Dynamic relations of uncertainty expectations: a conditional assessment of implied volatility indices

This study examines the spillover effects in international financial markets with respect to implied volatility indices. The use of the latter as the basis of integration analysis means that we test market participants’ expectations and not the actual price fluctuations. The empirical analysis, which includes all publicly available implied volatility indices, employs the dynamic conditional correlation model of Engle (2002) and its findings suggest that there is significant integration of investors’ expectations about future uncertainty. Furthermore, by accounting for the dynamic volatility of implied volatility inter-dependencies, we are able to reveal possible shifts in conditional correlations of market expectations over time. More specifically, our findings show a slight increase in the conditional correlations for all the volatility indices under review over the years and prove that in periods of turbulence in the financial markets the conditional correlations across implied volatility indices increase.     

Pricing American options under stochastic volatility and stochastic interest rates

We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters, this method can accurately price options with time-to-maturity up to several years. The main advantage of our approach over existing methods lies in its straightforward extension to models with stochastic volatility and stochastic interest rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility, and correlations on the American put price.     

Characteristics of permanent and transitory returns in oil-sensitive emerging stock markets: The case of GCC countries

The estimates suggest that for both return components there exists a statistically significant high volatility regime for all the Gulf Cooperation Council (GCC) stock markets and the oil market. On the other hand, the results for the low volatility state of both components are mixed. The individual GCC markets vary in terms of sensitivity to volatility and its duration; with Saudi Arabia and Oman having the highest overall return volatility. All the GCC markets are much less volatile than that of the more open, crisis-ridden, oil-exporting Mexico. All GCC returns move in the same direction, whether in terms of total return, fundamentals or fads under both volatility regimes. The correlations between themselves and with Mexico, the oil price and the Morgan Stanley Capital International Index (MSCI) returns are weak compared to the correlations among stock returns of Germany, Japan UK and the US [Bhar, R., Hamori, S., 2004. Empirical characteristics of the permanent and transitory components of stock returns: analysis in a Markov-switching heteroscedasticity framework. Economics Letters 82, 157–165]. Mexico has considerably higher correlation with both MSCI and the oil price than all the GCC countries.     

Pricing a CDO on stochastically correlated underlyings

In this paper, we propose a method to price collateralized debt obligations (CDO) within Merton's structural model on underlyings with a stochastic mean-reverting covariance dependence. There are two key elements in our development, first we reduce dimensionality and complexity using principal component analysis on the assets' covariance matrix. Second, we approximate this continuous multidimensional structure using a tree method. Trinomial-tree models can be developed for both the principal components and the eigenvalues assuming the eigenvectors are constant over time and the eigenvalues are stochastic. Our method allows us to compute the joint default probabilities for k defaults of stochastically correlated underlyings and the value of CDOs in a fast manner, without having lost much accuracy. Furthermore we provide a method based on moments to estimate the parameters of the model.     

Dynamics of foreign exchange implied volatility and implied correlation surfaces

The prices of currency options expressed in terms of their implied volatilities and the implied correlations between foreign exchange rates at a given point in time depend on option delta and time to maturity. Implied volatilities and implied correlations likewise may thus be represented as a surface. It is well known that these surfaces exhibit both skew/smile features and term structure effects and their shapes fluctuate substantially over time. Using implied volatilities on three currency pairs as well as historical implied correlation values between them, we study the nature of these fluctuations by applying a Karhunen-Loève decomposition that is a generalization of a principal component analysis. We demonstrate that the largest share in the dynamics of these surfaces' fluctuations may be explained by exactly the same three factors, providing evidence of strong interdependences between implied correlation and implied volatility of global currency pairs.     

