Risk minimization in stochastic volatility models: model risk and empirical performance

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.     

Conditional currency hedging

I propose a simple and robust approach to hedge currency risk that can be directly applied by international investors in diverse asset classes. Compared to current mean-variance approaches, it is robust to overfitting and thus better anticipates risk-minimizing currency positions for global equity, bond, and commodity investors out of sample. Furthermore, correlations among currencies, equities, and commodities can be predicted by lagged implied foreign exchange volatility. This allows investors to dynamically adjust their hedges, resulting in significantly lower risk compared to other hedging alternatives while maintaining or even improving Sharpe ratio, particularly during crisis periods.     

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.     

Option pricing and hedging under a stochastic volatility Lévy process model

In this paper, we discuss a stochastic volatility model with a Lévy driving process and then apply the model to option pricing and hedging. The stochastic volatility in our model is defined by the continuous Markov chain. The risk-neutral measure is obtained by applying the Esscher transform. The option price using this model is computed by the Fourier transform method. We obtain the closed-form solution for the hedge ratio by applying locally risk-minimizing hedging.     

Arbitrage-free interval and dynamic hedging in an illiquid market

This paper derives two pricing PDEs for a general European option under liquidity risk. We provide two modified hedges: one hedge replicates a short option and the other replicates a long option inclusive of liquidity costs under continuous rebalancing. We identify an arbitrage-free interval by calculating the costs of the two hedges. Unlike in a setting with infinite overall transaction costs, the overall liquidity cost in our model is proved to be finite even under continuous rebalancing. Numerical results on option pricing and the moments of hedge errors of Black–Scholes and our modified hedges are also presented.     

A Futures Duration-Convexity Hedging Method

A duration-based hedge ratio is the conventional method to hedge against price changes of a fixed-income instrument. However, the relationship between bond prices and interest rates is nonlinear, creating a convexity effect. Moreover, term structure changes often are nonparallel in nature, which causes imperfect hedges for the duration-based hedging model. One solution to these problems is to dynamically change the duration-based hedge ratio; however, this procedure is costly and is not effective when jumps in prices occur. A superior solution is to develop a two-instrument hedge ratio that simultaneously hedges both duration and convexity effects. This paper first presents such a two-instrument hedge ratio and then we examine its effectiveness. The simulation results show that this duration-convexity hedge ratio is vastly superior to alternative hedge ratio methods for both simple and complex changes in the term structure.     

Do commodities make effective hedges for equity investors?

The purpose of this paper is to evaluate whether commodities are effective hedges for equity holders. We employ three different methodologies to calculate time varying hedge ratios. First, we examine time-varying hedge ratios and how much portfolio risk can be reduced relative to a long position in the S&P 500. We calculate hedge ratios from realized variances and covariances; second, we estimate a recursive multivariate GARCH (BEKK) model and calculate the hedge ratios from the estimated covariances; and thirdly, we calculate the hedge ratios by estimating recursive OLS regressions. The results of our paper are very clear. First, commodities are not effective hedges for the S&P 500. Equity market investors and asset managers looking for a way to manage and reduce portfolio risk will be well advised to search for alternative hedges for the S&P 500 than commodities. Second, our results do not support the claim that commodities were a good hedge for the equity market during the financial crisis.     

Basis risk in static versus dynamic longevity-risk hedging

This paper provides a tractable, parsimonious model for assessing basis risk in longevity and its effect on the hedging strategies of Pension Funds and annuity providers. Basis risk is captured by a single parameter, that measures the co-movement between the portfolio and the reference population’s longevity. The paper sets out the static, full and customized swap-hedge for an annuity, and compares it with a dynamic, partial, and index-based hedge. We calibrate our model to the UK and Scottish populations. The effectiveness of static versus dynamic strategies depends on the rebalancing frequency of the second, on the relative costs, and on basis risk, which does not affect fully-customized, static hedges. We show that appropriately calibrated dynamic hedging strategies can still be reasonably effective, even at low rebalancing frequencies.     

