Does implied volatility of currency futures option imply volatility of exchange rates?

By investigating currency futures options, this paper provides an alternative economic implication for the result reported by Stein [Overreactions in the options market, Journal of Finance 44 (1989) 1011–1023] that long-maturity options tend to overreact to changes in the implied volatility of short-maturity options. When a GARCH process is assumed for exchange rates, a continuous-time relationship is developed. We provide evidence that implied volatilities may not be the simple average of future expected volatilities. By comparing the term–structure relationship of implied volatilities with the process of the underlying exchange rates, we find that long-maturity options are more consistent with the exchange rates process. In sum, short-maturity options overreact to the dynamics of underlying assets rather than long-maturity options overreacting to short-maturity options.     

A note on Black–Scholes implied volatility

An approximate formula for the Black–Scholes implied volatility is given by means of an asymptotic representation of the Black–Scholes formula. This representation is based on a variable change that reduces the number of meaningful variables from five to three. It is stated clearly which is the family of functions we are going to work, specially the inverse of the normal accumulative function. Estimates for the error in the resulting approximate formulas for both the option value and the volatility are obtained as well.     

Forecasting volatility of SSEC in Chinese stock market using multifractal analysis

In this paper, taking about 7 years’ high-frequency data of the Shanghai Stock Exchange Composite Index (SSEC) as an example, we propose a daily volatility measure based on the multifractal spectrum of the high-frequency price variability within a trading day. An ARFIMA model is used to depict the dynamics of this multifractal volatility (MFV) measures. The one-day ahead volatility forecasting performances of the MFV model and some other existing volatility models, such as the realized volatility model, stochastic volatility model and GARCH, are evaluated by the superior prediction ability (SPA) test. The empirical results show that under several loss functions, the MFV model obtains the best forecasting accuracy.     

Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black-Scholes model with transaction costs

This paper deals with the problem of discrete time option pricing using the fractional Black-Scholes model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained. In addition, the relation between scaling and implied volatility smiles is discussed.     

Dynamics of implied volatility surfaces from random matrix theory

We analyze the dynamics of the implied volatility surface of KOSPI 200 futures options from random matrix theory. To extract the informative data, we use random matrix criteria. Implied volatility data have a colossal eigenvalue, and the order of eigenvalues in a noisy regime is distinguishably smaller than a random matrix theory prediction. We discern the marketwide knowledge of the implied volatility surface movement such as the level, skew, and smile effect. These dynamics has the ergodic property and long range autocorrelation. We also study the relationship between the three implied volatility surface dynamics and the underlying asset dynamics, and confirm the existence of leverage effect even in the short time interval.     

The fractional volatility model: An agent-based interpretation

Based on the criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, some features of the model are reviewed and extended to account for leverage effects. Using agent-based models, one tries to find which agent strategies and (or) properties of the financial institutions might be responsible for the features of the fractional volatility model.     

Long-memory volatility in derivative hedging

The aim of this work is to take into account the effects of long memory in volatility on derivative hedging. This idea is an extension of the work by Fedotov and Tan [Stochastic long memory process in option pricing, Int. J. Theor. Appl. Finance 8 (2005) 381–392] where they incorporate long-memory stochastic volatility in option pricing and derive pricing bands for option values. The starting point is the stochastic Black–Scholes hedging strategy which involves volatility with a long-range dependence. The stochastic hedging strategy is the sum of its deterministic term that is classical Black–Scholes hedging strategy with a constant volatility and a random deviation term which describes the risk arising from the random volatility. Using the fact that stock price and volatility fluctuate on different time scales, we derive an asymptotic equation for this deviation in terms of the Green's function and the fractional Brownian motion. The solution to this equation allows us to find hedging confidence intervals.     

Asymmetric volatility, volatility clustering, and herding agents with a borrowing constraint

Recent empirical research has documented asymmetric volatility and volatility clustering in stock markets. We conjecture that a limit of arbitrage due to a borrowing constraint and herding behavior by investors are related to these phenomena. This study conducts simulation analyses on a spin model where borrowing constrained agents imitate their nearest neighbors but switch their strategies to a different one intermittently. We show that herding matters for volatility clustering while a borrowing constraint intensifies the asymmetry of volatility through the herding effect.     