国际原油价格对我国新能源行业股票价格的波动溢出效应研究

在大气污染日益严峻的情况下,新能源行业受政府大力支持和投资者青睐。新能源与原油一定程度上互为替代品,理论上国际原油价格必然对我国新能源行业股票价格有显著的波动溢出效应,但有些学者却持反对态度,认为我国股票市场对外还没有完全开放,新能源行业发展又很不成熟,所以该溢出效应很难显著。文章运用VAR- Asymmetric- BEKK模型进行比较研究得出：在未去除我国整体股市行情因素时,国际原油价格波动对我国新能源行业股票价格波动溢出效应不显著;而在去除我国整体股行情因素时,国际原油价格波动对我国新能源行业股票价格波动溢出效应在1%显著性水平下显著。表明存在从国际原油价格向我国新能源行业股票价格的波动溢出效应,只是该溢出效应被我国股市总体行情掩盖了。     

Volatility linkage across global equity markets

This paper analyzes the volatility linkage across the U.S., European, German, Japanese, and Swiss equity markets from 1999 to 2009. Both the unconditional and conditional correlations exhibit large fluctuations during the sample period. The results from the VAR analysis show an asymmetric two-way relation between the VIX and other market volatility indices, in which VIX has a larger impact in both the tranquil and crisis times. The structure of the volatility correlation before and during the recent global financial crisis does not show significant changes. In addition, robust test results from realized volatilities confirm the results from implied volatility indices.     

The role of gold futures in mitigating the impact of economic uncertainty on spot prices: Evidence from China

This paper studies the co-integration relationship and volatility spillover effect between China's gold futures and spot prices through the VECM-BEKK-GARCH model. Then, MSGARCH and DCCE-GARCH are applied to study the relationship among China's gold futures market, spot market price volatility and the stabilization effect in uncertain economic environments. This paper enriches the current research, providing gold market participants with hints to address economic uncertainty. The empirical results show that China's gold futures market has a weak stabilization effect on spot price volatility. In scenarios with uncertain economic information and uncertain macroeconomic changes, the correlation between gold futures and spot price volatility is reduced in China, and the role of gold futures in stabilizing the spot price weakens. Furthermore, with economic uncertainty, the fluctuation range of the gold futures price is greater than that of the spot price, with a tendency of more frequent fluctuations. This also means that the effectiveness of the futures market in regulating the spot price will be reduced, and gold market regulators need to stabilize the market through alternative methods to futures.     

Linkages between Indian and US financial markets: impact of global financial crisis and Eurozone debt crisis

This paper examines inter-linkages between Indian and US equity, foreign exchange and money markets using the vector autoregressive-multivariate GARCH-BEKK framework. We investigate the impact of global financial crisis (GFC) and Eurozone debt crisis (EZDC) on the conditional volatility and conditional correlation estimates derived from the multivariate GARCH model for Indian and US financial markets. Our results indicate that there is significant bidirectional causality-in-mean between the Indian stock market returns and the Rs./USD market returns, and significant unidirectional causality-in-mean from the US stock market returns to the Indian stock market returns. As regards volatility spillovers, we find that volatility in the Indian stock market rises in response to domestic as well as US financial market shocks but Indian financial market shocks do not impact the US markets. Further, impact of the recent crisis episodes on the covariance matrix is found to be significant. We find that volatility in the Indian and US financial markets significantly amplified during GFC. The conditional correlations across asset markets were significantly accentuated in the wake of the two crisis episodes. The impact of GFC on cross-market conditional correlations is higher for majority of the asset market pairs in comparison to the EZDC.     

Option pricing with Weyl–Titchmarsh theory

In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrödinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.     

Option pricing under fast-varying and rough stochastic volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.     

Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work. This revised version was published online in August 2006 with corrections to the Cover Date.     

