Equity Volatility Research Articles

On multilevel RBF collocation to solve nonlinear PDEs arising from endogenous stochastic volatility models

The main aim of this paper is the analytical and numerical study of a time-dependent second-order nonlinear partial differential equation (PDE) arising from the endogenous stochastic volatility model, introduced in [Bensoussan, A., Crouhy, M. and Galai, D., Stochastic equity volatility related to the leverage effect (I): equity volatility behavior. Applied Mathematical Finance, 1, 63–85, 1994]. As the first step, we derive a consistent set of initial and boundary conditions to complement the PDE, when the firm is financed by equity and debt. In the sequel, we propose a Newton-based iteration scheme for nonlinear parabolic PDEs which is an extension of a method for solving elliptic partial differential equations introduced in [Fasshauer, G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43, 423–438, 2002]. The scheme is based on multilevel collocation using radial basis functions (RBFs) to solve the resulting locally linearized elliptic PDEs obtained at each level of the Newton iteration. We show the effectiveness of the resulting framework by solving a prototypical example from the field and compare the results with those obtained from three different techniques: (1) a finite difference discretization; (2) a naive RBF collocation and (3) a benchmark approximation, introduced for the first time in this paper. The numerical results confirm the robustness, higher convergence rate and good stability properties of the proposed scheme compared to other alternatives. We also comment on some possible research directions in this field.

Understanding the Sources of Risk Underlying the Cross Section of Commodity Returns

We show that a model featuring an average commodity factor, a carry factor, and a momentum factor is capable of describing the cross-sectional variation of commodity returns. More parsimonious one- and two-factor models that feature only the average and/or carry factors are rejected. To provide an economic interpretation, we show that innovations in global equity volatility can price portfolios formed on carry, while innovations in a commodity-based measure of speculative activity can price portfolios formed on momentum. Finally, we characterize the relation between the factors and the investment opportunity set. Data and the Internet appendix are available at https://doi.org/10.1287/mnsc.2017.2840 This paper was accepted by Neng Wang, finance.

Volatility in equity markets and monetary policy rate uncertainty

Abstract Asset pricing models assume the risk-free rate to be a key factor for equity prices. Hence, there should be a strong link between monetary policy rate uncertainty and equity return volatility, both in theory and data. This paper uses regression-based projections for realized variance to examine the relationship between short horizon forecasts of equity variance and proxies for monetary policy rate uncertainty. By assessing various projection models for UK, US and euro area equity indices, we show that the proxies for monetary policy rate uncertainty have a significant and positive predictive power for the equity return variance. Adding monetary policy rate uncertainty variables can significantly improve forecasting models for equity variance and volatility at weekly, monthly and even quarterly horizons. The findings imply that market views of short-term interest rate developments may indeed be embedded in equity prices and their variations.

Equilibrium Implications of Interest Rate Smoothing

We introduce a macro-finance model in which monetary authorities adjust the money supply by targeting not only output and inflation but also the slope of the yield curve. We study the impact of McCallum-type rules on capital growth, the volatility of interest rates, the spread between long- and short-term rates, the persistence of monetary shocks and equity volatility. Our model supports the Federal Reserve’s choice to incorporate financial data in their policy decisions and expand the monetary base to decrease the nominal interest rate spread at the cost of lower expected long-term growth.

Central Bank Actions and Words: The Intraday Effects of FOMC Policy Communications on Individual Equity Volatility and Returns

AbstractWe compare intraday impacts of the release of Federal Reserve decision announcements and of Federal Open Market Committee minutes between 2004 and 2015 on 1,997 equity return and volatility series. We find that returns are unresponsive to either news release, but conditional volatility increases for both, manifesting immediately after each information release, and persisting for 30 minutes post‐announcement. These effects are larger for decisions than for minutes. On stratifying firms by trading intensity, we find most “high trading intensity” firms respond to these announcements, while “low trading intensity” firms are less affected. Our results show that traders respond, albeit differently, to both sets of information releases.

How Do Quasi‐Random Option Grants Affect CEO Risk‐Taking?

