Beyond cash-additive risk measures: when changing the numéraire fails

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on value-at-risk and tail value-at-risk on L ^p spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.     

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.     

The market price of risk and the equity premium: A legacy of the Great Depression?

By positing learning and a pessimistic initial prior, we build a model that disconnects a representative consumer's subjective attitudes toward risk from the high price of risk that a rational-expectations econometrician would deduce from financial market data. We follow Friedman and Schwartz [1963. A Monetary History of the United States, 1857-1960. Princeton University Press, Princeton, NJ] in hypothesizing that the Great Depression heightened fears of economic instability. We use a robustness calculation to elicit a pessimistic prior for a representative consumer and let him update beliefs via Bayes’ law. Learning eventually erases pessimism, but while it persists, pessimism contributes a volatile multiplicative component to the stochastic discount factor that a rational-expectation econometrician would detect. With sufficient initial pessimism, the model generates substantial values for the market price of risk and equity premium and predicts high Sharpe ratios and forecastable excess stock returns.     

On the exposure of insurance companies to sovereign risk—Portfolio investments and market forces

A sovereign debt crisis can have significant knock-on effects in the financial markets and put financial stability at risk. This paper focuses on the transmission of sovereign risk to insurance companies as some of the largest institutional investors in the sovereign bond market. We use a firm level panel dataset that covers large insurance companies, banks and non-financial firms from nine countries over the time period from 1 January 2008–1 May 2013. We find significant and robust transmission effects from sovereign risk to domestic insurers. The impact on insurers is not significantly different from that on banks but larger than for non-financial firms. We find that systemically important insurers are more closely linked to the domestic sovereign. Based on European data, we show that risks in sovereign bond portfolios are an important driver of insurer risk, which is not reflected in current insurance regulation (incl. Solvency II in Europe).     

The price of carbon risk: Evidence from China’s bond market

Using a 2009–2019 sample of Chinese bond issuers, we examine the effect of carbon risk on bond financing costs. Relative to low carbon risk issuers, high carbon risk issuers have substantially larger bond credit spreads, mainly because their credit risk is greater and they invest the funds in non-green projects. This positive relationship is more pronounced for issuers with financing constraints, those not making a green transition and those in cities with stringent environmental regulations. We find a reversed effect during the COVID-19 pandemic. However, China’s carbon peak and carbon neutral goals have renewed the focus on carbon risk. Carbon risk also causes bond issuers to scale back production and negatively affects their likelihood of receiving long-term financial support. Our findings suggest that investors consider carbon risk and charge a corresponding risk premium.     

Why do we smile? On the determinants of the implied volatility function

We report simple regressions and Granger causality tests in order to understand the pattern of implied volatilities across exercise prices. We employ all calls and puts transacted between 16:00 and 16:45 on the Spanish IBEX-35 index from January 1994 to April 1996. Transaction costs, proxied by the bid–ask spread, seem to be a key determinant of the curvature of the volatility smile. Moreover, time to expiration, the uncertainty associated with the market and the relative market momentum are also important variables in explaining the smile.     

Can habit formation under complete market integration explain the cross-section of international equity risk premia?

In this article, we analyse whether the prominent habit formation model of Campbell and Cochrane (1999) can explain the cross-section of the G7 equity risk premia when formulated under the assumption of complete capital market integration. We test the conditional covariance representation of the model using a combined GARCH and GMM approach in the spirit of Bali (2008) and find that in comparison to the CAPM and the standard power utility CCAPM the habit model has superior explanatory power. It explains more than 90% of the cross-sectional variation in risk premia. Overall, our findings suggest that global consumption-based recession indicators and not returns of reference portfolios are key risk factors driving equity risk premia.     

Are regime-shift sources of risk priced in the market?

In this paper we develop a discrete-time pricing model for European options where the log-return of the underlying asset is subject to discontinuous regime shifts in its mean and/or volatility which follow a Markov chain. The model allows for multiple regime shifts whose risk cannot be hedge out and thus must be priced in option market. The paper provides estimates of the price of regime-shift risk coefficients based on a joint estimation procedure of the Markov regime-switching process of the underlying stock and the suggested option pricing model. The results of the paper indicate that bull-to-bear and bear-to-crash regime shifts carry substantial prices of risk. Risk averse investors in the markets price these regime shifts by assigning higher transition (switching) probabilities to them under the risk neutral probability measure than under the physical. Ignoring these sources of risk will lead to substantial option pricing errors. In addition, the paper shows that investors also price reverse regime shifts, like the crash-to-bear and bear-to-bull ones, by assigning smaller transition probabilities under the risk neutral measure than the physical. Finally, the paper evaluates the pricing performance of the model and indicates that it can be successfully employed, in practice, to price European options.     

No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale numéraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale numéraire under the original probability \(P\) becomes a local martingale numéraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.     

Does the market understand the ex ante risk of expropriation by controlling shareholders?

