Multivariate dependence of implied volatilities from equity options as measure of systemic risk

Abstract

This paper presents a methodology to examine the multivariate tail dependence of the implied volatility of equity options as an early warning indicator of systemic risk within the financial sector. Using non-parametric methods of estimating changes in the dependence structure in response to common shocks affecting individual risk profiles, possible linkages during periods of stress are quantifiable while recognizing that large shocks are transmitted across financial markets differently than small shocks. Before and during the initial phase of the financial crisis, we find that systemic risk increased globally as early as February 2007 — months before the unraveling of the U.S. subprime mortgage crisis and long before the collapse of Lehman Brothers. The average (multivariate) dependence among a global sample of banks and insurance companies increased by almost 30% while joint tail risk declined by about the same order of magnitude, indicating that co-movements of large changes in equity volatility were more likely to occur and responses to extreme shocks became more differentiated as distress escalated. The key policy consideration flowing from our analysis is that complementary measures of joint tail risk at high data frequency are essential to the robust measurement of systemic risk, which could enhance market-based early warning mechanisms as part of macroprudential surveillance.

Introduction

The global financial crisis demonstrated the importance of establishing effective early warning systems for identifying system-wide vulnerabilities to sources of financial sector distress and adopting suitable defenses against the impact of systemic risk events. Such events can arise from shocks to individual or collective arrangements – both institutional and market-based – that could either lead directly to material financial distress and/or significantly amplify its consequences (with adverse effects on the real economy). Thus, any disruptions to the flow of financial services due to an impairment of all or parts of the financial system would be deemed material and systemically relevant if there were the potential of financial instability to trigger serious negative spillovers to the real economy.¹

Systemic risk is also an integral element in the design and implementation of macroprudential surveillance. Macroprudential surveillance aims to limit, mitigate or reduce systemic risk, thereby minimizing the incidence and impact of disruptions in the provision of key financial services that can have adverse consequences for the real economy (and broader implications for economic growth). The traditional approach to financial stability analysis concentrates analytical efforts on vulnerabilities to individual failures, assuming that the financial system is in equilibrium and adjusts when it experiences a shock. As opposed to this conventional approach, the potential build-up of systemic vulnerabilities warrants a comprehensive monitoring of on-going developments in areas where the impact of disruptions to financial stability is deemed most severe and wide-spread — and especially in areas of economic significance to both the financial sector and the real economy (Table 1).

Ideally, systemic risk measures should support, or be linked to, macroprudential surveillance by providing information on the build-up of system-wide vulnerabilities in both the time and cross-sectional dimensions with an acceptable level of accuracy and latency. Efforts aimed at preventing the propagation of individual or joint distress of firms and/or the failure markets that are deemed systemically important have resulted in a multi-faceted approach comprising complementary measures in areas of regulatory policies, supervisory scope, and resolution arrangements with a view towards enhancing the resilience of the financial sector while avoiding impairment to efficient activities that do not cause and/or amplify system-wide stress in any meaningful manner.²

While there is still no consistent theory of systemic risk measurement, existing approaches can be broadly distinguished based on several core principles underpinning two general approaches (see Box 1). The “contribution approach” defines the propensity of individual failure to pose a threat to financial stability in the absence of close substitutes due to the nature, scope, size, scale, concentration of its activities, or its connectedness with other financial institutions FSB (2010 and 2012). In contrast, the “participation approach” assumes that a firm experiences losses from a single (or multiple) large shock(s) to concentrated activity that represents a common exposure whose impact under adverse conditions exceeds the firm's loss bearing capacity. In the case of the former, firms contribute to systemic risk from individual failures that propagate material financial distress or activities via intra- and inter-sectoral linkages to other institutions and markets (especially within conglomerate structures). Moreover, the initial effect of direct and indirect exposures to a failing institution (e.g., defaults on liabilities and/or asset fire sales) can also escalate to cause spillover effects to previously unrelated institutions and markets as a result of greater uncertainty or the reassessment of financial risk (i.e., changes in general risk appetite and/or the market price of risk). Conversely, the participation in systemic risk occurs via an institution's credit and market risk exposures affected by the adverse impact of other financial institutions. Table 2 in Box 1 below shows the distinguishing features of both approaches.

The distinction of measurement approaches also reflects varying channels of risk transmission that influence assumptions on tail dependence. Most approaches thus far have focused on determining the contribution of financial institutions to systemic risk, including the assessment of spillover and contagion effects between institutions within and across different sectors and national boundaries. Among the main channels that facilitate the transmission of shocks, the combination of interconnectedness and asset liquidation has become most relevant for the modeling of dependence that takes into account measures of joint tail risk, i.e., multiple institutions and/or markets experience a high-severity but low-probability event. Claims by creditors, counterparties, investors, or other market participants (“direct linkages”), as well as common exposures to certain asset classes, industry sectors, and markets (“indirect linkages”) establish relationships that can exacerbate contagion effects, especially when extreme shocks trigger actions by counterparties and market participants that are outside the historical experience. Excessive leverage and maturity mismatches can amplify these effects if the sudden disposal of large asset positions of an institution in distress significantly disrupts trading and/or causes significant losses for other firms with similar holdings due to increases in asset and funding liquidity risk.

