Continuous-time Asset Pricing Research Articles

Secured and unsecured debt in creditor-friendly bankruptcy

This article develops a continuous-time asset pricing model for valuing corporate securities in the presence of both secured and unsecured debt. We consider a framework where creditors dominate the negotiation process. This is consistent with the increasing influence of creditors in bankruptcy. We show that the unsecured creditors are incentivized to liquidate the firm prematurely relative to the first-best threshold. However, if the firm’s liquidation value is very low, it should complement its secured debt with unsecured debt as a form of insurance to avoid early liquidations. Our results have important implications for the debt structure and the resolution of financial distress of modern firms with substantial intangible assets.

An explosion time characterization of asset price bubbles

AbstractIn a standard continuous time asset pricing model, this paper provides an explosion time characterization of asset price bubbles that extends the existing characterization theorems in the literature from diffusion processes to general semimartingales (which can include jumps). This characterization has a nice economic interpretation, not emphasized in the existing literature.

Inference on Risk Premia in Continuous-Time Asset Pricing Models

We develop and implement asymptotic theory to conduct inference on continuous-time asset pricing models using individual equity returns sampled at high frequencies over an increasing time horizon. We study the identification and estimation of risk premia for the continuous and jump components of risks. Our results generalize the Fama-MacBeth two-pass regression approach from the classical discrete-time factor setting to a continuous-time factor model with general dynamics for the factors, idiosyncratic components and factor loadings, while accounting for the fact that the inputs of the second-pass regression are themselves estimated in the first pass. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Ambiguity, long-run risks, and asset prices in continuous time

This paper extends the long-run risks (LRR) models coined by Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2012) with Epstein and Zin’s (1989) recursive preferences into the LRR models with Chen and Epstein’s (2002) recursive multiple-priors utility, which captures the investor’s ambiguity on uncertainties. Our empirical analysis shows that considering ambiguity on uncertainties, in particular, large ambiguity on the long-run risks, in the LRR models can significantly improve the fitting performance of the equity premium and the risk-free rate when using a low relative risk aversion coefficient.

Downside risk measurement in regime switching stochastic volatility

Risk measurement is important to firms to enable management of risks, and ensure profitability during different firm and market events. In particular, downside risk is an important risk measure as it is a coherent risk measure, and it is also compatible with industry risk management approaches such as stop losses. Whilst regime switching models have been used for downside risk measurement, the regime switching models for stochastic volatility dynamics have been limited and so restrict risk measurement. In this paper we propose a new regime switching model that incorporates non-trivial stochastic volatility dynamics, hence we are able to measure risk more realistically. We derive the downside risk measure associated with our regime switching model, for risk measurement including and excluding jump risk. We prove that the regime switching model converges to the underlying continuous time asset pricing model, hence our risk measurement is consistent. We provide a discretisation for the variance risk process, which is locally consistent and enables computational implementation. We also provide numerical experiments to illustrate our method.

Bayesian inference of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme

Using recent developments in econometrics and computational statistics we consider the estimation of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme. To address the problem, we adopt throughout the paper an exact discretization approach. A flow sampling scheme arises, for example, naturally in modelling asset prices in continuous time since the time integral over successive observations defines the observable increments of asset log-prices. Exact discretization delivers an ARIMA(1,1,1) model for log-prices with a fractional driving noise. Building on the resulting exact discretization formulae and covariance function, a new Markov Chain Monte Carlo scheme is proposed and apply it to investigate the properties of both the time and frequency domain likelihoods/posteriors. For the exact discrete model, we adopt a general sampling interval of length h. This allows us to determine the optimal choice of h independent of the sample size. To illustrate the methods, with no ambition to a comprehensive data analysis, we use high frequency stock price data showing the relevance of aggregation over time issues in modelling asset prices.

Ambiguity, Long-Run Risks, and Asset Prices in Continuous Time

Ambiguity, Long-Run Risks, and Asset Prices in Continuous Time

Drift in Transaction‐Level Asset Price Models

We study the effect of drift in pure‐jump transaction‐level models for asset prices in continuous time, driven by point processes. The drift is assumed to arise from a non‐zero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, that is, the cumulative number of transactions. This link reveals a mechanism by which properties of intertrade durations (such as heavy tails and long memory) can have a strong impact on properties of average returns, thereby potentially making it extremely difficult to determine long‐term growth rates or to reliably detect an equity premium. We focus on a basic univariate model for log price, coupled with general assumptions on the point process that are satisfied by several existing flexible models, allowing for both long memory and heavy tails in durations. Under our pure‐jump model, we obtain the limiting distribution for the suitably normalized log price. This limiting distribution need not be Gaussian and may have either finite variance or infinite variance. We show that the drift can affect not only the limiting distribution for the normalized log price but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of durations) affects the rate of convergence of estimators of the growth rate and can invalidate standard hypothesis tests for that growth rate. As a remedy to these problems, we propose a new ratio statistic that behaves more robustly and employ subsampling methods to carry out inference for the growth rate. Our analysis also sheds some new light on two long‐standing debates as to whether stock returns have long memory or infinite variance.