ABSTRACTWe examine how an increase in stock option grants affects CEO risk‐taking. The overall net effect of option grants is theoretically ambiguous for risk‐averse CEOs. To overcome the endogeneity of option grants, we exploit institutional features of multiyear compensation plans, which generate two distinct types of variation in the timing of when large increases in new at‐the‐money options are granted. We find that, given average grant levels during our sample period, a 10% increase in new options granted leads to a 2.8% to 4.2% increase in equity volatility. This increase in risk is driven largely by increased leverage.

Structural GARCH: The Volatility-Leverage Connection

We propose a new model of volatility where financial leverage amplifies equity volatility by what we call the “leverage multiplier”. The exact specification is motivated by standard structural models of credit; however, our parametrization departs from the classic Merton (1974) model and is, as we show, flexible and accurate enough to capture environments where the firm’s asset volatility is stochastic, asset returns can jump, and asset shocks are non-normal. As a result, our model also provides estimates of daily asset returns and asset volatility. In addition, our specification nests both a standard GARCH and the Merton model, which allows for a simple statistical test of how leverage interacts with equity volatility. Empirically, the Structural GARCH model outperforms a standard GARCH model for approximately 75% of the financial firms we analyze. We then apply the Structural GARCH model to two empirical applications: the leverage effect and systemic risk measurement.

The role of jumps and leverage in forecasting volatility in international equity markets

We analyze the importance of jumps and the leverage effect on forecasts of realized volatility in a large cross-section of 18 international equity markets, using daily realized measures data from the Oxford-Man Realized Library, and two widely employed empirical models for realized volatility that allow for jumps and leverage. Our out-of-sample forecast evaluation results show that the separation of realized volatility into a continuous and a discontinuous (jump) component is important for the S&P 500, but of rather limited value for the remaining 17 international equity markets that we analyze. Only for 6 equity markets are significant and sizable forecast improvements realized at the one-step-ahead horizon, which, nevertheless, deteriorate quickly and abruptly as the prediction horizon increases. The inclusion of the leverage effect, on the other hand, has a much larger impact on all 18 international equity markets. Forecast gains are not only highly significant, but also sizeable, with gains remaining significant for forecast horizons of up to one month ahead.

Interest rate volatility and risk management: Evidence from CBOE Treasury options

This paper investigates US Treasury market volatility and develops new ways of dealing with the underlying interest rate volatility risk. We adopt an innovative approach which is based on a class of model-free interest rate volatility (VXI) indices we derive from options traded on the CBOE. The empirical analysis indicates substantial interest rate volatility risk for medium-term instruments which declines to the levels of the equity market only as the tenor increases to 30 years. We show that this risk appears to be priced in the market and has a significant time-varying relationship with equity volatility risk. US Treasury market volatility is appealing from an investment diversification perspective since the VXI indices are negatively correlated with the levels of interest rates and of equity market implied volatility indices, respectively. Although VXI indices are affected by macroeconomic and monetary news, they are only partially spanned by information contained in the yield curve. Motivated by our results on the magnitude and the nature of interest rate volatility risk and by the phenomenal recent growth of the equity volatility derivative market, we propose the use of our VXI indices as benchmarks for monitoring, securitizing, managing and trading interest rate volatility risk. As a first step in this direction, we describe a framework of one-factor equilibrium models for pricing VXI futures and options on the basis of empirically favored mean-reverting jump-diffusions.

Assessing the impact of an EU financial transactions tax on asset volatility: An event study

Under the context of EMU debt and financial crisis, we assess the impact of EMU's announcement of a Financial Transactions tax (FTT) on bond and equity volatilities for seven countries, namely Germany, France (core EMU), Greece, Italy, Ireland, Portugal and Spain (periphery EMU). In the absence of historical data on volume and volatility of transactions of such a tax, we utilize the event study methodology. The selected event date considering the FTT announcement was found significant for core EMU's equity portfolio and periphery EMU's bond portfolio. Moreover, under GARCH models, we found that the announcement effect of FTT increases the volatility of both core EMU's equity portfolio as well as periphery EMU's bond portfolio.

