An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs

We introduce a jump-diffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the S&P500 index. We consider a delta-hedging strategy (DHS) for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM), assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (P&L) of the DHS under both models. We find that, under the log-normal model of Black–Scholes–Merton, the actual PDF of the P&L can be well approximated by the chi-squared distribution with specific parameters. We derive an approximation for the P&L volatility in the DM and JDM. We show that, under both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply mean–variance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the P&L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the P&L volatility can be reduced by increasing the hedging frequency.     

Hedging structured credit products during the credit crisis: A horse race of 10 models

Pricing and hedging structured credit products poses major challenges to financial institutions. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went under? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.     

Dynamic volatility spillovers and investment strategies between the Chinese stock market and commodity markets

Building on the increased interest in the volatility spillover effects between Chinese stock market and commodity markets, this paper investigates the dynamic volatility spillovers of Chinese stock market and Chinese commodity markets based on the volatility spillover index under the framework of TVP-VAR. The result shows that there is a highly dependent relationship between the stock market and commodity markets. On average, the Chinese stock market is the net recipient of spillover, non-ferrous metals and chemical industry have a very obvious spillover impact on the stock market. The degree of total volatility spillover is different in different periods. After major crisis events, the volatility correlation between markets increases. Since the outbreak of COVID-19, the spillover effect of the stock market on the commodity market has been significantly enhanced. Then optimal portfolio weights and hedge ratios are calculated for portfolio diversification and risk management. The result shows that the ability of most commodities to hedge against risks is significantly reduced when the crisis occurs; NMFI (precious metals) and CRFI (grain) still have good hedging ability after the crisis, but the effectiveness of hedging risk is relatively low. Besides, the combination of CRFI and SHCI (the Shanghai composite index) is the most effective for risk reduction.     

Dynamic connectedness and optimal hedging strategy among commodities and financial indices

This study looks at the best portfolio strategy for mitigating the risk associated with the MSCI ACWI & Frontier Markets Index, as well as the volatility spillovers between commodity markets and certain financial markets. Therefore, we empirically explore the connectedness among three financial indicators and five product groups using the framework of Diebold and Yilmaz (2012), which is based on a vector autoregressive process and variance decomposition of prediction errors, between 31 May 2002 and 30 July 2021. We also investigate the best hedging instrument(s) for the MSCI ACWI & Frontier Markets Global Index by combining the Asymmetric Dynamic Conditional Correlation (ADCC) model with the risk reduction index and the hedging ratios. Our empirical findings highlight the importance of volatility spillover effects across financial markets, which is not the case for commodity markets with low volatility externalities. Furthermore, the first markets appear to be net transmitters of volatility, whereas the second markets appear to be net receivers. Using the approach of Kroner and Sultan (1993), we show that the least risk portfolio is a portfolio that combines the MSCI ACWI & Frontier Markets Global Index with financial indices related to socially responsible and irresponsible investing.     

Hedging with futures: Efficacy of GARCH correlation models to European electricity markets

European electricity markets have been subject to a broad deregulation process in the last few decades. We analyse hedging policies implemented through different hedge ratios estimation. More specifically we compare naïve, ordinary least squares, and GARCH conditional variance and correlations models to test if GARCH models lead to higher variance reduction in a context of high time varying volatility as the case of electricity markets. Our results show that the choice of the hedge ratio estimation model is central on determining the effectiveness of futures hedging to reduce the portfolio volatility.     

Learning minimum variance discrete hedging directly from the market

Option hedging is a critical risk management problem in finance. In the Black–Scholes model, it has been recognized that computing a hedging position from the sensitivity of the calibrated model option value function is inadequate in minimizing variance of the option hedge risk, as it fails to capture the model parameter dependence on the underlying price (see e.g. Coleman et al., J. Risk, 2001, 5(6), 63–89; Hull and White, J. Bank. Finance, 2017, 82, 180–190). In this paper, we demonstrate that this issue can exist generally when determining hedging position from the sensitivity of the option function, either calibrated from a parametric model from current option prices or estimated nonparametricaly from historical option prices. Consequently, the sensitivity of the estimated model option function typically does not minimize variance of the hedge risk, even instantaneously. We propose a data-driven approach to directly learn a hedging function from the market data by minimizing variance of the local hedge risk. Using the S&P 500 index daily option data for more than a decade ending in August 2015, we show that the proposed method outperforms the parametric minimum variance hedging method proposed in Hull and White [J. Bank. Finance, 2017, 82, 180–190], as well as minimum variance hedging corrective techniques based on stochastic volatility or local volatility models. Furthermore, we show that the proposed approach achieves significant gain over the implied BS delta hedging for weekly and monthly hedging.     