Reducing reporting risk: Designating foreign currency forward contracts as cash flow hedges

We present an example that compares the effects on earnings of designating a foreign currency forward contract as either a cash-flow or fair-value hedge of a foreign currency denominated receivable. Entities engaging in exchange transactions not denominated in their functional currency frequently enter into foreign currency forward contracts in order to mitigate their foreign exchange rate risk exposure. The aggregate effect on earnings of the transaction gain or loss on the foreign currency receivable and the gain or loss on the forward contract is known on the date the forward contract is initiated. The effect on each period’s earnings during the term of a forward contract designated as a cash-flow hedge is also known on the date the contract is initiated; whereas the effect on each periods’ earnings from a fair-value hedge cannot be determined until the respective balance sheet dates. Therefore, designating forward contracts as cash-flow hedges may suppress volatility in reported earnings compared to designating forward contracts as fair-value hedges. In addition, the reporting risk (the amount of uncertainty surrounding the pending measure of an item to be reported in the financial statements) is lower when a forward contract is designated as a cash-flow hedge relative to designating it as a fair-value hedge. This suggests foreign currency forward contracts designated as cash-flow hedges are more consistent with the purpose of hedge accounting: to mitigate the effects on earnings of applying different measurement criteria for the hedge and the hedged item.     

Static versus dynamic hedges: an empirical comparison for barrier options

We conduct an empirical comparison of static versus dynamic hedges of barrier options. Using more than five years of data, we compare a number of static hedges from the literature with dynamic hedges based on the local volatility model. The main result is that the variability of profit-and-loss distributions from certain static hedges is significantly smaller than that of dynamic hedges and robust to changing market scenarios. Furthermore, these static hedges are able to provide a robust tracking of barrier options’ sensitivities. This article reflects the authors’ personal opinion and not necessarily the opinion of their employers.     

Hedging with Two Futures Contracts: Simplicity Pays

We propose to use two futures contracts in hedging an agricultural commodity commitment to solve either the standard delta hedge or the roll‐over issue. Most current literature on dual‐hedge strategies is based on a structured model to reduce roll‐over risk and is somehow difficult to apply for agricultural futures contracts. Instead, we propose to apply a regression based model and a naive rules of thumb for dual‐hedges which are applicable for agricultural commodities. The naive dual strategy stems from the fact that in a large sample of agricultural commodities, De Ville, Dhaene and Sercu (2008) find that GARCH‐based hedges do not perform as well as OLS‐based ones and that we can avoid estimation error with such a simple rule. Our semi‐naive hedge ratios are driven from two conditions: omitting exposure to spot price and minimising the variance of the unexpected basis effects on the portfolio values. We find that, generally, (i) rebalancing helps; (ii) the two‐contract hedging rules do better than the one‐contract counterparts, even for standard delta hedges without rolling‐over; (iii) simplicity pays: the naive rules are the best one–for corn and wheat within the two‐contract group, the semi‐naive rule systematically beats the others and GARCH performs worse than OLS for either one‐contract or two‐contract hedges and for soybeans the traditional naive rule performs nearly as well as OLS. These conclusions are based on the tests on unconditional variance ( Diebold and Mariano, 1995 ) and those on conditional risk ( Giacomini and White, 2006 ).     

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.     

Property and Inflation: The Hedging Characteristics of U.K. Commercial Property, 1967–1994

This paper examines the statistical similarities between U.K. commercial property capital and rental values and the price level. Our aim is to determine whether commercial property is an inflation hedge and, if so, what type of inflation it hedges against. To answer these questions, we use both a multivariate unobserved components model and structural vector autoregressions. We find that commercial property is an inflation hedge but only a weak one. More specifically, we find that property offers some form of partial hedge against changes in the underlying inflation rate but not to either temporary or permanent changes to the price level. We also find that capital values offer a stronger hedge than rental values and that industrial and retail property account for most of this hedging capacity. We find no evidence that property responds differently to high or low inflation but we do find capital and rental values respond more to unexpected inflation than anticipated price changes.     

Empirical evidence concerning incentives to hedge transaction and translation exposures

We investigate Swedish firms’ use of financial hedges against foreign exchange exposure. Our survey data lets us distinguish between translation exposure and transaction exposure hedging. Survey responses indicate that over 50% of the sampled firms employ financial hedges, and that transaction exposure is more frequently hedged than is translation exposure. The likelihood of using financial hedges increases with firm size and exposure, and liquidity constraints are important in explaining transaction exposure hedging. Importantly, the existence of loan covenants accounts for translation exposure hedging, suggesting that firms hedge translation exposure to avoid violating loan covenants.     

