Peter Christoffersen

Evaluating Interval Forecasts

A complete theory for evaluating interval forecasts has not been worked out to date. Most of the literature implicitly assumes homoskedastic errors even when this is clearly violated and proceed by merely testing for correct unconditional coverage. Consequently, the author sets out to build a consistent framework for conditional interval forecast evaluation, which is crucial when higher-order moment dynamics are present. The new methodology is demonstrated in an application to the exchange rate forecasting procedures advocated in risk management. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Optimal Prediction Under Asymmetric Loss

Prediction problems involving asymmetric loss functions arise routinely in many fields, yet the theory of optimal prediction under asymmetric loss is not well developed. We study the optimal prediction problem under general loss structures and characterize the optimal predictor. We compute the optimal predictor analytically in two leading cases. Analytic solutions for the optimal predictor are not available in more complicated cases, so we develop numerical procedures for computing it. We illustrate the results by forecasting the GARCH(1,1) process which, although white noise, is non-trivially forecastable under asymmetric loss.

Which GARCH Model for Option Valuation

Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a time series of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the risk-neutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returns-based inference, we find that our option-based objective function favors a relatively parsimonious model. Specifically, when evaluated out-of-sample, our analysis favors a model that, besides volatility clustering, only allows for a standard leverage effect.

Option Valuation with Long-Run and Short-Run Volatility Components

This paper presents a new model for the valuation of European options, in which the volatility of returns consists of two components. One of these components is a long-run component, and it can be modeled as fully persistent. The other component is short-run and has a zero mean. Our model can be viewed as an affine version of Engle and Lee (1999), allowing for easy valuation of European options. The model substantially outperforms a benchmark single-component volatility model that is well-established in the literature, and it fits options better than a model that combines conditional heteroskedasticity and Poissonnormal jumps. The component model’s superior performance is partly due to its improved ability to model the smirk and the path of spot volatility, but its most distinctive feature is its ability to model the volatility term structure. This feature enables the component model to jointly model long-maturity and short-maturity options.

Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach

International equity markets are characterized by nonlinear dependence and asymmetries. We propose a new dynamic asymmetric copula model to capture long-run and short-run dependence, multivariate nonnormality, and asymmetries in large cross-sections. We find that copula correlations have increased markedly in both developed markets (DMs) and emerging markets (EMs), but they are much lower for EMs than for DMs. Tail dependence has also increased but its level is still relatively low for EMs. We propose new measures of dynamic diversi?cation bene?ts that take into account higher order moments and nonlinear dependence. The bene?fits from international diversi?cation have reduced over time, drastically so for DMs. EMs still offer signi?cant diversi?cation bene?ts, especially during large market downturns.

Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option Prices

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. We investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources: realized volatilities, S&P500 returns, and an extensive panel of option data. The three sources of data all point to the same conclusion: the best volatility specification is one with linear rather than square root diffusion for variance. This model captures the stylized facts in realized volatilities, it performs well in fitting various samples of index returns, andit has the lowest option implied volatility mean squared error in and out of sample. The Author 2010. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.

From Inflation to Growth: Eight Years of Transition

This paper reexamines growth in transition using panel data to 1997. It suggests that output has been strongly affected by export market growth; that inflation has been associated with weaker output only above a threshold inflation rate; that structural reform has been associated with weaker output initially, but that it stimulates higher growth thereafter; and that rapid disinflation has been associated with output losses only in the presence of pegged exchange rates.

Capturing Option Anomalies with a Variance-Dependent Pricing Kernel

We develop a GARCH option model with a new pricing kernel allowing for a variance premium. While the pricing kernel is monotonic in the stock return and in variance, its projection onto the stock return is nonmonotonic. A negative variance premium makes it U shaped. We present new semiparametric evidence to confirm this U-shaped relationship between the risk-neutral and physical probability densities. The new pricing kernel substantially improves our ability to reconcile the time-series properties of stock returns with the cross-section of option prices. It provides a unified explanation for the implied volatility puzzle, the overreaction of long-term options to changes in short-term variance, and the fat tails of the risk-neutral return distribution relative to the physical distribution. The Author 2013. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com., Oxford University Press.

Value–at–Risk Models

In this chapter, we build first a univariate and then a multivariate filtered historical simulation (FHS) model for financial risk management. Both the univariate and multivariate methods simulate future returns from a model using historical return innovations. While the former relies on portfolio returns filtered by a dynamic variance model, the latter uses individual or base asset return innovations from dynamic variance and correlation models. The univariate model is suitable for passive risk management or risk measurement whereas the multivariate model is useful for active risk management such as optimal portfolio allocation. Both models are constructed in such a way as to capture the stylized facts in daily asset returns and to be simple to estimate. The FHS approach enables the risk manager to easily compute Value-at-Risk and other risk measures including Expected Shortfall for various investment horizons that are conditional on current market conditions. The chapter also lists various alternatives to the suggested FHS approach.

Option-Implied Measures of Equity Risk

Equity risk measured by beta is of great interest to both academics and practitioners. Existing estimates of beta use historical returns. Many studies have found option-implied volatility to be a strong predictor of future realized volatility. We find that option-implied volatility and skewness are also good predictors of future realized beta. Motivated by this finding, we establish a set of assumptions needed to construct a beta estimate from option-implied return moments using equity and index options. This beta can be computed using only option data on a single day. It is therefore potentially able to reflect sudden changes in the structure of the underlying company. Copyright 2011, Oxford University Press.