time to apply a standard econometric toolkit to estimating and forecasting using a multifractal model. An empirical investigation of four daily currency series shows that MSM performs well in comparison with leading forecasting models. In the data, MSM has a higher likelihood for all currencies, and the improvement is statistically significant. Since both MSM and generalized autoregressive conditional heteroskedasticity (GARCH) models have the same number of parameters, the multifractal is also preferred by standard selection criteria. Out-of-sample, MSM matches the accuracy of GARCH forecasts at very short horizons, such as one day, and provides substantially better forecasts at longer horizons.