Global Currency Hedging

Over the period 1975 to 2005, the U.S. dollar (particularly in relation to the Canadian dollar), the euro, and the Swiss franc (particularly in the second half of the period) moved against world equity markets. Thus, these currencies should be attractive to risk-minimizing global equity investors despite their low average returns. The risk-minimizing currency strategy for a global bond investor is close to a full currency hedge, with a modest long position in the U.S. dollar. There is little evidence that risk-minimizing investors should adjust their currency positions in response to movements in interest differentials.     

Cross-commodity hedging for illiquid futures: Evidence from China's base metal futures market

This paper evaluates the effectiveness of cross-commodity hedging between China's base metal spot and futures markets, using daily data of metal spot and futures prices in the Shanghai Futures Exchange. The main findings suggest that, compared to unhedged spot portfolios, a naïve hedge increases risk exposure, while static and dynamic hedges can significantly reduce the risk of holding spot assets. Zinc futures and nickel futures outperform other base metal futures in individually hedging lead spot and tin spot respectively, while copper futures constitute a moderately optimal instrument to hedge both lead and tin spot assets.     

INTEREST RATE RISK,MARKET VALUE,AND HEDGING FINANCIAL PORTFOLIOS

This paper derives theoretical hedge ratios for the financial portfolio that preserve its present value in the presence of interest rate risk. From a practical point of view and for any given portfolio, the existence of the financial futures market allows the investor to employ any of a number of different hedges, each of which approximately satisfies the theoretical condition. The theory indicates that wealth-preserving hedges depend on the interest elasticities (durations) of the spot assets and liabilities contained in the portfolio, portfolio leverage, and the interest elasticity (duration) of the financial instrument underlying the futures contract that is employed in constructing the hedge. Also, hedges designed to maintain net interest margin or net cash flow do not minimize exposure to interest rate risk.     

Klugman,S.A., Panjer,H.H., and Willmot,G.E., 1998, Los flwfodels: From Data to Decisions

Abstract

Basis risk is an important consideration when hedging longevity risk with instruments based on longevity indices, since the longevity experience of the hedged exposure may differ from that of the index. As a result, any decision to execute an index-based hedge requires a framework for (1) developing an informed understanding of the basis risk, (2) appropriately calibrating the hedging instrument, and (3) evaluating hedge effectiveness. We describe such a framework and apply it to a U.K. case study, which compares the population of assured lives from the Continuous Mortality Investigation with the England and Wales national population. The framework is founded on an analysis of historical experience data, together with an appreciation of the contextual relationship between the two related populations in social, economic, and demographic terms. Despite the different demographic profiles, the case study provides evidence of stable long-term relationships between the mortality experiences of the two populations. This suggests the important result that high levels of hedge effectiveness should be achievable with appropriately calibrated, static, index-based longevity hedges. Indeed, this is borne out in detailed calculations of hedge effectiveness for a hypothetical pension portfolio where the basis risk is based on the case study. A robustness check involving populations from the United States yields similar results.     

Using the short-lived arbitrage model to compute minimum variance hedge ratios: application to indices,stocks and commodities

The short-lived arbitrage model has been shown to significantly improve in-sample option pricing fit relative to the Black–Scholes model. Motivated by this model, we imply both volatility and virtual interest rates to adjust minimum variance hedge ratios. Using several error metrics, we find that the hedging model significantly outperforms the traditional delta hedge and a current benchmark hedge based on the practitioner Black–Scholes model. Our applications include hedges of index options, individual stock options and commodity futures options. Hedges on gold and silver are especially sensitive to virtual interest rates.     

BASIS VOLATILITY: IMPLICATIONS FOR HEDGING

Most hedges placed in futures markets must be lifted before contract expiration, which necessitates incurring “basis risk.” The focus of this paper is on quantifying such risk as a function of the timing of a hedge, its duration, distance from contract expiration, hedge life, and other market-observable variables. The development of basis-risk profiles provides a hedger with estimates of hedging risks that reasonably can be expected before the actual placement of hedges, thus serving as a useful input in the hedging decision.