We examine how the market values operating assets in the presence of time-varying ex ante risk that these assets may be tunneled away. We analyze pairs of Chinese publicly listed firms and their non-listed parents and examine the market valuation of current assets (cash balances, trade receivables, receivables due from the controlling shareholders, inventories) and fixed assets on the publicly listed firm's balance sheet. Our results show that in periods when the risk of tunneling from the publicly listed firm to its controlling shareholder increases, operating assets that are easy to tunnel (cash and receivables due from the controlling shareholder) are valued at larger discounts, while operating assets that are not easy to tunnel (trade receivables, inventories, fixed assets) are not valued at such discounts.     

Edgeworth type expansion of ruin probability under Lévy risk processes in the small loading asymptotics

This paper presents an asymptotic expansion of the ultimate ruin probability under Lévy insurance risks as the loading factor tends to zero. The expansion formula is obtained via the Edgeworth type expansion for compound geometric distributions. We give higher-order expansion of the ruin probability, any order of which is available in explicit form, and discuss a certain type of validity of the expansion. We shall also give applications to evaluation of the VaR-type risk measure due to ruin, and the scale function of spectrally negative Lévy processes.     

Asymptotics of the probability of minimizing ‘down-side’ risk under partial information

We consider minimizing the probability of falling below a target growth rate of the wealth process up to a time horizon T in an incomplete market model under partial information and then study the asymptotic behavior of the minimizing probability as T → ∞. This problem is closely related to an ergodic risk-sensitive stochastic control problem under partial information in the risk-averse case. Indeed, in our main theorem we relate the former problem to the latter as its dual. As a result we obtain an explicit expression for the limit value of the former problem in the case of linear Gaussian models.     

Does information risk affect the implied cost of equity capital? An analysis of PIN and adjusted PIN

Using a unique dataset of Korean listed companies for which trade initiators are correctly identifiable, we estimate bias-free PIN (probability of informed trading) that is no longer subject to the trade misspecification problem and test whether it is related to expected returns. Unlike prior studies, we find that bias-free AdjPIN, the adjusted PIN purged of a liquidity component, is positively related to implied cost of equity. Our findings suggest that the errors in PIN variables hamper a proper identification of PIN pricing in prior studies.     

Managerial market timing under credit risk: How do timed buybacks and stock issuances influence the value of long-term shareholders?

Using discounted cash flow valuation together with a Merton-style credit risk model this paper quantifies the effects of credit risk on timed equity buybacks and issuances. Assumed managers act in best interest of their long-term shareholders and do have superior information, the potential value gain for long-term shareholders out of market timing with and without credit risk is evaluated. As demonstrated in a real-world example, credit risk is important. Ignoring credit risk results in value gains of buybacks motivated by smaller (larger) mispricing levels being underestimated (overestimated), while value gains of issuances driven by lower (larger) mispricings are overestimated (underestimated). These effects are stronger for firms with higher debt levels.     

Is consumption risk priced in the stock market?

The paper examines whether the risk in the consumption of stockholders caused by incomplete consumption insurance is priced in the cross-section of average stock returns. Using Taylor series expansion of the average marginal utility of consumption, we show that the risk in the consumption of stock market participants can be decomposed into two components, insurable (hedgeable using financial assets) and uninsurable (caused by incomplete consumption insurance) consumption risks. We argue that the growth rate of average consumption may be viewed as a proxy for the insurable component of consumption risk, while the growth rates of the rescaled higher-order cross-sectional consumption distribution moments may be regarded as a multivariate proxy for uninsurable risk in consumption. Exploiting microlevel household quarterly consumption data from the US Consumer Expenditure Survey, we find that both components of consumption risk are significantly priced when the limited stock market participation is taken into account. Neither the insurable and uninsurable components of consumption risk nor the Fama–French risk factors are rejected as capturing important components of systematic risk when tested against each other in an integrated multifactor asset pricing model.     

News announcements and price discovery in the RMB–USD market

Review of Quantitative Finance and Accounting - This study investigates the impact of improved central parity quotations in the Chinese Renminbi on price discovery in onshore (USD/CNY) and offshore...     

Does knowing the volatility states affect the market risk premium?

Mayfield (J Financ Econ 73:465–496, 2004) has devised a method for estimating the market risk premium, based on a variant of Merton’s ICAPM wherein volatility is specified as a two-state Markov process. In this study, we assess Mayfield’s key assumption that investors know the current volatility state with certainty, via empirical testing of the assumption of exogenous Markov-switching in Mayfield’s model. We detect strong evidence of endogenous switching. This indicates that investors infer the current volatility state, as opposed to simply observing it. We also find that the risk premium estimates are affected by the switching type.     

Option pricing under stochastic volatility and tempered stable Lévy jumps

The purpose of this paper is to introduce a stochastic volatility model for option pricing that exhibits Lévy jump behavior. For this model, we derive the general formula for a European call option. A well known particular case of this class of models is the Bates model, for which the jumps are modeled by a compound Poisson process with normally distributed jumps. Alternatively, we turn our attention to infinite activity jumps produced by a tempered stable process. Then we empirically compare the estimated log-return probability density and the option prices produced from this model to both the Bates model and the Black–Scholes model. We find that the tempered stable jumps describe more precisely market prices than compound Poisson jumps assumed in the Bates model.