Dependence measures can deliver important insights about contagion effects that are essential to the analysis of systemic risk to financial stability. Asset price returns and changes in individual default risk convey information about the behavior of each firm (and its implications for investor perception) given that other variables, including the reaction of markets and other firms, are unchanged, i.e., they reflect the marginal behavior of each institution in isolation. However, the system-wide perspective of aggregate changes of these risk factors requires information about their dependence structure. Assuming that risk factors are not fully correlated, it is reasonable to account for differences in the dependence structure of risk profiles across multiple entities in which the impact of each firm on aggregate default risk is lower than the appropriate percentile for that risk in isolation.

For analyzing dependence under stress conditions, consideration also needs to be given to the possibility that one risk factor might increase the likelihood of other risk factors that are external or internal to a firm (with common shocks affecting multiple firms at the same time). Such association might deviate significantly from historically observed relationships, especially when the severity of shocks is far removed from central expectations. While it is generally straightforward to examine the risk profile of an individual firm in response to a particular shock, combining multiple risk factors (and the extent to which firms might affect each other in terms of default risk) tends to complicate a reliable assessment of financial stability.³ The combination of risk factors, such as the joint increase of default risk, engenders a systemic risk event if exposures from direct and/or indirect linkages are significant enough to cause either material impairments of several financial institutions (by threatening their financial condition and/or competitive position) or disruptions to critical functions of the sector and/or financial system. Moreover, joint default risk within a system varies over time and depends on the individual firm's likelihood to cause and/or propagate shocks arising from the adverse change in one or more risk factors. Default risk may be weakly correlated under normal economic circumstances but highly correlated in times of distress (as correlations become very high in downturn conditions).

Recent systemic risk measures have gone beyond standard econometric procedures that characterize dependence as central tendencies. The measurement of joint tail risks from common shocks is fundamental to the measurement of systemic risk (and associated early warning mechanisms). However, conventional correlation measures fail to address the conjunctural dimensions influencing tail risk (arising from both cross-sectional and time-varying changes of determinants of co-movement) and capture their impact on the reliable and consistent implementation of measurement methods of systemic risk. These shortcomings reflect the general danger of underestimating the practical problems of applying concepts aimed at representing the relation between multiple entities under normal circumstances, reflecting the time-varying interdependence, to generate measures of systemic risk from the joint realization of extremes as a result of contagion effects.

As a result, the literature on tail dependence within systemic risk measurement approaches has grown considerably in response to greater demand placed on the ability to develop a better understanding of the interlinkages between firms and its implications for financial stability (see Table 3). Some of the more prominent measurement approaches to systemic risk, such as CoVaR (Adrian & Brunnermeier, 2008), CoRisk (Chan-Lau, 2010), Systemic Expected Shortfall (SES) (Acharya et al., 2009, Acharya et al., 2010, Acharya et al., 2012) (as well as extensions thereof, such as the Distress Insurance Premium (DIP) by Huang, Zhou, and Zhu (2010)), SRISK (Brownlees & Engle, 2011), Joint Probability of Distress (Segoviano & Goodhart, 2009), and Systemic CCA (International Monetary Fund (IMF), 2009b, Gray and Jobst, 2010, Gray and Jobst, 2010, chap. 1, Gray and Jobst, 2011b, Gray et al., 2010, Jobst and Gray, 2013), have focused on the "contribution approach" by including an implicit or explicit treatment of the dependence structure as an essential element to the determination of joint default risk or expected losses.⁴ There are also several studies on network analysis and agent-based models that are closer related to the "participation approach" by modeling how inter-linked asset holdings matter in the generation and propagation of systemic risk (Allen et al., 2010, Espinosa-Vega and Solé, 2011, Organization for Economic Cooperation and Development (OECD), 2012). Haldane and Nelson (2012) underscore this observation by arguing that networks can produce non-linearity and unpredictability with the attendant extreme (or fat-tailed) events.⁵ The specification of dependence between firms helps identify common vulnerabilities to risks as a result of the (assumed) collective behavior under common distress in keeping with a macroprudential perspective.

Most of these approaches have placed the emphasis squarely on modeling dependence in response to extreme changes in market conditions (without controlling for firm-to-firm relationships) if one or more firms were to experience material distress. For instance, Acharya, Pedersen, et al. (2012) estimate potential losses as Marginal Expected Shortfall (MES) of individual banks in the event of a systemic crisis, which is defined as the situation when the aggregate equity capital of sample banks falls below some fraction of aggregate assets.⁶ Thus, the MES specifies historical expected losses, conditional on a firm having breached some high systemic risk threshold based on its historical equity returns. Adjusting MES by the degree of firm-specific leverage and capitalization yields the Systemic Expected Shortfall (SES).⁷ This method, however, generates a purely empirical measure of linear and bivariate dependence rather than a closed-form solution. It does not consider interaction between subsets of banks and is limited to cases when the entire banking sector is undercapitalized. Brownlees and Engle (2011) apply the same definition for a systemic crisis and formulate a capital shortfall measure (“SRISK index”) that is similar to SES; however, they provide a close-form specification of extreme value dependence underpinning MES by modeling the correlations between the firm and market returns using the Dynamic Conditional Correlation (DCC)-GARCH (Engle, 2001, Engle, 2002) model to estimate these quantities on a weekly basis using daily equity returns.⁸ Also Huang, Zhou, and Zhu (2009); Huang et al. (2010) derive correlation of equity returns via DCC-GARCH as statistical support to motivate the specification of the dependence structure for default probabilities within a system of firms.⁹ Both CoVaR and CoRisk follow the same logic of deriving a bivariate measure of dependence between a firm's financial performance and an extreme deterioration of market conditions (or that of its peers). For instance, the CoVaR (CoRisk) for a certain firm is defined as the Value-at-Risk (VaR) (as a measure of extreme default risk) of the whole sector (firm) conditional on a particular institution being in distress.¹⁰ However, these approaches are not based on structural models and rely on quantile regressions as a way of determining an empirical measure of linear sensitivity (i.e., conditional dependence).

