Modeling Copper Prices

This paper introduces a general class of stochastic volatility models that can serve as a basis for modeling and estimating simultaneous equation systems involving multivariate volatility processes. The class consists of processes obtained as monotone polynomial transformations of so-called stochastic autoregressive volatility (SARV) models. The class permits a flexible modeling of volatility and avoids strong distributional assumptions. Most of the standard stochastic volatility models are incorporated in the framework, including the lognormal autoregressive model and stochastic volatility generalizations of GARCH and EGARCH. General conditions for strict statinarity, ergodicity, and the existence of finite lower-order unconditional moments for SARV are established. Closed-form expressions for the lowerorder moments constitute the basis for the proposed generalized method of moments estimation procedure. As an illustration we consider an informationdriven model of the return volatility–trading volume system for financial assets. Parameters with structural interpretation are estimated and an extended Kalmanfilter procedure allows for the construction of volatility forecasts based on the bivariate return-volume series. Empirical results are based on daily NYSE data for IBM over 1973–1991.

Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps

ABSTRACTWe introduce a novel class of stochastic volatility models, which can utilize and relate many high‐frequency realized volatility (RV) measures to latent volatility. Instrumental variable methods provide a unified framework for estimation and testing. We study parameter inference problems in the proposed framework with nonstationary stochastic volatility and exogenous predictors in the latent volatility process. Identification‐robust methods are developed for a joint hypothesis involving the volatility persistence parameter and the autocorrelation parameter of the composite error (or the noise ratio). For inference about the volatility persistence parameter, projection techniques are applied. The proposed tests include Anderson‐Rubin‐type tests and their point‐optimal versions. For distributional theory, we provide finite‐sample tests and confidence sets for Gaussian errors, establish exact Monte Carlo test procedures for non‐Gaussian errors (possibly heavy‐tailed), and show asymptotic validity under weaker assumptions. Simulation results show that the proposed tests outperform the asymptotic test regarding size and exhibit excellent power in empirically realistic settings. The proposed inference methods are applied to IBM's price and option data (2009–2013). We consider 175 different instruments (IVs) spanning 22 classes and analyze their ability to describe the low‐frequency volatility. IVs are compared based on the average length of the proposed identification‐robust confidence intervals. The superior instrument set mostly comprises 5‐min HF realized measures, and these IVs produce confidence sets which show that the volatility process is nearly unit‐root. In addition, we find RVs with higher frequency yield wider confidence intervals than RVs with slightly lower frequency, indicating that these confidence intervals adjust to absorb market microstructure noise. Furthermore, when we consider irrelevant or weak IVs (jumps and signed jumps), the proposed tests produce unbounded confidence intervals. We also find that both RV and BV measures produce almost identical confidence intervals across all 14 subclasses, confirming that our methodology is robust in the presence of jumps. Finally, although jumps contain little information regarding the low‐frequency volatility, we find evidence that there may be a nonlinear relationship between jumps and low‐frequency volatility.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