Commodity futures hedging,risk aversion and the hedging horizon

This paper examines the impact of management preferences on optimal futures hedging strategy and associated performance. Applying an expected utility hedging objective, the optimal futures hedge ratio is determined for a range of preferences on risk aversion, hedging horizon and expected returns. Empirical results reveal substantial hedge ratio variation across distinct management preferences and are supportive of the hedging policies of real firms. Hedging performance is further shown to be strongly dependent on underlying preferences. In particular, hedgers with high risk aversion and short horizon reduce hedge portfolio risk but achieve inferior utility in comparison to those with low aversion.     

A Markov regime switching approach for hedging energy commodities

This paper estimates constant and dynamic hedge ratios in the New York Mercantile Exchange oil futures markets and examines their hedging performance. We also introduce a Markov regime switching vector error correction model with GARCH error structure. This specification links the concept of disequilibrium with that of uncertainty (as measured by the conditional second moments) across high and low volatility regimes. Overall, in and out-of-sample tests indicate that state dependent hedge ratios are able to provide significant reduction in portfolio risk.     

Volatility spillovers between oil prices and the stock market under structural breaks

This paper employs univariate and bivariate GARCH models to examine the volatility of oil prices and US stock market prices incorporating structural breaks using daily data from July 1, 1996 to June 30, 2013. We endogenously detect structural breaks using an iterated algorithm and incorporate this information in GARCH models to correctly estimate the volatility dynamics. We find no volatility spillover between oil prices and US stock market when structural breaks in variance are ignored in the model. However, after accounting for structural breaks in the model, we find strong volatility spillover between the two markets. We compute optimal portfolio weights and dynamic risk minimizing hedge ratios to highlight the significance of our empirical results which underscores the serious consequences of ignoring these structural breaks. Our findings are consistent with the notion of cross-market hedging and sharing of common information by financial market participants in these markets.     

Model risk of contingent claims

Paralleling regulatory developments, we devise value-at-risk and expected shortfall type risk measures for the potential losses arising from using misspecified models when pricing and hedging contingent claims. Essentially, P&L from model risk corresponds to P&L realized on a perfectly hedged position. Model uncertainty is expressed by a set of pricing models, each of which represents alternative asset price dynamics to the model used for pricing. P&L from model risk is determined relative to each of these models. Using market data, a unified loss distribution is attained by weighing models according to a likelihood criterion involving both calibration quality and model parsimony. Examples demonstrate the magnitude of model risk and corresponding capital buffers necessary to sufficiently protect trading book positions against unexpected losses from model risk. A further application of the model risk framework demonstrates the calculation of gap risk of a barrier option when employing a semi-static hedging strategy.     

Empirical pricing kernels

This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a time-varying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using S&P 500 index option data and a stochastic volatility model for the S&P 500 return process. We find that the EPK exhibits counter cyclical risk aversion over S&P 500 return states. We also find that hedging performance is significantly improved when we use hedge ratios based the EPK rather than a time-invariant pricing kernel.     

Benefits from Asia-Pacific Mutual Fund Investments with Currency Hedging

This study presents empirical evidence on the efficiency and effectiveness of hedging U.S.-based international mutual funds with an Asia-Pacific investment objective. The case for active currency risk management is examined for a passive and a selective hedge, which is constructed with currency futures in the major currencies. Both static and dynamic hedging models are used to estimate the risk-minimizing hedge ratio. The results show that currency hedging improves the performance of internationally diversified mutual funds. Such hedging is beneficial even when based on prior optimal hedge ratios. Further, efficiency gains from hedging, as measured by the percent change in the Sharpe Index, are greatest under a selective portfolio strategy that is implemented with an optimal constant hedge ratio.     

Is gold the best hedge and a safe haven under changing stock market volatility?

We evaluate the role of gold and other precious metals relative to volatility (Volatility Index (VIX)) as a hedge (negatively correlated with stocks) and safe haven (negatively correlated with stocks in extreme stock market declines) using data from the US stock market. Using daily data from November 1995 to November 2010, we find that gold, unlike other precious metals, serves as a hedge and a weak safe haven for US stock market. However, we find that VIX serves as a very strong hedge and a strong safe haven during our sample period. We also find that in periods of extremely low or high volatility, gold does not have a negative correlation with the US stock market. Our results show that VIX is a superior hedging tool and serves as a better safe haven than gold during our sample period. We highlight the practical significance of our results for financial market participants by conducting a portfolio analysis.     

