Variance trading and market price of variance risk

This paper develops a new approach for variance trading. We show that the discretely-sampled realized variance can be robustly replicated under very general conditions, including when the price can jump. The replication strategy specifies the exact timing for rebalancing in the underlying. The deviations from the optimal schedule can lead to surprisingly large hedging errors. In the empirical application, we synthesize the prices of the variance contract on S&P 500 index over the period from 01/1990 to 12/2009. We find that the market variance risk is priced, its risk premium is negative and economically very large. The variance risk premium cannot be explained by the known risk factors and option returns.     

Skewness preference,mean–variance and the demand for put options

This paper compares the mean–variance and the mean–variance–skewness approaches to modelling expected utility. Attention is focused on a problem encountered in risk management: determining the optimal demand for a put option hedging the return on an asset with a negatively skewed return distribution. It is demonstrated theoretically that incorporating positive skewness preference into the decision‐maker's objective function typically produces a reduction in the demand for put options when compared with the mean–variance solution. A state‐dependent example is provided to illustrate how a mean–variance–skewness objective can result in a significant reduction in the optimal amount of crop insurance demanded when compared with the mean–variance solution. Copyright © 1999 John Wiley & Sons, Ltd.     

Variance swaps,non-normality and macroeconomic and financial risks

This paper studies the determinants of the variance risk premium and discusses the hedging possibilities offered by variance swaps. We start by showing that the variance risk premium responds to changes in higher order moments of the distribution of market returns. But the uncertainty that determines the variance risk premium – the fear by investors to deviations from normality in returns – is also strongly related to a variety of macroeconomic and financial risks associated with default, employment growth, consumption growth, stock market and market illiquidity risks. We conclude that the variance risk premium reflects the market willingness to pay for hedging against these financial and macroeconomic sources of risk. An out-of-sample asset allocation exercise shows that the inclusion of the variance swap reduces the modified value-at-risk with respect to a portfolio holding exclusively the equity market portfolio.     

Option hedging theory under transaction costs

The problem of option hedging in the presence of proportional transaction costs can be formulated as a singular stochastic control problem. Hodges and Neuberger [1989. Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8, 222–239] introduced an approach that is based on maximization of the expected utility of terminal wealth. We develop a new algorithm to solve the corresponding singular stochastic control problem and introduce a new approach to option hedging which is closer in spirit to the pathwise replication of Black and Scholes [1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654]. This new approach is based on minimization of a Black–Scholes-type measure of pathwise risk, defined in terms of a market delta, subject to an upper bound on the hedging cost. We provide an efficient backward induction algorithm for the problem of cost-constrained risk minimization, whose associated singular stochastic control problem is shown to be equivalent to an optimal stopping problem. This algorithm is then modified to solve the singular stochastic control problem associated with utility maximization, which cannot be reduced to an optimal stopping problem. We propose to choose an optimal parameter (risk-aversion coefficient or Lagrange multiplier) in either approach by minimizing the mean squared hedging error and demonstrate that with this “best” choice of the parameter, both approaches have similar performance. We also discuss the different notions of risk in both approaches and propose a volatility adjustment for the risk-minimization approach, which is analogous to that introduced by Zakamouline [2006. European option pricing and hedging with both fixed and proportional transaction costs. Journal of Economic Dynamics and Control 30, 1–25] for the utility maximization approach, thereby providing a unified treatment of both approaches.     

Input Hedging,Output Hedging,and Market Power

We argue that commodity input hedging is different from commodity output hedging. Output hedging can be detrimental to “sector play.” Furthermore, firms with market power that hedge outputs have incentives to over‐produce and distort market prices. In rational markets, such hedging will be expensive and we expect to see a negative relationship between hedging and market power in “output industries” but not in “input industries.” We test these predictions on a sample of S&P500 firms from 2001 to 2005. Our results support both hypotheses. Placebo tests show that the same empirical regularities do not apply to currency hedging. Finally, our empirical framework, which differentiates between hedging inputs and hedging outputs, can also help in reconciling conflicting results in prior studies.     

