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.     

Contemporaneous aggregation and long-memory property of returns and volatility in the Korean stock market

The principal objective of this study is to determine whether the long-memory property is real or a spurious result caused by contemporaneous aggregation. In order to assess the presence of long memory in returns and volatility, two different long-memory detection techniques (modified R/S analysis and the GPH test) were applied to the KOSPI 50 index and its 50 constituent individual stock prices. According to the empirical evidence gleaned from the two long-memory tests, we conclude that there exists significant evidence for the long-memory property in volatility in both the market index and in a majority of individual stocks. These findings indicate that the observed evidence of the long-memory feature in volatility of index series is not spurious, and that we can reject the hypothesis that spurious long-memory evidence in the volatility of index series is the consequence of contemporaneous aggregation. However, this conclusion should be considered cautiously, given that a considerable number of the individual stock volatilities in square returns strongly show a short-memory property, as the level of significance in statistical decisions is lowered to the 1% level.     

Forecasting volatility of fuel oil futures in China: GARCH-type,SV or realized volatility models?

In most previous works on forecasting oil market volatility, squared daily returns were taken as the proxy of unobserved actual volatility. However, as demonstrated by Andersen and Bollerslev (1998) [22], this proxy with too high measurement noise could be perfectly outperformed by a so-called realized volatility (RV) measure calculated by the cumulative sum of squared intraday returns. With this motivation, we further extend earlier works by employing intraday high-frequency data to compare the performance of three typical volatility models in the daily out-of-sample volatility forecasting of fuel oil futures on the Shanghai Futures Exchange (SHFE): the GARCH-type, stochastic volatility (SV) and realized volatility models. By taking RV as the proxy of actual daily volatility and then computing forecasting errors, we find that the realized volatility model based on intraday high-frequency data produces significantly more accurate volatility forecasts than the GARCH-type and SV models based on daily returns. Furthermore, the SV model outperforms many linear and nonlinear GARCH-type models that capture long-memory volatility and/or the asymmetric leverage effect in volatility. These results also prove that abundant volatility information is available in intraday high-frequency data, and can be used to construct more accurate oil volatility forecasting models.     

Modeling long-range memory trading activity by stochastic differential equations

We propose a model of fractal point process driven by the nonlinear stochastic differential equation. The model is adjusted to the empirical data of trading activity in financial markets. This reproduces the probability distribution function and power spectral density of trading activity observed in the stock markets. We present a simple stochastic relation between the trading activity and return, which enables us to reproduce long-range memory statistical properties of volatility by numerical calculations based on the proposed fractal point process.     

A study about the existence of the leverage effect in stochastic volatility models

The empirical relationship between the return of an asset and the volatility of the asset has been well documented in the financial literature. Named the leverage effect or sometimes risk-premium effect, it is observed in real data that, when the return of the asset decreases, the volatility increases and vice versa.Consequently, it is important to demonstrate that any formulated model for the asset price is capable of generating this effect observed in practice. Furthermore, we need to understand the conditions on the parameters present in the model that guarantee the apparition of the leverage effect.In this paper we analyze two general specifications of stochastic volatility models and their capability of generating the perceived leverage effect. We derive conditions for the apparition of leverage effect in both of these stochastic volatility models. We exemplify using stochastic volatility models used in practice and we explicitly state the conditions for the existence of the leverage effect in these examples.     

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.     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

“Exact” and Approximate Methods for Bayesian Inference: Stochastic Volatility Case Study

We conduct a case study in which we empirically illustrate the performance of different classes of Bayesian inference methods to estimate stochastic volatility models. In particular, we consider how different particle filtering methods affect the variance of the estimated likelihood. We review and compare particle Markov Chain Monte Carlo (MCMC), RMHMC, fixed-form variational Bayes, and integrated nested Laplace approximation to estimate the posterior distribution of the parameters. Additionally, we conduct the review from the point of view of whether these methods are (1) easily adaptable to different model specifications; (2) adaptable to higher dimensions of the model in a straightforward way; (3) feasible in the multivariate case. We show that when using the stochastic volatility model for methods comparison, various data-generating processes have to be considered to make a fair assessment of the methods. Finally, we present a challenging specification of the multivariate stochastic volatility model, which is rarely used to illustrate the methods but constitutes an important practical application.     

Role of scaling in the statistical modelling of finance

Modelling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an autoregressive strategy. Results are fully consistent with volatility clustering and with the multiscaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.      

The non-random walk of stock prices: the long-term correlation between signs and sizes

We investigate the random walk of prices by developing a simple model relating the properties of the signs and absolute values of individual price changes to the diffusion rate (volatility) of prices at longer time scales. We show that this benchmark model is unable to reproduce the diffusion properties of real prices. Specifically, we find that for one hour intervals this model consistently over-predicts the volatility of real price series by about 70%, and that this effect becomes stronger as the length of the intervals increases. By selectively shuffling some components of the data while preserving others we are able to show that this discrepancy is caused by a subtle but long-range non-contemporaneous correlation between the signs and sizes of individual returns. We conjecture that this is related to the long-memory of transaction signs and the need to enforce market efficiency.     

Asymmetry and long-memory volatility: Some empirical evidence using GARCH

This paper investigates the asymmetry and long-memory volatility behavior of the Malaysian Stock Exchange daily data over a period of 1991–2005. The long-spanning data set enable us to examine piecewise before, during and after the economic crisis encountered in the Malaysian stock market. The daily index returns are adjusted for infrequent trading effect and the estimated Hurst's parameter allows us to rank the market efficiency across the periods. The leverage effect, clustering volatility and long-memory behavior of the volatility are fitted by the asymmetry GARCH models and GARCH with the inclusion of realized volatility at the final period. Across the periods, the results show the mixture of symmetry and asymmetry GARCH modeling.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

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.     

Volatilities, traded volumes, and the hypothesis of price increments in derivative securities

A detrended fluctuation analysis (DFA) is applied to the statistics of Korean treasury bond (KTB) futures from which the logarithmic increments, volatilities, and traded volumes are estimated over a specific time lag. In this study, the logarithmic increment of futures prices has no long-memory property, while the volatility and the traded volume exhibit the existence of the long-memory property. To analyze and calculate whether the volatility clustering is due to a inherent higher-order correlation not detected by with the direct application of the DFA to logarithmic increments of KTB futures, it is of importance to shuffle the original tick data of future prices and to generate a geometric Brownian random walk with the same mean and standard deviation. It was found from a comparison of the three tick data that the higher-order correlation inherent in logarithmic increments leads to volatility clustering. Particularly, the result of the DFA on volatilities and traded volumes can be supported by the hypothesis of price changes.     

Heavy-tailed value-at-risk analysis for Malaysian stock exchange

This article investigates the comparison of power-law value-at-risk (VaR) evaluation with quantile and non-linear time-varying volatility approaches. A simple Pareto distribution is proposed to account the heavy-tailed property in the empirical distribution of returns. Alternative VaR measurement such as non-parametric quantile estimate is implemented using interpolation method. In addition, we also used the well-known two components ARCH modelling technique under the assumptions of normality and heavy-tailed (student-t distribution) for the innovations. Our results evidenced that the predicted VaR under the Pareto distribution exhibited similar results with the symmetric heavy-tailed long-memory ARCH model. However, it is found that only the Pareto distribution is able to provide a convenient framework for asymmetric properties in both the lower and upper tails.     

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.     

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.     

Stochastic arbitrage return and its implication for option pricing

The purpose of this work is to explore the role that random arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility “smile” can also be explained in terms of random arbitrage opportunities.