Narrow Results

We have investigated the leverage effects of three major financial markets within a time frame from 2000 to 2012 throughout the 2008 financial crisis. First, dividing the considered time into four consecutive periods, we find the leverage effects of markets exhibiting similar pattern at various periods. Second, splitting the yield data into the positive-return and negative-return series, we find these two series always show anti-leverage effect. The anti-leverage effect of negative-return series usually dominates over the positive one, reflecting people at most times are more sensitive to bad news. However, we observe anomalous behavior in approaching the outbreak of crisis, where the positive-return series shows stronger anti-leverage effect, i.e. people become more sensitive to good news instead. Such phenomenology can persist till after the crisis for an immature market, as opposed to a mature market where it disappears before the end of crisis without external intervene. Our results afford insight into the micro-emotion of various financial markets swept through by the financial crisis.

This study develops a GARCH-type model, i.e., the variance-gamma GARCH (VG GARCH) model, based on the two major strands of option pricing literature. The first strand of the literature uses the variance-gamma process, a time-changed Brownian motion, to model the underlying asset price process such that the possible skewness and excess kurtosis on the distributions of asset returns are considered. The second strand of the literature considers the propagation of the previously arrived news by including the feedback and leverage effects on price movement volatility in a GARCH framework. The proposed VG GARCH model is shown to obey a locally risk-neutral valuation relationship (LRNVR) under the sufficient conditions postulated by Duan (1995). This new model provides a unified framework for estimating the historical and risk-neutral distributions, and thus facilitates option pricing calibration using historical underlying asset prices. An empirical study is performed comparing the proposed VG GARCH model with four competing pricing models: benchmark Black–Scholes, ad hoc Black–Scholes, normal NGARCH, and stochastic volatility VG. The performance of the VG GARCH model versus these four competing models is then demonstrated.

We revisit robust replication theory of volatility derivatives and introduce a broader class which may be considered as the second generation of volatility derivatives. One of them is a swap contract on the quadratic covariation between an asset price and the model-free implied variance (MFIV) of the asset. It can be replicated in a model-free manner and its fair strike may be interpreted as a model-free measure for the covariance of the asset price and the realized variance. The fair strike is given in a remarkably simple form, which enable to compute it from the Black–Scholes implied volatility surface. We call it the model-free implied leverage (MFIL) and give several characterizations. In particular, we show its simple relation to the Black–Scholes implied volatility skew by an asymptotic method. Further to get an intuition, we demonstrate some explicit calculations under the Heston model. We report some empirical evidence from the time series of the MFIV and MFIL of the Nikkei stock average.

We conduct an empirical study using the quantile-based correlation function to uncover the temporal dependencies in financial time series. The study uses intraday data for the S&P500 stocks from the New York Stock Exchange (NYSE). After establishing an empirical overview, we compare the quantile-based correlation function to stochastic processes from the GARCH family and find striking differences. This motivates us to propose the quantile-based correlation function as a powerful tool to assess the agreements between stochastic processes and empirical data.

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well-known stylized fact is usually referred to as the leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts for leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics according to a certain threshold. It can be seen as a continuous-time version of the self-exciting threshold autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Parameters estimated on the daily prices of 351 stocks of NYSE and S&P 500, on different time windows, show consistent empirical evidence for leverage effects. Mean-reversion effects are also detected, most markedly in crisis periods.

This paper uses ARCH models to examine if there is a leverage effect and also to test if A- and B-share holdings have different risks in Chinese stock markets before and after B-share markets open to domestic investors in February 2001. The empirical results suggest that leverage effect was not present and shocks have symmetric impact on the volatility of Chinese B-share stock returns in both periods and A-share returns in Period I. Thus GARCH model would be a better model to fit the Chinese B-share stock returns than EGARCH or GJR-GARCH model. But EGARCH or GJR-GARCH model fits recent (Period II) A-share markets data better than GARCH model. Another finding of this paper is that holding A- or B-share bears different risk in returns in the two Chinese markets. Furthermore, news or shocks have a larger impact on volatility of B-share returns in Period I than in Period II.