Modelling financial volatility in the presence of abrupt changes

The volatility of financial instruments is rarely constant, and usually varies over time. This creates a phenomenon called volatility clustering, where large price movements on one day are followed by similarly large movements on successive days, creating temporal clusters. The GARCH model, which treats volatility as a drift process, is commonly used to capture this behaviour. However research suggests that volatility is often better described by a structural break model, where the volatility undergoes abrupt jumps in addition to drift. Most efforts to integrate these jumps into the GARCH methodology have resulted in models which are either very computationally demanding, or which make problematic assumptions about the distribution of the instruments, often assuming that they are Gaussian. We present a new approach which uses ideas from nonparametric statistics to identify structural break points without making such distributional assumptions, and then models drift separately within each identified regime. Using our method, we investigate the volatility of several major stock indexes, and find that our approach can potentially give an improved fit compared to more commonly used techniques.     

Forecasting volatility in Shanghai and Shenzhen markets based on multifractal analysis

This paper analyzes the multifractality in Shanghai and Shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis. We find that the main source of multifractality is long-range correlations of large and small fluctuations. Then, we introduce a multifractal volatility measure (MV) and find that by taking MV as daily conditional volatility, the simulated series displayed similar “stylized facts” to the original daily return series. By capturing the dynamics of MV using the ARFIMA model, we find that the out-of-sample forecasting performance of the ARFIMA-MV model is better than some GARCH-class models and the ARFIMA-RV model under some criteria of loss function.     

Analysis of the efficiency of the Shanghai stock market: A volatility perspective

By applying the rolling window method, we investigate the efficiency of the Shanghai stock market through the dynamic changes of local Hurst exponents based on multifractal detrended fluctuation analysis. We decompose the realized volatility into continuous sample paths and jump components and analyze their long-range correlations of decomposing components. Our results reveal that the efficiency of the Shanghai stock market improved greatly based on the time-varying Hurst exponents.     

Modeling natural gas market volatility using GARCH with different distributions

In this paper, we model natural gas market volatility using GARCH-class models with long memory and fat-tail distributions. First, we forecast price volatilities of spot and futures prices. Our evidence shows that none of the models can consistently outperform others across different criteria of loss functions. We can obtain greater forecasting accuracy by taking the stylized fact of fat-tail distributions into account. Second, we forecast volatility of basis defined as the price differential between spot and futures. Our evidence shows that nonlinear GARCH-class models with asymmetric effects have the greatest forecasting accuracy. Finally, we investigate the source of forecasting loss of models. Our findings based on a detrending moving average indicate that GARCH models cannot capture multifractality in natural gas markets. This may be the plausible explanation for the source of model forecasting losses.     

Stochastic volatility of financial markets as the fluctuating rate of trading: An empirical study

We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log-returns) for a given number of trades N is found to be approximately Gaussian. The probability distribution of N for a given time interval Δt is non-Poissonian and has an exponential tail for large N and a sharp cutoff for small N. Combining these two distributions produces a non-trivial distribution of log-returns for a given time interval Δt, which has exponential tails and a Gaussian central part, in agreement with empirical observations.     

Portfolio management under sudden changes in volatility and heterogeneous investment horizons

We analyze the implications for portfolio management of accounting for conditional heteroskedasticity and sudden changes in volatility, based on a sample of weekly data of the Dow Jones Country Titans, the CBT-municipal bond, spot and futures prices of commodities for the period 1992–2005. To that end, we first proceed to utilize the ICSS algorithm to detect long-term volatility shifts, and incorporate that information into PGARCH models fitted to the returns series. At the next stage, we simulate returns series and compute a wavelet-based value at risk, which takes into consideration the investor's time horizon. We repeat the same procedure for artificial data generated from semi-parametric estimates of the distribution functions of returns, which account for fat tails. Our estimation results show that neglecting GARCH effects and volatility shifts may lead to an overestimation of financial risk at different time horizons. In addition, we conclude that investors benefit from holding commodities as their low or even negative correlation with stock and bond indices contribute to portfolio diversification.     