Financial Intermediation and the Market Price of Risk: Theory and Evidence

We theoretically and empirically show financial intermediaries affect the price of risk of the overall market. We first study a continuous-time joint dynamic contracting and asset pricing model, whereby households must contract with and incentivize a financial intermediary to access the financial markets. We show the market price of risk in the intermediated economy is decreasing in the households’ absolute- and increasing in the households’ downside-risk aversions over the intermediary’s wealth. To test this prediction, we use volatilities and credit spreads of the financial sector to empirically proxy for these two risk aversions respectively. Time series regressions and cointegration tests show these two variables affect the equity market Sharpe ratio in the sign directions as predicted by the model. Furthermore, we show short-run shocks to these two variables is a priced factor with a large negative risk price in the cross-section of size, book-to-market, momentum, and also 135 equity and non-equity portfolios. Our theory model shows the role of optimal contracts for portfolio choices and asset pricing in an intermediated economy. The empirical results support our theory predictions, and show uncertainty and downside risks of the financial sector affect the overall market in both the time series and cross-section.

Realized Volatility and Multipower Variation

This paper reviews basic notions of return variation in the context of a continuous-time arbitrage-free asset pricing model and discusses some of their applications. We first define return variation in the infeasible continuous-sampling case. Then we introduce realized measures obtained from high-frequency observations which provide consistent and asymptotically normal estimates of the underlying return variation. The paper discusses applications of these measures for reduced-form volatility modeling and forecasting as well as testing for the presence of jumps.

DYNAMIC RISK MEASURES AND G-EXPECTATION

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

LIMIT LAWS IN TRANSACTION-LEVEL ASSET PRICE MODELS

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility and nontrading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. For the weak fractional cointegration case, we obtain the asymptotic distribution of the ordinary least squares estimator of the cointegrating parameter based on data sampled from an equally spaced discretization of calendar time, and we justify a feasible method of hypothesis testing for the cointegrating parameter based on the correspondingt-statistic. In the strong fractional cointegration case, we obtain the limiting distribution of a continuously averaged tapered estimator as well as other estimators of the cointegrating parameter, and we find that the rate of convergence can be affected by properties of intertrade durations. In particular, the persistence of durations (hence of volatility) can affect the degree of cointegration. We also obtain the rate of convergence of several estimators of the cointegrating parameter in the standard cointegration case. Finally, we consider the properties of the ordinary least squares estimator of the regression parameter in a spurious regression, i.e., in the absence of cointegration.

Empirical Asset Pricing with Nonlinear Risk Premia

We introduce a new model for the joint dynamics of the S&P 100 index and the VXO implied volatility index. The nonlinear specification of the variance process is designed to simultaneously accommodate extreme persistence and strong mean reversion. This grants superior forecasting power over the standard (linear) specifications for implied variance forecasting. We obtain statistically significant predictions in an out-of-sample exercise spanning several market crashes starting 1986 and including the recent subprime crisis. The model specification is possible through a simple continuous-time no-arbitrage asset pricing framework that combines semi-analytic pricing with a nonlinear specification for the market price of risk.

Human Capital and Asset Pricing

From the optimal behavior of arbitrary number of consumer-investors who act as to maximize their life-time utility of consumption and leisure, this paper derives a continuous-time intertemporal asset pricing model with stochastic human capital, leisure, consumption, and investment opportunities. Explicit demand functions for assets are derived and it is shown that, with heterogeneous human capital, the classic fund separation results may no longer hold. The human capital is modeled to include not only its marketed benefits but also its non-market benefits by the use of aggregate labor income and three macro variables in labor statistics: the average weekly employment hours, the unemployment rate, and the labor participation rate. Three flexibility parameters are introduced to explicitly account for the hidden unemployment, the self-employment, and the time spent on job-related activity such as work-commute and so on. Although a formal and comprehensive test is left as a future research task, two pieces of evidence are provided. First, a quick revisit of the equity premium puzzle shows that my model is plausible to explain the observed equity premium with the actual data. Then a preliminary test using returns of the 25 Fama-French size B/M portfolios shows that my model captures about 75% of the total variation in the cross-section returns and that the unemployment rate plays a critical role in explaining the cross-section returns.