The tumultuous process of creating exchanges in the ongoing transition to a market economy is necessarily affecting sectors which produce a commodity typically traded on exchanges metal. In Moscow alone, three large special metal exchanges have recently begun operation the "Metals Exchange" (June 1991), the "Moscow Nonferrous Metals Exchange" (July 1991), and "Russian Metal" (RTSB November 1991). Specialized metal exchanges have been formed in the Ukraine, Kazakhstan, and other nations of the CIS, and nearly all of the large exchanges have set up special sections for metal products. There is already a trading market for such products, and a keen competition is developing between the different exchanges. The lead in this market will be taken by the exchange that can offer its stockholders and members the best prospects for growth and provide its brokers with the best opportunities for gain. It should be noted that that our leading metal exchanges bear little resemblance to classic international metal exchanges such as the London Metals Exchange (LME). The latter is a typical exchange and is one of the largest commodity trading centers in the world. Thus, the founders and managers of the exchanges in our country are very interested in the mechanism and organizational principles of its operation. The London Metals Exchange was organized in 1882 as the company "Metal Market and Exchange Company, Ltd." The exchange is a wholesale market for goods where deals are made for metal of a certain quantity at market prices. The exchange is a unique barometer of prices on the world nonferrous metals market. It is used to judge and establish the quality of metal and sets forth standard conditions for exchange contracts. The stock capital is 20,000 pounds sterling and is divided into 1000 ordinary shares costing 20 pounds each. The shareholders are all members of the exchange and must own at least two shares to enjoy the full rights of membership. The members of the exchange (about 100 companies) are divided into two categories: individual (principal) members, who have the right to conclude deals in their name; member-representatives (plenipotentiaries), who can make deals in the name of companies or corporations they represent. There is a small group of individual members who are in a priviledged position by virtue of the ability to conclude deals in the trading pit on their personal accounts and, for a fee, execute orders from non-priviledged members and third parties. The commissions thus earned range from 0.1 to 0.5% of the sale price in the case of copper and from 0.25 to 1% of the same in the case of lead and zinc. Non-priviledged members of the exchange generally pay lower brokerage fees than third parties. Shareholders and members of the exchange must be natives and residents of England. Individual members must be at least 21 years of age, as must the member-representatives and directors of the shareholding companies. All company shareholders must be English citizens, while at least 2/3 of the directors must also be citizens and residents of the country. The members of the exchange are chosen by the board of the exchange company by simple majority vote with at least 3/4 of the board present. New members are chosen by open ballot, but a secret ballot can be requested by any member of the board. The features of operation of our exchange stand in sharp contrast to the LME. First of all, a very limited number of metals is traded on the London exchange, and all of them are nonferrous metals. Our metal exchanges as most present domestic exchanges are of a universal character. Not only are all types of metal products traded, but so are many other goods. The exchanges are in essence large trade centers. And although most of the largest domestic exchanges began to specialize after prices were freed and a new law governing exchanges was enacted, they remain more diverse than the LME. This is due first of all to the lack of other channels for the sale of large quantities of metal products at market prices. In addition, the worsening shortage of all of the most important groups of commodities (at least for the short term) makes it almost mandatory that deals be concluded by barter which in turn requires broadening of the range of goods that can be traded.

A New Class of Discrete-Time Stochastic Volatility Model with Correlated Errors

Returns of risky assets and their time-dependent volatility are often jointly modelled by stochastic volatility models (SVMs). Over the last few decades, several SVMs have been proposed to adequately capture the features of the relationship between the return and its volatility. The earliest SVM considers a hierarchical model, with current return as a function of the current latent volatility, which is further modelled as an auto-regressive process. In an attempt to make SVM more appropriate for complex realistic market behaviour, leverage parameter was introduced, which however led to violation of martingale difference property of the risky part of return (a necessary mean-zero condition that prevents arbitrage opportunities under mild regularity conditions). Subsequently, alternative SVMs had been developed and are currently in use. In this article, we propose mean-corrections for several generalizations of SVM with leverage that capture the complex market behaviour as well as satisfy martingale difference property of returns. We also establish a few theoretical results to characterize the key desirable features of these models, and present comparison with other popular competitors. Furthermore, three real-life examples (CITI bank stock price, euro–USD rate and S&P 500 index returns) have been used to demonstrate the performance of this new class of SVMs.

This paper provides comparative results on prices, hedging strategies and risks for local risk-minimisation and mean-variance hedging for a class of stochastic volatility models. A pricing and hedging framework is presented for both approaches with detailed analysis undertaken for the well-known Heston and Stein/Stein stochastic volatility models. These illustrate important quantitative differences between the two approaches.