There are only few models that include multivariate (firm-by-firm) dependence through either a closed-form specification or the simulation of joint probabilities using historically informed measures of association. For instance, the Systemic CCA framework generates a multivariate density of joint expected losses based on the non-parametric dependence structure of sample firms for any level of statistical confidence. In addition, the measure of dependence is also linked to a semi-structural model of default risk since the input variables are individual expected losses of firms based on their implied put option values for a given debt level, market capitalization, and equity price volatility.¹¹ Similar to the empirical copula approach in the Systemic CCA, the JPoD measure uses a multivariate entropy measure to determine joint default risk using the implied default probability in credit default swap (CDS) spreads.¹² Thus, both approaches go beyond CoVaR, CoRisk and SES, which examine only the bivariate and marginal effect of extreme value dependence, by covering all available data that can be usefully integrated to assess the system-wide sensitivity of individual default risk.

Against this background, we present a forward-looking approach to the quantification of systemic risk based on market-implied connectedness. The suggested approach generates estimates of the dependence structure between a sample of large global financial institutions, reflecting their time-varying interdependence (which applies to all states of the world) and the potential of contagion effects (in situations of extreme institutional and/or market stress).¹³ More specifically, these estimates are derived from non-parametric methods of measuring changes in the dependence structure in response to common shocks affecting individual risk profiles implied by equity prices. The dependence measure is scale-invariant, which means that is not affected by different levels of inputs (such as implied volatilities in our case) but by their relative changes. They include the average probability of co-movement (“entropy correlation”) and the impact of very large negative shocks to the dependence structure (“tail dependence”), which help quantify possible linkages during periods of stress.

The presented methodology straddles both systemic risk measurement approaches (see Box 1 above). It assumes that market prices include information on investor perceptions of changes in aggregate risk, which is affected by the implications of individual distress for other institutions (“contribution approach”) and the susceptibility of firms to general macro-financial shocks impacting common exposures (“participation approach”). Any linkages between firms are considered endogenous to the dependence structure derived from market prices and change dynamically, which removes the assessment from the identification of specific sources of contagion. Since large shocks are transmitted across financial entities and markets differently than small shocks (and these differences vary over time), this analysis of higher moments and their non-linear dependence also deliver important insights into amorphous nature of systemic risk.

The presented methodology focuses on joint equity price dynamics during negative shock events as early warning indicators. Equity represents the most junior claim on firms, and thus, can be considered most sensitive to changes in the expectation of future asset performance (and the likelihood of default). In particular, equity derivatives contain valuable forward-looking information and “fat tails” implied by equity prices would indicate that market perception of downside equity risk has increased disproportionately.

Thus, equity prices in cash and derivatives markets reflect even marginal changes in expectations of default risk. This observation becomes even more important during times of stress, when the ability to apply options as forward-looking measures to hedge the downside risk of equity comes at a premium.¹⁴ Higher moments of equity price dynamics also better account for non-linearities of changes in default risk if large risk exposures become more commonplace than standard models suggest. Based on their research on the relation between the volatility skew (volatility measured at different strike prices) implied by equity options and CDS spreads. Hull, Nelken, and White (2004) find that if the probability distribution of the implied volatility underpinning option prices is negatively skewed (left-tailed) so is the implied underlying asset distribution, which suggests greater default risk if debt service remains unchanged. Moreover, the implicit put option of firm default increases disproportionately to the decline of the market value. This means, that accounting for higher moments of equity options can deliver important insights about significant changes in asset values of firms as a leading indicator, resulting in a higher probability of default, and thus, higher spreads. Fat tails would indicate that market perception of downside equity risk has increased.

The paper is structured as follows. The next section reviews the shortcomings of the standard approach of measuring dependence and introduces the rationale for the specification of two alternative methods — entropy correlation and tail dependence, which redress these shortcomings against the background of the general objectives of measuring systemic risk. The third section of the paper provides key findings from applying these non-parametric dependence measures to a sample of globally active banks and insurance companies and discusses their implications for systemic risk analysis during the first phase of the financial crisis until the end of 2008. The final section summarizes the paper and provides a view regarding the effective use of early warning indicators for macroprudential surveillance.