The relationship between equity and bond returns: An empirical investigation

The correlation between equity and corporate debt is ambiguous. News affecting the value of a firm's assets induces a positive correlation, whereas an increase in the volatility of a firm's assets induces a negative correlation. We examine the conditional correlation between these two securities. While the average correlation is positive, the conditional correlation increases with credit risk, and decreases with equity volatility. Our results are consistent with the thesis that the equity bond relation is dependent on the potential wealth transfer between stock and debt holders. Nevertheless, this relation seems to break down during periods of extreme market uncertainty.

The Effect of Mergers on US Bank Risk in the Short Run and in the Long Run

We examine changes in risk following US bank mergers in the period 1981-2014. Short-run increases in acquirer risk following mergers occur only in the first few mergers undertaken by the same acquirer, and only in systematic risk. The equity volatility of acquirers does not increase. Using a new approach to measure the long-run effect we find that these results persist, consistent with banks maintaining a constant level of total equity risk in the long run. Constant acquirer risk means that all diversification benefits of the mergers are dissipated. We measure the loss of diversification associated with mergers and find it to be 40% of the risk level in 1981. Almost all of this occurred prior to 2004. In addition, there has been a large increase in correlations between the largest banks, much of which has come from sources other than mergers. The results are inconsistent with these mergers being motivated by the ‘too big to fail’ put. They suggest that if one wanted to reduce the risk of the banking system by demerging major banks one would have to reach back to the structure that existed before 2004. Simply reversing recent mergers would not have much effect on stock market measures of risk.

The Factor Structure in Equity Options

Principal component analysis of equity options on Dow-Jones …rms reveals a strong factor structure. The …rst principal component explains 77% of the variation in the equity volatility level, 77% of the variation in the equity option skew, and 60% of the implied volatility term structure across equities. Furthermore, the …rst principal component has a 92% correlation with SP G12; G13.

CEO Inside Debt and Risk Taking: Evidence From Property–Liability Insurance Firms

AbstractWe examine the incentive effects of CEO inside debt holdings (pensions and deferred compensation) on risk taking using the sample of U.S. publicly traded property–liability insurers. To represent managerial risk taking, we employ value at risk (VaR) and expected shortfall (ES), which capture extreme movements in the lower tail of insurer stock return distribution. We also estimate firm default risk, equity volatilities, and insurance‐related risk as alternative measures of risk taking. We document that inside debt represents a significant component of CEOs’ compensation in the insurance industry. We find that there is a significant and negative relationship between CEO inside debt holdings and risk‐taking behavior. The results suggest that the structure of executive debt‐like compensation could be a potential method of reducing managers’ risk‐taking incentives.

Modelling long memory in volatility in sub-Saharan African equity markets

This study examines the long memory properties in the second moments of the return series in the equity markets of Ghana, Kenya, Nigeria and South Africa. Using 5219 daily observations of thin-trading adjusted total equity market return data, we find the presence of long-range dependency in the second moments of the return innovations in all the four countries’ equity markets in the full sample. To isolate spurious long memory, we perform structural breaks analysis to guide us in splitting the data for further examination. We find that all the four countries exhibited two structural breaks each during the sample period, and long memory identified in these two markets were not influenced by the structural breaks. This finding may have an influence on portfolio diversification and risk management. The long memory characteristics in the conditional volatility may also provide useful information to market participants in pricing long-term derivative contracts.

Accounting Standards and Pension Investment Decisions

This study analyzes the ‘real’ effects of accounting standards in the context of defined benefit pension plans. Specifically, we examine IAS 19R, which increases expected pension-induced equity volatility by eliminating the so-called ‘corridor method’ of accounting for actuarial gains and losses. Supported by interview evidence, we predict that IAS 19R leads firms to reconsider their pension investment decisions. Using matched samples of treatment and control firms, results from difference-in-differences tests indicate that firms affected by IAS 19R adoption significantly shift their pension assets from equities into bonds, relative to control firms. This effect is attenuated in the cross-section for firms with larger and better funded pension plans. Supplementary analyses suggest that this shift in pension investment is mainly due to IAS 19R’s changes in the accounting for actuarial gains and losses (the ‘OCI method’). These results are robust to a number of sensitivity tests, although endogeneity concerns cannot be fully ruled out. Our study informs accounting standard setters of potential shifts in firms’ real economic activities due to concerns about pension-induced equity volatility.