The connections between stock market volatility and returns are studied within the context of a general equilibrium framework. The framework rules out a priori any purely statistical relationship between volatility and returns by imposing uncorrelated innovations. The main model generates a two-factor structure for stock market volatility along with time-varying risk premiums on consumption and volatility risk. It also generates endogenously a dynamic leverage effect (volatility asymmetry), the sign of which depends upon the magnitudes of the risk aversion and the intertemporal elasticity of substitution parameters.

This paper studies the dynamics of cryptocurrency volatility using a stochastic volatility model with simultaneous and correlated jumps in returns and volatility. We estimate the model using an efficient sequential learning algorithm that allows for learning about multiple unknown model parameters simultaneously, with daily data on four popular cryptocurrencies. We find that these cryptocurrencies have quite different volatility dynamics. In particular, they exhibit different return-volatility relationships: While Ethereum and Litecoin show a negative relationship, Chainlink displays a positive one and interestingly, Bitcoin’s one changes from negative to positive in June 2016. We also provide evidence that the sequential learning algorithm helps better detect large jumps in the cryptocurrency market in real time. Overall, incorporating volatility jumps helps better capture the dynamic behavior of highly volatile cryptocurrencies.

This research utilizes the Autoregressive Moving Average–General Autoregressive Conditional Heteroskedasticity (ARMA–GARCH) and Autoregressive Moving Average–Exponential General Autoregressive Conditional Heteroskedasticity (ARMA–EGARCH) in studying the spillover and leverage effects of returns and volatilities of seven equity exchange-traded notes (ETNs) and their tracked stock indices. This study finds positive returns transmissions between the two investment instruments. Unilateral influence and bilateral relationships also exist that may help investors in finding investment clues to approximate possible movements of ETNs about stock indices and vice versa. This paper also observes negative returns and volatility transmissions that may caution traders in the possible reversal of movement of the other instrument. Disinvestments, transfer of allocation, and inverse investing strategies are some of the possible reasons attributable to this negative relation.

This chapter resolves an inconclusive issue in the empirical literature about the relationship between downside risk and stock returns for Asian markets. This study demonstrates that the mixed signs on the risk coefficient stem from the fact that the excess stock return series is assumed to be stationary with a short memory, which is inconsistent with the downside risk series featuring a long memory process. After we appropriately model the long memory property of downside risk and apply a fractional difference to downside risk, the evidence consistently supports a significant and positive risk–return relation. This holds true for downside risk not only in the domestic market but also across markets. The evidence suggests that the risk premium is higher if the risk originates in a dominant market, such as the US. These findings are robust even when we consider the leverage effect, value-at-risk feedback, and the long memory effect in the conditional variance.

In literature, a GARCH-jump mixture model, namely, the GARCH-jump model with autoregressive conditional jump intensity (GARJI) model, in which two conditional independent processes, i.e., a diffusion and a compounded Poisson process are used to account for stock price movements caused by normal and extreme event news arrivals, individually, is developed by Chan and Maheu (2002, 2004) to describe the volatility clustering and leverage effect phenomenon. The resulting model is less efficient and provides only ex post filter for the probability of the occurrences of large price movements. A more informative and parsimonious model, however, the VG NGARCH model, is proposed and calibrated in this study. Being an extension of the variance-gamma model developed by Madan et al. (1998), the proposed VG NGARCH model incorporates an autoregressive structure on the conditional shape parameters, which describes the news arrival rates of different impact sizes on the price movements, and an ex ante prediction for the occurrences of large price movements is provided. The performance of the proposed VG NGARCH model is compared to the GARJI model based on daily stock prices of five component financial companies in S&P 500, namely, Bank of America, Wells Fargo, J.P. Morgan Chase, CitiGroup, and AIG, respectively, from January 3, 2006 to December 31, 2009. The goodness of fit of the VG NGARCH model and its ability to predict the probabilities of large price movements are demonstrated by comparing with the benchmark GARJI model.

In this chapter, we provide a basic introduction to two fascinating instruments that have mesmerised Wall Street for a while, that of call and put. In their complexities hide great opportunities for trading profits but also great treacherous ways that lead to destruction for those unwary. Together with futures and swaps, these derivatives form the basic building blocks of derivatives jungle. Knowing how they are priced and how their prices may be used to infer information on the market is important.