Investigating Impacts of Self-Exciting Jumps in Returns and Volatility: A Bayesian Learning Approach

The paper proposes a new class of continuous-time asset pricing models where whenever there is a negative jump in asset returns, it is simultaneously passed on to diffusion variance and the jump intensity, generating co-jumps of prices and volatility and jump clustering. To properly deal with parameter uncertainty and hindsight bias, we employ a Bayesian learning approach, which generates all quantities necessary for sequential real-time model analysis. Empirical study using S&P 500 index returns shows that volatility jumps at the same time as negative jumps in asset returns mainly through jumps in diffusion volatility. We find weak evidence of jump clustering. Learning and parameter uncertainty are shown to have important implications for risk management, option pricing and volatility forecasting.

Calibration of Stock Betas from Skews of Implied Volatilities

We develop call option price approximations for both the market index and an individual asset using a singular perturbation of a continuous-time capital asset pricing model in a stochastic volatility environment. These approximations show the role played by the asset's beta parameter as a component of the parameters of the call option price of the asset. They also show how these parameters, in combination with the parameters of the call option price for the market, can be used to extract the beta parameter. Finally, a calibration technique for the beta parameter is derived using the estimated option price parameters of both the asset and market index. The resulting estimator of the beta parameter is not only simple to implement but has the advantage of being forward looking as it is calibrated from skews of implied volatilities.

MCMC Methods for Continuous-Time Financial Econometrics

Publisher Summary This chapter describes various Markov Chain Monte Carlo (MCMC) methods for exploring the posterior distributions generated by continuous-time asset pricing models. The MCMC methods are particularly well suited for continuous-time finance applications for several reasons. MCMC is a unified estimation procedure, which simultaneously estimates both parameters and latent variables. MCMC directly computes the distribution of the latent variables and parameters given the observed data and allows the researcher to quantify estimation and model risk. Estimation risk is the inherent uncertainty present in estimating parameters or state variables, while model risk is the uncertainty over model specification. The simplest MCMC algorithm is called the Gibbs sampler, which requires one to conveniently draw from the complete set of conditional distributions. In many cases, implementing the Gibbs sampler requires drawing random variables from standard continuous distributions such as normal, t, beta, or gamma or discrete distributions such as binomial, multinomial, or Dirichlet. The Griddy Gibbs sampler is an approximation that can be applied to approximate the conditional distribution by a discrete set of points. The Metropolis–Hastings algorithm allows the functional form of the density to be nonanalytic, where one only has to evaluate the true density at two given points. Random-walk Metropolis is the original algorithm considered by Metropolis et al. in 1953, and it is the mirror image of the independence Metropolis–Hastings algorithm.

Market selection of constant proportions investment strategies in continuous time

This paper studies the wealth dynamics of investors holding self-financing portfolios in a continuous-time model of a financial market. Asset prices are endogenously determined by market clearing. We derive results on the asymptotic dynamics of the wealth distribution and asset prices for constant proportions investment strategies. This study is the first step towards a theory of continuous-time asset pricing that combines concepts from mathematical finance and economics by drawing on evolutionary ideas.

Empirical Asset Pricing with Nonlinear Risk Premia

In this paper we introduce a simple continuous-time asset pricing framework, based on general multi-dimensional diffusion processes, that combines semi-analytic pricing with a nonlinear specification for the market price of risk. Our framework guarantees existence of weak solutions of the nonlinear SDEs under the physical measure, thus allowing to work with nonlinear models for the real world dynamics not considered in the literature so far. It emerges that the additional flexibility in the time series modelling is econometrically relevant: a nonlinear stochastic volatility diffusion model for the joint time series of the S&P 100 and the VXO implied volatility index data shows superior forecasting power over the standard specifications for implied and realized variance forecasting.

A structural model of debt pricing with creditor-determined liquidation

This paper develops a continuous time asset pricing model of debt and equity in a framework where equityholders decide when to default but creditors decide when to liquidate. This framework is relevant for environments where creditors exert a significant influence on the timing of liquidation, such as those of countries with creditor-friendly bankruptcy regimes, or in the case of secured debt. The interaction between the decisions of equityholders and creditors introduces an agency problem whereby equityholders default too early and creditors subsequently liquidate too early. Our model allows us to assess quantitatively how this problem affects the timing of default and liquidation, optimal capital structure, and spreads.