Continuous-time Asset Pricing Model 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.

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.

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.

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.

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.

Saddlepoint approximations for affine jump-diffusion models

Affine jump-diffusion (AJD) processes constitute a large and widely used class of continuous-time asset pricing models that balance tractability and flexibility in matching market data. The prices of e.g., bonds, options, and other assets in AJD models are given by extended pricing transforms that have an exponential-affine form; these transforms have been characterized in great generality by Duffie et al. [2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376]. Calculating model prices requires inversion of these transforms, and this has limited the application of AJD models to the comparatively small subclass for which the transforms are available in closed form. This article seeks to widen the scope of AJD models amenable to practical application through approximate transform inversion techniques. More specifically, we develop the use of saddlepoint approximations for AJD models. These approximations facilitate the calculation of prices in AJD models whose transforms are not available explicitly. We derive and test several alternative saddlepoint approximations and find that they produce accurate prices over a wide range of parameters.

The Valuation of Corporate Debt With Default Risk

This article values equity and corporate debt by taking into account the fact that in practice the default point differs from the liquidation point and that it might be in the creditors' interest to delay liquidation. The article develops a continuous time asset pricing model of debt restructuring which explicitly considers the inalienability of human capital. The study finds that even though in general the creditors will not liquidate the firm on the incidence of default, but nevertheless would liquidate the firm prematurely relative to the first best threshold. This agency problem leads to the breakdown of the capital structure irrelevance result. 1. Theoretical Foundation The literature on the pricing of defaultable bonds started with Merton (1974) who applied the contingent claims valuation insight of Black and Scholes (1973) to the pricing of corporate bonds. He obtained closed form valuation expressions for (zero-coupon) risky bonds, by taking the lower reorganization boundary as given. Thus in his model, default occurred at maturity if the value of the firm was less than the payment promised to the bondholders. Furthermore, at maturity the compensation received by the creditors is fixed and they receive the minimum of the value of the firm or the contracted payment. In the event of a default, the control of the firm is transferred to the creditors and this default point can also be interpreted as a liquidation point with zero bankruptcy costs. Thus, in Merton's model, both the lower reorganization boundary and the compensation received by the bondholders is taken to be exogenous. In practice, however, the compensation received by the creditors may vary continuously with the value of the firm if the firm is in financial distress. Furthermore, Merton does not make the distinction between default and liquidation. There have been a number of extensions of the Merton model. Black and Cox (1976) incorporate bond indenture provisions. Jones, Mason and Rosenfeld (1984) consider mul- tiple issue of callable coupon debt. In both of these models however the induced lower boundary at which the firm is liquidated is taken to be exogenous. Further, default is again tantamount to bankruptcy in these models. Leland (1993) and Leland and Toft (1996) extend Merton's analysis by endogenising the lower reorganization boundary. By assuming that the equityholders will always issue equity to prevent a default, they obtain expressions for the values of risky debt via the smooth-pasting condition. Thus in their models, the equityholders keep on issuing equity (when necessary) to avoid a default until the value of equity falls to zero. Thus default occurs when the value of equity falls to zero and this default point is again synonymous

Jumps in Real-Time Financial Markets: A New Nonparametric Test and Jump Dynamics

This paper introduces a new nonparametric jump test for continuous-time asset pricing models. It distinguishes jump arrival times and realized jump sizes in asset prices as precisely as at intra-day levels. We demonstrate the likelihood of misclassification of jumps in discrete data becomes negligible when we use high-frequency returns. We explore real-time jump dynamics using intra-day U.S. individual equity prices through the test, and find empirical evidence that jump arrivals are associated with both pre-scheduled earnings announcements and unscheduled real-time news release.

Jumps in Financial Markets: A New Nonparametric Test and Jump Clustering

We introduce a new nonparametric jump test for continuous-time asset pricing models. It distinguishes actual arrivals and different sizes of jumps. Asymptotic distribution of the test statistics is provided, and we demonstrate that it is desirable to use high-frequency data. We explore dynamic jump intensity structure through the test, and find empirical evidence of jump clustering in foreign currency exchange markets. Its implication for financial risk management, in particular value at risk, is also discussed.

Corporate Walkout Decisions and the Value of Default

We present a continuous-time asset pricing model of the levered firm where shareholders select not only the timing but also the form of abandonment. Shareholders can walk out of the firm either by (i) defaulting on their debt obligations or (ii) selling their shares to alternative operators of the technologies, as in a corporation sale. The structural model relates shareholders' ex-post choice to both technological and financial factors. Considering that operators' technological supremacy is not universal, we obtain that whereas default necessarily involves an inefficient timing of ownership transfer, corporation sales do not. Then, the likelihood of default being chosen instead of a corporation sale increases with (i) the degree of leverage displayed by the firm and (ii) its technological supremacy. By ignoring corporation sales, existing defaultable bond pricing models have thus a tendency to exaggerate risk premia and underestimate the borrowing ability (debt capacity) of firms.

The Effect of VaR Based Risk Management on Asset Prices and the Volatility Smile

Value‐at‐risk (VaR) has become the standard criterion for assessing risk in the financial industry. Given the widespread usage of VaR, it becomes increasingly important to study the effects of VaR based risk management on the prices of stocks and options. We solve a continuous‐time asset pricing model, based on Lucas (1978) and Basak and Shapiro (2001), to investigate these effects. We find that the presence of risk managers tends to reduce market volatility, as intended. However, in some cases VaR risk management undesirably raises the probability of extreme losses. Finally, we demonstrate that option prices in an economy with VaR risk managers display a volatility smile.

The Effect of VaR Based Risk Management on Asset Prices and the Volatility Smile

Value-at-Risk (VaR) has become the standard criterion for assessing risk in the financial industry. Given the widespread usage of VaR based risk management, it becomes increasingly important to study the effects on the stock market and the option market of these constraints. We therefore introduce a continuous-time asset pricing model, based on Lucas (1978) and Basak and Shapiro (2001), in order to investigate the impact of the VAR risk managers on general equilibrium prices We find that agents with a VAR constraint affect the equilibrium prices remarkably and tend to reduce market volatility, as intended. However, in some cases VAR risk management can undesirably raise the probability of extreme losses. Finally, we demonstrate that option prices in an economy with VAR risk managers display a volatility smile as deserved in practice.

Strategic Debt Service

ABSTRACTWhen firms experience financial distress, equity holders may act strategically, forcing concessions from debtholders and paying less than the originally‐contracted interest payments. This article incorporates strategic debt service in a standard, continuous time asset pricing model, developing simple closed‐form expressions for debt and equity values. We find that strategic debt service can account for a substantial proportion of the premium on risky corporate debt. We analyze the efficiency implications of strategic debt service, showing that it can eliminate both direct bankruptcy costs and agency costs of debt.

An analysis of some aspects of regulatory risk and the required rate of return for public utilities

This paper uses a modification of the continuous time asset pricing model of Cox, Ingersoll, and Ross to analyze the effect of regulatory risk on the cost of capital. Analysis shows that random errors in setting the allowed rate of return can either increase or decrease the cost of capital depending on other regulatory parameters. However, the analysis suggests that regulatory risk is not material.