Quadratic hedging schemes for non-Gaussian GARCH models

We propose different schemes for option hedging when asset returns are modeled using a general class of GARCH models. More specifically, we implement local risk minimization and a minimum variance hedge approximation based on an extended Girsanov principle that generalizes Duan׳s (1995) delta hedge. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization hedging strategies with respect to a pricing kernel. These approaches are investigated in the context of non-Gaussian driven models. Furthermore, we analyze these methods for non-Gaussian GARCH diffusion limit processes and link them to the corresponding discrete time counterparts. A detailed numerical analysis based on S&P 500 European call options is provided to assess the empirical performance of the proposed schemes. We also test the sensitivity of the hedging strategies with respect to the risk neutral measure used by recomputing some of our results with an exponential affine pricing kernel.     

On new variance approximations for linear models with inequality constraints

In this paper, we examine the estimation of linear models subject to inequality constraints with a special focus on new variance approximations for the estimated parameters. For models with one inequality restriction, the proposed variance formulas are exact. The variance approximations proposed in this paper can be used in regression analysis, Kalman filtering, and balancing national accounts, when inequality constraints are to be incorporated in the estimation procedure.     

Estimation of the Option Prime: Microsimulation of Backward Stochastic Differential Equations

A mathematical statistical model is needed to obtain an option prime and create a hedging strategy. With formulas derived from stochastic differential equations, the primes for US Dollar/Chilean Pesos currency options using a prime calculator are obtained. Furthermore, a backward simulation of the option prime trajectory is used with a numerical method created for backward stochastic differential equations. The use of statistics in finance is highly important in order to develop complex products.     

股指期货的最优套期保值率实证研究——基于沪深300指数期货仿真交易视角

在分析和比较常用的几种股指期货最优套期保值比率确定模型的基础上,基于风险最小化模型框架,利用沪深300指数期货合约模拟运行以来的样本数据,通过最小二乘回归模型、向量自回归模型、误差修正模型以及广义自回归条件异方差模型四种估计方法,对其最优套期保值比率进行了实证测算和绩效比较,提出了相应的政策建议和投资策略。     

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters.     

International trade and hedging in economies in transition

This paper develops a general equilibrium framework to analyze risk management policies in economies in transition. By cross-hedging against real exchange rate risk exposures, these economies can increase their gains from international trade. We suggest that countries with emerging forward markets can gradually introduce the risk sharing markets, as limiting resources may prevent them from introducing complete hedging markets in the first place. Thus the growing demand for risk management instruments can be gradually met and it would be welfare enhancing. Economies in transition benefit when hedging devices are offered by financial markets, irrespective of whether the hedging instruments are de facto perfect or not.     

Some Recent Developments in Futures Hedging

The use of futures contracts as a hedging instrument has been the focus of much research. At the theoretical level, an optimal hedge strategy is traditionally based on the expected–utility maximization paradigm. A simplification of this paradigm leads to the minimum–variance criterion. Although this paradigm is quite well accepted, alternative approaches have been sought. At the empirical level, research on futures hedging has benefited from the recent developments in the econometrics literature. Much research has been done on improving the estimation of the optimal hedge ratio. As more is known about the statistical properties of financial time series, more sophisticated estimation methods are proposed. In this survey we review some recent developments in futures hedging. We delineate the theoretical underpinning of various methods and discuss the econometric implementation of the methods.     

Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances

In this paper we derive the exact risk (under quadratic loss) of pre-test estimators of the prediction vector and of the error variance of a linear regression model with spherically symmetric disturbances. The pre-test in question is one of the validity of a set of exact linear restrictions on the model's coefficient vector. We demonstrate how the known results for the model with normal disturbances can be extended to this broader case. We also show that the critical value of unity results in a minimum of the risk of the pre-test estimator of the error variance. To illustrate the results we assume multivariate Student-t regression disturbances and numerically evaluate the derived expressions.     

Quasi Score is more Efficient than Corrected Score in a Polynomial Measurement Error Model

We consider a polynomial regression model, where the covariate is measured with Gaussian errors. The measurement error variance is supposed to be known. The covariate is normally distributed with known mean and variance. Quasi score (QS) and corrected score (CS) are two consistent estimation methods, where the first makes use of the distribution of the covariate (structural method), while the latter does not (functional method). It may therefore be surmised that the former method is (asymptotically) more efficient than the latter one. This can, indeed, be proved for the regression parameters. We do this by introducing a third, so-called simple score (SS), estimator, the efficiency of which turns out to be intermediate between QS and CS. When one includes structural and functional estimators for the variance of the error in the equation, SS is still more efficient than CS. When the mean and variance of the covariate are not known and have to be estimated as well, one can still maintain that QS is more efficient than SS for the regression parameters.     