Better cross hedges with composite hedging? Hedging equity portfolios using financial and commodity futures

Unless a direct hedge is available, cross hedging must be used. In such circumstances portfolio theory implies that a composite hedge (the use of two or more hedging instruments to hedge a single spot position) will be beneficial. The study and use of composite hedging has been neglected; possibly because it requires the estimation of two or more hedge ratios. This paper demonstrates a statistically significant increase in out-of-sample effectiveness from the composite hedging of the Amex Oil Index using S&P500 and New York Mercantile Exchange crude oil futures. This conclusion is robust to the technique used to estimate the hedge ratios, and to allowance for transactions costs, dividends and the maturity of the futures contracts.     

Optimal hedge ratio and hedging effectiveness of stock index futures: evidence from India

In a free capital mobile world with increased volatility, the need for an optimal hedge ratio and its effectiveness is warranted to design a better hedging strategy with future contracts. This study analyses four competing time series econometric models with daily data on NSE Stock Index Futures and S&P CNX Nifty Index. The effectiveness of the optimal hedge ratios is examined through the mean returns and the average variance reduction between the hedged and the unhedged positions for 1-, 5-, 10- and 20-day horizons. The results clearly show that the time-varying hedge ratio derived from the multivariate GARCH model has higher mean return and higher average variance reduction across hedged and unhedged positions. Even though not outperforming the GARCH model, the simple OLS-based strategy performs well at shorter time horizons. The potential use of this multivariate GARCH model cannot be sublined because of its estimation complexities. However, from a cost of computation point of view, one can equally consider the simple OLS strategy that performs well at the shorter time horizons.     

A regime-switching real-time copula GARCH model for optimal futures hedging

A regime-switching real-time copula GARCH (RSRTCG) model is suggested for optimal futures hedging. The specification of RSRTCG is to model the margins of asset returns with state-dependent real-time GARCH and the dependence structure of asset returns with regime switching copula functions. RSRTCG is faster in adjusting to the new level of volatility under different market regimes which is a regime-switching multivariate generalization of the state-independent univariate real-time GARCH. RSRTCG is applied to cross hedge the price risk of S&P 500 sector indices with crude oil futures. The empirical results show that RSRTCG possesses superior hedging performance compared to its nested non-real-time or state-independent copula GARCH models based on the criterion of percentage variance reduction, utility gain, model confidence set, model combination strategy, risk-adjusted return and reward-to-semivariance ratio.     

股指期货与股指期权套期保值组合的Delta中性动态模拟——基于沪深300股指期货仿真交易的分析

沪深300股指期货的推出,在为市场提供套保工具和流动性的同时,也带来新的风险。本文运用同一指数的股指期权与股指期货组成多种动态套期保值组合,分析股指期货的风险对冲策略。结果表明所构造的组合都能为股指期货提供有效的套期保值;不论多头股指期货还是空头股指期货,保护性策略的风险控制能力更强。     

Currency hedging for single-currency equity portfolios: Does cross-asset risk matter?

Foreign investors who are fully invested in a single-currency domestic equity portfolio are exposed to domestic equity risk, but also to currency risk. The standard approach to hedging the currency risk optimally is to estimate a single optimal hedge ratio, but this approach hedges only exchange rate risk, not cross-asset risk. We provide an alternative approach that estimates two optimal hedge ratios to adjust the currency exposures—one associated with the domestic currency and one associated with the foreign currency—and hedges both exchange rate risk and cross-asset risk. This alternative approach can significantly reduce risk.     

Quadratic hedging in affine stochastic volatility models

We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as in Carr et al. (Math Finance 13:345–382, 2003).      

Delta-Hedged Gains and the Negative Market Volatility Risk Premium

We investigate whether the volatility risk premium is negativeby examining the statistical properties of delta-hedged optionportfolios (buy the option and hedge with stock). Within a stochasticvolatility framework, we demonstrate a correspondence betweenthe sign and magnitude of the volatility risk premium and themean delta-hedged portfolio returns. Using a sample of S&P500 index options, we provide empirical tests that have thefollowing general results. First, the delta-hedged strategyunderperforms zero. Second, the documented underperformanceis less for options away from the money. Third, the underperformanceis greater at times of higher volatility. Fourth, the volatilityrisk premium significantly affects delta-hedged gains, evenafter accounting for jump fears. Our evidence is supportiveof a negative market volatility risk premium.