Clustering of volatility in variable diffusion processes

Increments in financial markets have anomalous statistical properties including fat-tailed distributions and volatility clustering (i.e., the autocorrelation functions of return increments decay quickly but those of the squared increments decay slowly). One of the central questions in financial market analysis is whether the nature of the underlying stochastic process can be deduced from these statistical properties. We have shown previously that a class of variable diffusion processes has fat-tailed distributions. Here we show analytically that such models also exhibit volatility clustering. To our knowledge, this is the first case where clustering of volatility is proven analytically in a model.Our results are compatible with the viewpoint that variable diffusion processes are possible models for financial markets.     

Asymmetric conditional volatility in international stock markets

Recent studies show that a negative shock in stock prices will generate more volatility than a positive shock of similar magnitude. The aim of this paper is to appraise the hypothesis under which the conditional mean and the conditional variance of stock returns are asymmetric functions of past information. We compare the results for the Portuguese Stock Market Index PSI 20 with six other Stock Market Indices, namely the SP 500, FTSE 100, DAX 30, CAC 40, ASE 20, and IBEX 35. In order to assess asymmetric volatility we use autoregressive conditional heteroskedasticity specifications known as TARCH and EGARCH. We also test for asymmetry after controlling for the effect of macroeconomic factors on stock market returns using TAR and M-TAR specifications within a VAR framework. Our results show that the conditional variance is an asymmetric function of past innovations raising proportionately more during market declines, a phenomenon known as the leverage effect. However, when we control for the effect of changes in macroeconomic variables, we find no significant evidence of asymmetric behaviour of the stock market returns. There are some signs that the Portuguese Stock Market tends to show somewhat less market efficiency than other markets since the effect of the shocks appear to take a longer time to dissipate.     

Stock price dynamics and option valuations under volatility feedback effect

According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price–dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonstrate a mechanism by which the market price of diffusion return risk, or an equity risk-premium, affects option prices and empirically illustrate how to identify that mechanism using forward-looking information on option contracts. Our theoretical and empirical results support the relevance of the volatility feedback effect. Overall, the results indicate that the prevailing practice of ignoring the time-varying dividend yield in option pricing can lead to oversimplification of the stock market dynamics.     

Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

The expOU stochastic volatility model is capable of reproducing fairly well most important statistical properties of financial markets daily data. Among them, the presence of multiple time scales in the volatility autocorrelation is perhaps the most relevant which makes appear fat tails in the return distributions. This paper wants to go further on with the expOU model we have studied in Ref. [J. Masoliver, J. Perelló, Quant. Finance 6 (2006) 423] by exploring an aspect of practical interest. Having as a benchmark the parameters estimated from the Dow Jones daily data, we want to compute the price for the European option. This is actually done by Monte Carlo, running a large number of simulations. Our main interest is to “see” the effects of a long-range market memory from our expOU model in its subsequent European call option. We pay attention to the effects of the existence of a broad range of time scales in the volatility. We find that a richer set of time scales brings the price of the option higher. This appears in clear contrast to the presence of memory in the price itself which makes the price of the option cheaper.     

Time-varying volatility in Malaysian stock exchange: An empirical study using multiple-volatility-shift fractionally integrated model

This article investigated the influences of structural breaks on the fractionally integrated time-varying volatility model in the Malaysian stock markets which included the Kuala Lumpur composite index and four major sectoral indices. A fractionally integrated time-varying volatility model combined with sudden changes is developed to study the possibility of structural change in the empirical data sets. Our empirical results showed substantial reduction in fractional differencing parameters after the inclusion of structural change during the Asian financial and currency crises. Moreover, the fractionally integrated model with sudden change in volatility performed better in the estimation and specification evaluations.     

Multiple time scales and the empirical models for stochastic volatility

The most common stochastic volatility models such as the Ornstein–Uhlenbeck (OU), the Heston, the exponential OU (ExpOU) and Hull–White models define volatility as a Markovian process. In this work we check the applicability of the Markovian approximation at separate times scales and will try to answer the question which of the stochastic volatility models indicated above is the most realistic. To this end we consider the volatility at both short (a few days) and long (a few months) time scales as a Markovian process and estimate for it the coefficients of the Kramers–Moyal expansion using the data for Dow-Jones Index. It has been found that the empirical data allow to take only the first two coefficients of expansion to be non-zero that define form of the volatility stochastic differential equation of Itô. It proved to be that for the long time scale the empirical data support the ExpOU model. At the short time scale the empirical model coincides with ExpOU model for the small volatility quantities only.