信贷与资产价格关系研究——来自中国房地产市场的证据

货币政策与资产价格之间的关系一直以来都是学术界研究的重点,但从房地产信贷视角分析房地产价格的文献却相对较少。本文基于多元MGARCH—BEKK模型和GRACH均值方程模型分析了房地产信贷、货币供应量与房地产价格的波动相关性以及它们的各种波动对房地产价格的影响。研究发现,房地产信贷增长的波动能影响房地产价格的增长,而货币供应量的波动,对房地产价格增长影响不显著。同时实证分析显示在对房价的调控中,房地产信贷的调控是抑制房价波动的一个工具选项。以银行信贷为主的货币供应量已经不能全面反映社会的流动性状况。与货币供应量相比,社会融资总量指标与实体经济指标的联系更加紧密。     

Implied integrated variance and hedging

If the volatility is stochastic, stock price returns and European option prices depend on the time average of the variance, i.e. the integrated variance, not on the path of the volatility. Applying a Bayesian statistical approach, we compute a forward-looking estimate of this variance, an option-implied integrated variance. Simultaneously, we obtain estimates of the correlation coefficient between stock price and volatility shocks, and of the parameters of the volatility process. Due to the convexity of the Black–Scholes formula with respect to the volatility, pricing and hedging with Black–Scholes-type formulas and the implied volatility often lead to inaccuracies if the volatility is stochastic. Theoretically, this problem can be avoided by using Hull–White-type option pricing and hedging formulas and the integrated variance. We use the implied integrated variance and Hull–White-type formulas to hedge European options and certain volatility derivatives.     

Volatility and covariation of financial assets: A high-frequency analysis

Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance–covariance matrix of n assets. We propose a Kalman-filter-based methodology that allows us to deconstruct price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.     

Volatility and correlation in international stock markets and the role of exchange rate fluctuations

This paper develops a direct, explicit model for the role of exchange rate fluctuations in international stock markets and examines how and to what extent volatility and correlations in equity markets are influenced by exchange rate fluctuations. Evidence presented in this paper indicates that a higher foreign exchange rate variability mostly increases local stock market volatility but decreases volatility for the US stock market. The extent to which stock market volatility is influenced by foreign exchange variability is greater for local markets than for the US market, due to the fact that exchange rate changes are more strongly correlated with local equity market returns than the US market returns. We find that a higher exchange rate fluctuation marginally decreases the US/local equity market correlation. While exchange rate fluctuations held a relatively large fraction of the variation in local stock market returns, there was no significant influence on the US/local equity market correlation.     

Correlations among cryptocurrencies: Evidence from multivariate factor stochastic volatility model

This paper is the first study to apply the multivariate factor stochastic volatility model (MFSVM) for analyzing the correlations among six cryptocurrencies. We use MFSVM with the Bayesian estimation procedure for the period from August 8, 2015, to January 1, 2020. According to the findings, there is a significant positive correlation between price volatility values of Bitcoin and Litecoin. Besides, the volatility values of Ethereum have a positive correlation with both Ripple and Stellar. There is also a positive correlation between the volatility values of Ripple and Dash. These findings are robust to consider different correlation networks. The evidence implies that Bitcoin is mainly related to Litecoin, but Ethereum is associated with other cryptocurrencies.     

Theory and evidence on the dynamic interactions between sovereign credit default swaps and currency options

Using sovereign CDS spreads and currency option data for Mexico and Brazil, we document that CDS spreads covary with both the currency option implied volatility and the slope of the implied volatility curve in moneyness. We propose a joint valuation framework, in which currency return variance and sovereign default intensity follow a bivariate diffusion with contemporaneous correlation. Estimation shows that default intensity is much more persistent than currency return variance. The market price estimates on the two risk factors also explain the well-documented evidence that historical average default probabilities are lower than those implied from credit spreads.     

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

Empirical properties of large covariance matrices

The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following. The spectrum of the covariance is very static, except for the top three to 10 eigenvalues, and decay exponentially fast toward zero. The mean spectrum and spectral density show no particular feature that would separate ‘meaningful’ from ‘noisy’ eigenvalues. The spectrum of the correlation is more static, with three to five eigenvalues that have distinct dynamics. The mean projector of rank k on the leading subspace shows that a large part of the dynamics occurs in the eigenvectors. Together, this implies that the reduction of the covariance to a few leading static eigenmodes misses most of the dynamics. Finally, all the analysed properties of the dynamics of the covariance and correlation are similar. This indicates that a covariance estimator correctly evaluates both volatilities and correlations, and separate estimators are not required.