On valuing and hedging European options when volatility is estimated directly

We quantify the effects on contingent claim valuation of using an estimator for the unknown volatility σ of a geometric Brownian motion (GBM) process. The theme of the paper is to show what difficulties can arise when failing to account for estimation risk. Our narrative uses a direct estimator of volatility based on the sample standard deviation of increments of the underlying Brownian motion. After replacing the direct estimator into the GBM, we derive the resulting distribution function of the approximated GBM for any time point. This allows us to present post-estimation distributions and valuation formulae for an assortment of European contingent claims that are in accord with many of the basic properties of the underlying risk-neutral process, and yet better reflect the additional uncertainties and risks that exist in the Black-Scholes-Merton paradigm.     

我国原油价格与外国原油价格的波动溢出效应——基于DCC-MGARCH模型分析

本文利用扩展的4维(E)DCC-MGARCH(1,1)模型,分析了四个原油市场(Brent、WTI、Dubai、China)之间的相互波动溢出效应。研究表明,Brent、WTI原油市场对我国市场均有显著的单向波动溢出效应,WTI原油市场比Brent原油市场对我国原油市场的波动溢出效应更明显。我国和美国原油市场波动都是暂时的,而Brent原油市场波动性是持久的。我国原油市场对Dubai原油市场有单向的波动溢出效应。结果还显示,Brent与WTI原油市场有双向波动溢出效应,Brent的波动溢出效应小于WTI波动溢出效应。     

Optimally thresholded realized power variations for Lévy jump diffusion models

Thresholded Realized Power Variations (TPVs) are one of the most popular nonparametric estimators for general continuous-time processes with a wide range of applications. In spite of their popularity, a common drawback lies in the necessity of choosing a suitable threshold for the estimator, an issue which so far has mostly been addressed by heuristic selection methods. To address this important issue, we propose an objective selection method based on desirable optimality properties of the estimators. Concretely, we develop a well-posed optimization problem which, for a fixed sample size and time horizon, selects a threshold that minimizes the expected total number of jump misclassifications committed by the thresholding mechanism associated with these estimators. We analytically solve the optimization problem under mild regularity conditions on the density of the underlying jump distribution, allowing us to provide an explicit infill asymptotic characterization of the resulting optimal thresholding sequence at a fixed time horizon. The leading term of the optimal threshold sequence is shown to be proportional to Lévy’s modulus of continuity of the underlying Brownian motion, hence theoretically justifying and sharpening selection methods previously proposed in the literature based on power functions or multiple testing procedures. Furthermore, building on the aforementioned asymptotic characterization, we develop an estimation algorithm, which allows for a feasible implementation of the newfound optimal sequence. Simulations demonstrate the improved finite sample performance offered by optimal TPV estimators in comparison to other popular non-optimal alternatives.     

On generalisations of the log-Normal distribution by means of a new product definition in the Kapteyn process

We discuss the modification of the Kapteyn multiplicative process using the q$q$-product of Borges [E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340 (2004) 95]. Depending on the value of the index q$q$ a generalisation of the log-Normal distribution is yielded. Namely, the distribution increases the tail for small (when q<1$q<1$) or large (when q>1$q>1$) values of the variable upon analysis. The usual log-Normal distribution is retrieved when q=1$q=1$, which corresponds to the traditional Kapteyn multiplicative process. The main statistical features of this distribution as well as related random number generators and tables of quantiles of the Kolmogorov–Smirnov distance are presented. Finally, we illustrate the validity of this scenario by describing a set of variables of biological and financial origin.     

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.     

Comparing simulation models for market risk stress testing

The subprime crisis has reminded us that effective stress tests should not only combine subjective scenarios with historical data, but also be probabilistic. In this paper, we combine three hypothetical shocks, of varying degrees, with more than six years of daily data on USD-INR and Euro-INR. Our objective is to compare six simulation-based stress models for foreign exchange positions. We find that while volatility-weighted historical simulation is the best model for volatility persistence, jump diffusion based Monte Carlo simulation is better at capturing correlation breakdown. Loss estimates from very fat-tailed distributions are not sensitive to the severity of stress scenarios.     

Modeling electricity spot prices using mean-reverting multifractal processes

We discuss stochastic modeling of volatility persistence and anti-correlations in electricity spot prices, and for this purpose we present two mean-reverting versions of the multifractal random walk (MRW). In the first model the anti-correlations are modeled in the same way as in an Ornstein–Uhlenbeck process, i.e. via a drift (damping) term, and in the second model the anti-correlations are included by letting the innovations in the MRW model be fractional Gaussian noise with H<1/2$H<1/2$. For both models we present approximate maximum likelihood methods, and we apply these methods to estimate the parameters for the spot prices in the Nordic electricity market. The maximum likelihood estimates show that electricity spot prices are characterized by scaling exponents that are significantly different from the corresponding exponents in stock markets, confirming the exceptional nature of the electricity market. In order to compare the damped MRW model with the fractional MRW model we use ensemble simulations and wavelet-based variograms, and we observe that certain features of the spot prices are better described by the damped MRW model. The characteristic correlation time is estimated to approximately half a year.     

The Physical,Chemical and Biological Effects by Room Temperature Ceramic Far‐infrared Ray Emitting Material Irradiated Water: A Pilot Study

Far infrared ray (FIR) is non‐ionizing electromagnetic radiation with wavelengths of 4‐16 μm. Ceramic far infrared ray emitting materials (cFIR) are sources of FIR that exhibit only non‐thermal effects at room temperature. Certain physical, chemical and biological effects of cFIR irradiation were investigated in this study that heretofore has not been well characterized. We demonstrated that cFIR irradiation reduced the size of water clusters, and significantly increased the freezing temperature of water. We also observed an increase in the volatility of a complex mixture of alcohol, water and solutes and recorded elevated total phenol in green tea infusions prepared using cFIR irradiated water. The effects of cFIR irradiated water on living cells were also investigated. The MC3T3‐E1 murine osteoblast cells grown in the presence of cFIR irradiated water exhibited anti‐oxidative effects on H₂O₂‐mediated toxicity, as evidenced by late stage increase in intracellular alkaline phosphatase. The presence of cFIR irradiated water also resulted in significant decrease in COX‐2 production in the chondrosarcoma cell line, SW1353, in response to lipopolysaccharide induction. Based on our findings, the possible anti‐inflammatory effects of cFIR irradiated water and implications of our study with regard to bone and joint health were discussed.     

On dynamics of volatilities in nonstationary GARCH models

This paper precisely characterizes asymptotic behaviors of the volatilities in nonstationary GARCH(1, 1) models. After suitable renormalization, it is shown that the volatility converges in distribution to a non-degenerate limit. This provides more insight into the dynamics of volatilities in nonstationary GARCH models.