An optimal stochastic control framework for determining the cost of hedging of variable annuities

An implicit partial differential equation (PDE) method is used to determine the cost of hedging for a Guaranteed Lifelong Withdrawal Benefit (GLWB) variable annuity contract. In the basic setting, the underlying risky asset is assumed to evolve according to geometric Brownian motion, but this is generalised to the case of a Markov regime switching process. A similarity transformation is used to reduce a pricing problem with K regimes to the solution of K coupled one dimensional PDEs, resulting in a considerable gain in computational efficiency. The methodology developed is flexible in the sense that it can calculate the cost of hedging for a variety of different withdrawal strategies by investors. Cases considered here include both optimal withdrawal strategies (i.e. strategies which generate the highest possible cost of hedging for the insurer) and sub-optimal withdrawal strategies in which the policy holder׳s decisions depend on the moneyness of the embedded options. Numerical results are presented which demonstrate the sensitivity of the cost of hedging (given the withdrawal specification) to various economic and contractual assumptions.     

The LIML estimator has finite moments!

The Limited Information Maximum Likelihood estimator of the vector of coefficients of a structural equation in a simultaneous equation model is the vector that defines the linear combination maximizing the effect variance relative to the error variance. If this “eigenvector” solution is normalized by setting a designated coefficient equal to 1, the second-order moment of the estimator may be unbounded. However, the second-order moment is finite if the normalization sets the sample error variance of the linear combination equal to 1.     

The incompatibility of achieving a fully specified linear model with assuming that residual dependent-variable variance is random

The residual dependent-variable variance in experiments is not “random error”, as it is often assumed to be, but merely “unaccounted for variance”, because what is random is inexplicable in terms of any possible set of independent-variables and this is something that ultimately is only empirically determinable. So, if there is any unaccounted for dependent-variable variance, an experiment’s set of independent-variables is certainly under-specified and perhaps mis-specified because of the confounding of variables included in this set by causally relevant variables not included in the set. Thus, the proper first empirical test of any linear model is whether it leaves any residual dependent-variable variance, and if it does then none of its independent variables can yet logically justifiably be claimed to predict or causally explain any of the dependent-variable variance whatsoever.     

Cross-hedging minimum return guarantees: Basis and liquidity risks

We reveal pitfalls in the hedging of insurance contracts with a minimum return guarantee on the underlying investment, e.g. an external mutual fund. We analyze basis risk entailed by hedging the guarantee with a dynamic portfolio of proxy assets for the funds. We also take account of liquidity risk which arises since the insurer may need to advance funds for performing the hedge. Based on a least-squares Monte Carlo simulation, we study the economic implications of basis and liquidity risks. We demonstrate that both risks may be surprisingly high and show how the design of the contract and the hedging strategy may help to alleviate them.     

Bivariate garch estimation of the optimal commodity futures Hedge

Six different commodities are examined using daily data over two futures contract periods. Cash and futures prices for all six commodities are found to be well described as martingales with near-integrated GARCH innovations. Bivariate GARCH models of cash and futures prices are estimated for the same six commodities. The optimal hedge ratio (OHR) is then calculated as a ratio of the conditional covariance between cash and futures to the conditional variance of futures. The estimated OHRs reveal that the standard assumption of a time-invariant OHR is inappropriate. For each commodity the estimated OHR path appears non-stationary, which has important implications for hedging strategies.     

Asymptotics of SIMEX-based variance estimation

In this paper, we study the asymptotic properties of simulation extrapolation (SIMEX) based variance estimation that was proposed by Wang et al. (J R Stat Soc Series B 71:425–445, 2009). We first investigate the asymptotic normality of the parameter estimator in general parametric variance function and the local linear estimator for nonparametric variance function when permutation SIMEX (PSIMEX) is used. The asymptotic optimal bandwidth selection with respect to approximate mean integrated squared error (AMISE) for nonparametric estimator is also studied. We finally discuss constructing confidence intervals/bands of the parameter/function of interest. Other than applying the asymptotic results so that normal approximation can be used, we recommend a nonparametric Monte Carlo algorithm to avoid estimating the asymptotic variance of estimator. Simulation studies are carried out for illustration.