The motivation of this paper is to identify the effect of treatment charge (TC) on LME (London Metal Exchange) copper prices. It is a fundamental variable as a supply side factor, because it is related to the smelting process and reflects the level of concentrates market tightness. To examine this question carefully, the regression model is applied. This paper finds a statistically significant negative link between TC and LME copper prices. It is found that a 10% increase in TC of copper decreases in copper return by 1.8%. Subsequently, the vector autoregression (VAR) model is introduced to consider the impact of TC to copper prices as a permanent effect. It is found that the negative impact of the TC to copper returns dies out quickly. The statistical estimation in this article will provide a good reference for future study.

In the past years, there has been an extensive investigation of the class of stochastic volatility models for the evaluation of options and complex derivatives. These models have proven to be extremely useful in generalizing the classic Black–Scholes economy and accounting for discrepancies between observation and predictions in the simple log-normal, constant-volatility model. In this paper, we study the structure of an options market with a stochastic volatility that will eventually vanish (i.e., reaches zero) for very short periods of time with probability of one. We investigate the form of pricing measures in this situation, first in a simple binomial case, and then for a diffusion model, by constructing a weak approximation in discrete space and continuous time. The market described allows fleeting arbitrage opportunities, since a vanishing volatility prevents the construction of an equivalent measure, so that pricing contingent claims are, a priori, not obvious. Nevertheless, we can still produce a fair pricing equation. Let us note that this issue is not only of theoretical relevance, as the phenomenon of very low volatility has indeed been observed in the financial markets and the economy for quite a long time in the recent past.

This paper introduces a new class of stochastic volatility models for asset prices, the generalized Heston Nandi GARCH (GHN‐GARCH), with the primary objective of optimal dynamic asset allocation under expected utility theory for constant relative risk aversion investors. We study some of its theoretical properties, and demonstrate that the GHN‐GARCH class of models permits closed‐form solutions for optimal allocation and value function. We introduce and study in more detail an example of this class, the 4/2‐HN‐GARCH model, inspired by the continuous‐time 4/2 stochastic volatility model of Grasselli. A robust parameter estimation procedure is developed, and a numerical analysis is performed.

The aim of this thesis is to develop efficient valuation methods for financial contracts under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need to be priced and hedged under models that are flexible enough to describe the observed option prices at speeds close to real time. To do so, hundreds of vanilla options, which are quoted in terms of implied volatility, need to be calibrated to market prices quickly and accurately on a regular basis. With this in mind we develop efficient methods for the evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent options. Firstly, we derive a new numerical method for the classical problem of pricing vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier option pricing method with respect to speed appears to favour our newly developed method in the cases considered. Secondly, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. In order to achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than is possible using other common methods. Error estimates are presented for a wide range of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to estimate the expectations of path-dependent functionals, such as first-passage times and occupation times, under a class of stochastic volatility models with jumps. The method is based on a recursive approximation of the first-passage time probabilities and expected occupation times of Levy bridge processes that relies in part on a randomisation of the timeparameter. We derive the explicit form of the recursive approximation in the case of bridge processes corresponding to the class of Levy processes with mixed-exponential jumps, and

ABSTRACTWe study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of An(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).

Geometric Asian options are a type of option where the payoff depends on the geometric mean of the underlying asset over a certain period of time. This paper is concerned with the pricing of such options for the class of Volterra-Heston models, covering the rough Heston model. We are able to derive semi-closed formulas for the prices of geometric Asian options with fixed and floating strikes for this class of stochastic volatility models. These formulas require the explicit calculation of the conditional joint Fourier transform of the logarithm of the stock price and the logarithm of the geometric mean of the stock price over time. Linking our problem to the theory of affine Volterra processes, we find a representation of this Fourier transform as a suitably constructed stochastic exponential, which depends on the solution of a Riccati-Volterra equation. Finally, we provide a numerical study for our results in the rough Heston model.

We provide and analyze explicit estimators for a class of discretely observed continuous-time stochastic volatility models with jumps. In particular we consider the class of non-Gaussian Ornstein-Uhlenbeck based models, as introduced by Barndorff-Nielsen and Shephard.