Explaining asset pricing puzzles associated with the 1987 market crash

The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options, despite minimal changes in aggregate consumption. We explain these events within a general equilibrium framework in which expected endowment growth and economic uncertainty are subject to rare jumps. The arrival of a jump triggers the updating of agents' beliefs about the likelihood of future jumps, which produces a market crash and a permanent shift in option prices. Consumption and dividends remain smooth, and the model is consistent with salient features of individual stock options, equity returns, and interest rates.     

Attention allocation and international stock return comovement: Evidence from the Bitcoin market

We employ extreme Bitcoin returns as exogenous shock events to investigate the impact of investor attention allocation on worldwide stock return comovement. We find that (1) these shock events decrease worldwide stock return comovement, (2) there is an asymmetric effect in which a crash shock event has a greater impact on return comovement than a jump shock event, and (3) the impact of these shock events on equity comovement is more pronounced in emerging markets. Our results suggest that identifying extreme Bitcoin returns will benefit portfolio construction. Our results may be of considerable interest to investors, as well as to academics interested in portfolio diversification, asset comovement, and cryptocurrencies.     

Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks

Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black–Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black–Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at‐the‐money implied volatility is a state variable. We compute risk minimization hedging by modeling at‐the‐money Black–Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.     

Tails,Fears, and Risk Premia

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time‐varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high‐frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out‐of‐the‐money options and new model‐free implied variation measures for estimating the corresponding risk‐neutral expectations.     

Stock volatility and the crash of '87

This article analyzes the behavior of stock return volatilityusing daily data from 1885 through 1988. The October 1987 stockmarket crash was unusual in many ways. October 19 was the largestpercentage change in market value in over 29,000 days. Stockvolatility jumped dramatically during and after the crash. Nevertheless,it returned to lower, more normal levels more quickly than pastexperience predicted. I use data on implied volatilities fromcall option prices and estimates of volatility from futurescontracts on stock indexes to confirm this result.     

Effect of Analysts' Optimism on Estimates of the Expected Rate of Return Implied by Earnings Forecasts

Recent literature has used analysts' earnings forecasts, which are known to be optimistic, to estimate implied expected rates of return, yielding upwardly biased estimates. We estimate that the bias, computed as the difference between the estimates of the implied expected rate of return based on analysts' earnings forecasts and estimates based on current earnings realizations, is 2.84%. The importance of this bias is illustrated by the fact that several extant studies estimate an equity premium in the vicinity of 3%, which would be eliminated by the removal of the bias. We illustrate the point that cross‐sample differences in the bias may lead to the erroneous conclusion that cost of capital differs across these samples by showing that analysts' optimism, and hence, bias in the implied estimates of the expected rate of return, differs with firm size and with analysts' recommendation. As an important aside, we show that the bias in a value‐weighted estimate of the implied equity premium is 1.60% and that the unbiased value‐weighted estimate of this premium is 4.43%.     

Correlated implied volatility with jump and cross section of stock returns

I derive the option‐implied volatility allowing for nonzero correlation between price jump and diffusive risk to examine the information content of implied diffusive, jump risks and their implied covariance in the cross‐sectional variation of future returns. This study documents a strong predictive power of realized volatility and correlated implied volatility spread (RV ? IV_C) in the cross section of stock returns. The difference of realized volatility with the implied diffusive volatility (RV ? σ_C), jump risk (RV ? γ_C) and covariance (RV ? ICov) can forecast future returns. These RV ? σ_C and RV ? γ_C anomalies are robustly persistent even after controlling for market, size, book‐to‐market value, momentum and liquidity factors.     

Retrieving risk neutral moments and expected quadratic variation from option prices

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.     

Modelling conditional heteroscedasticity and jumps in Australian short-term interest rates

The present paper explores a class of jump–diffusion models for the Australian short‐term interest rate. The proposed general model incorporates linear mean‐reverting drift, time‐varying volatility in the form of LEVELS (sensitivity of the volatility to the levels of the short‐rates) and generalized autoregressive conditional heteroscedasticity (GARCH), as well as jumps, to match the salient features of the short‐rate dynamics. Maximum likelihood estimation reveals that pure diffusion models that ignore the jump factor are mis‐specified in the sense that they imply a spuriously high speed of mean‐reversion in the level of short‐rate changes as well as a spuriously high degree of persistence in volatility. Once the jump factor is incorporated, the jump models that can also capture the GARCH‐induced volatility produce reasonable estimates of the speed of mean reversion. The introduction of the jump factor also yields reasonable estimates of the GARCH parameters. Overall, the LEVELS–GARCH–JUMP model fits the data best.     

Information Risk and Fair Values: An Examination of Equity Betas

Using a sample of U.S. financial institutions, we exploit recent mandatory disclosures of financial instruments designated as fair value level 1, 2, and 3 to test whether greater information risk in financial instrument fair values leads to higher cost of capital. We derive an empirical model allowing asset‐specific estimates of implied betas, and find evidence that firms with greater exposure to level 3 financial assets exhibit higher betas relative to those designated as level 1 or level 2. We further find that this difference in implied betas across fair value designations is more pronounced for firms with ex ante lower‐quality information environments: firms with lower analyst following, lower market capitalization, higher analyst forecast errors, or higher analyst forecast dispersion. Overall, the results are consistent with a higher cost of capital for more opaque financial assets, but also suggest that differences in firms' information environments can mitigate information risk across the fair value designations.     