The effect of quantitative easing on the variance and covariance of the UK and US equity markets

We examine the impact on the variance-covariance structure of UK and US equity markets of the quantitative easing (QE) operations implemented by the Bank of England (BoE) and the Federal Reserve (Fed). While the theory of portfolio balance suggests that QE operations could affect markets other than those in which the operations occur, prior analysis of these other markets is scarce. We find that while QE operations in general reduced equity volatility, day to day operations generated spikes in volatility in UK equities. We also find that BoE operations increased the covariance between the UK and US equity markets.

Covering the World: Global Evidence on Covered Calls

Typical covered call strategies may be decomposed, using a risk and performance attribution methodology, into three components: equity exposure, short volatility exposure and equity timing. This paper applies that attribution methodology to covered calls on eleven global indexes. We find that the relative risk and return contributions of the three components are remarkably consistent across our cross-section of indexes. Across the board, the covered call’s equity exposure is responsible for most of the strategy’s risk and return, while the short volatility exposure has the highest Sharpe ratio of the strategy’s components. The returns from the equity timing exposure are statistically insignificant in all eleven indexes, yet this exposure contributes a relatively large amount of the strategy’s risk. These results provide further evidence that managing equity exposure in covered calls provides superior risk-adjusted returns. Further, a globally diversified portfolio of risk-managed covered calls may be viewed as a defensive alternative to global equity, providing similar returns with lower volatility and lower drawdowns.

Credit Default Swaps and Debt Overhang

We analyze the impact of credit default swaps (CDS) trading on firm investment, long-term debt financing, and valuation. In our model, the firm is endowed with a real option to initiate a project and enhance its future growth. Its creditors have access to CDS contracts that hedge them against default losses. We show that CDS protection increases the firm's pledgeable income, that is, the maximum amount of debt it can raise. However, at the same time CDS protection decreases asset growth and impedes project initiation. As a result, CDS trading could reduce firm value, and the negative effects are stronger when the firm is riskier, where shareholders have stronger bargaining power, and growth opportunities are less valuable. Using simulated cross-sections of firms, we find that CDS trading increases corporate default rates and deters investment. Altogether, CDS firms tend to have a lower firm value and more volatile equity returns than non-CDS firms.

Investigating Linkages between U.S. CDS Spreads and Both Equity Market Price and Equity Market Volatility Channels: A Quantile Cointegrating Regression Approach

We investigate the behavior of daily aggregate U.S. CDS spreads during almost six years. Existing linkages between credit and equity markets advocate the use of market price and volatility channels as explanatory factors. In particular, the evolution of CDS spreads is analyzed along with the level of the equity market and a well-chosen implied volatility index. We describe such robust (to spurious correlation) relationship with the quantile (cointegrating) regression approach. It allows for describing distributional patterns of CDS spreads in the light of the equity market’s influence. Going beyond a simple mean relationship, the quantile regression tool quantifies accurately the linkages between credit and market risks. Such powerful econometric approach handles extreme quantiles (i.e. extreme CDS values) and their behavior with respect to the equity market’s impact. Moreover, heteroskedastic patterns such as time-varying variance, but also autocorrelation, skewness and leptokurtosis are captured. Incidentally, the sensitivity of aggregate CDS spreads to equity market price and volatility channels vary across quantiles. Being quantile-dependent, the sensitivity of credit risk with respect to (equity) market risk exhibits asymmetric responses to equity market shocks. A complementing sub-period analysis investigates potential regime shifts in estimated quantile cointegrating regressions. Quantile cointegrating coefficients vary over time and quantiles, and exhibit different magnitudes across sub-periods. However, the nature of the full sample-specific impact of equity market on CDS spreads is mostly preserved across sub-periods. As a result, the assessment of credit risk with respect to the impact of market risk and the nature of corresponding linkages differ from one quantile to another. Such methodology can help enhance risk management processes, among which value-at-risk (VaR) and expected shortfall. In this light, a scenario analysis and risk signaling application are proposed for credit risk management prospects. Under specific risk levels, credit risky situations are described conditional on the equity market’s impact over time, and related expected aggregate CDS spreads are computed.