The Impact of Jumps in Volatility and Returns

This paper examines continuous‐time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood‐based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we find strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.     

Robust estimation of intraweek periodicity in volatility and jump detection

Opening, lunch and closing of financial markets induce a periodic component in the volatility of high-frequency returns. We show that price jumps cause a large bias in the classical periodicity estimators and propose robust alternatives. We find that accounting for periodicity greatly improves the accuracy of intraday jump detection methods. It increases the power to detect the relatively small jumps occurring at times for which volatility is periodically low and reduces the number of spurious jump detections at times of periodically high volatility. We use the series of detected jumps to estimate robustly the long memory parameter of the squared EUR/USD, GBP/USD and YEN/USD returns.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Using price distributions to estimate search costs

We show how the equilibrium restrictions implied by standard search models can be used to estimate search‐cost distributions using price data alone. We consider both sequential and non‐sequential search strategies, and develop estimation methodologies that exploit equilibrium restrictions to recover estimates of search‐cost heterogeneity that are theoretically consistent with the search models. We illustrate the method using online prices for several economics and statistics textbooks.     

New evidence on the implied-realized volatility relation

We consider the relation between the volatility implied in an option's price and the subsequently realized volatility. Earlier studies on stock index options have found biases and inefficiencies in implied volatility as a forecast of future volatility. More recently, Christensen and Prabhala find that implied volatility in at-the-money one-month OEX call options on the S&P 100 index in fact is an unbiased and efficient forecast of ex-post realized index volatility after the 1987 stock market crash. In this paper, the robustness of the unbiasedness and efficiency result is extended to a more recent period covering April 1993 to February 1997. As a new contribution, implied volatility is constructed as a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both puts and calls. We run a horse race between implied call, implied put, and historical return volatility. Several robustness checks, including a new simultaneous equation approach, underscore our conclusion, that implied volatility is an efficient forecast of realized return volatility.     

Exchange options under clustered jump dynamics

Exchange options are one of the most popular exotic options, and have important implications for many common financial arrangements and for implied beta as a measure of systematic risk. In this study, we extend the existing literature on exchange options to allow for clustered jump contagion dynamics in each single asset, as well as across assets, using the Hawkes jump-diffusion model. We derive the analytical pricing formulae, the Greeks, and the optimal hedging strategy via Fourier transforms. Using an illustrative numerical analysis, we present the relationship between the exchange option price and clustered jump intensities and jump sizes in the underlying assets. We discuss the managerial insights on financial arrangements with exchange option characteristics. Furthermore, we discuss the implications of incorporating clustered jumps into the estimation of implied beta with exchange options, in which the applications can be insightful and useful in finance practice.     

Is long memory necessary? An empirical investigation of nonnegative interest rate processes

This paper analyzes a class of nonnegative processes for the short-term interest rate. The dynamics of interest rates and yields are driven by the dynamics of the conditional volatility of the pricing kernel. We study Markovian interest rate processes as well as more general non-Markovian processes that display “short” and “long” memory. These processes also display heteroskedasticity patterns that are more general than those of existing models. We find that deviations from the Markovian structure significantly improve the empirical performance of the model. Certain aspects of the long memory effect can be captured with a (less parsimonious) short memory parameterization, but a simulation experiment suggests that the implied term structures corresponding to the estimated long- and short-memory specifications are very different. We also find that the choice of proxy for the short rate affects the estimates of heteroskedasticity patterns.     

How Crashes Develop: Intradaily Volatility and Crash Evolution

This paper explores whether affine models with volatility jumps estimated on intradaily S&P 500 futures data over 1983 to 2008 can capture major daily outliers such as the 1987 stock market crash. Intradaily jumps in futures prices are typically small; self‐exciting but short‐lived volatility spikes capture intradaily and daily returns better. Multifactor models of the evolution of diffusive variance and jump intensities improve fits substantially, including out‐of‐sample over 2009 to 2016. The models capture reasonably well the conditional distributions of daily returns and realized variance outliers, but underpredict realized variance inliers. I also examine option pricing implications.     

The complexity of price discovery in an efficient market: the stock market reaction to the Challenger crash

We provide evidence on the speed and accuracy of price discovery by studying stock returns and trading volume surrounding the crash of the space shuttle Challenger. While the event was widely observed, it took several months for an esteemed panel to determine which of the mechanical components failed during the launch. By contrast, in the period immediately following the crash, securities trading in the four main shuttle contractors seemingly singled out the firm that manufactured the faulty component. We show that price discovery occurred without large trading profits and that much of the price discovery occurred during a trading halt of the firm responsible for the faulty component. Finally, although we document what are arguably quick and accurate movements of the market, we are unable to detect the actual manner in which particular informed traders induced price discovery.     

Implied Risk Aversion Parameter from Option Prices

This paper estimates the representative investor's coefficient of relative risk aversion using option price data. Estimation is carried out using the method of simulated moments. Employing the following assumptions: a) agents have constant proportional risk averse preferences, b) complete markets exist, and c) asset returns are distributed lognormally, an objective function is constructed within the equivalent martingale measure framework. Unlike the case of equity markets, the implied risk aversion parameter from option prices is quite low and stays between zero and one.