HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

This paper presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor’s theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk‐free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.     

Minimum variance cross hedging under mean‐reverting spreads,stochastic convenience yields,and jumps: Application to the airline industry

Exchange traded futures contracts often are not written on the specific asset that is a source of risk to a firm. The firm may attempt to manage this risk using futures contracts written on a related asset. This cross hedge exposes the firm to a new risk, the spread between the asset underlying the futures contract and the asset that the firm wants to hedge. Using the specific case of the airline industry as motivation, we derive the minimum variance cross hedge assuming a two‐factor diffusion model for the underlying asset and a stochastic, mean‐reverting spread. The result is a time‐varying hedge ratio that can be applied to any hedging horizon. We also consider the effect of jumps in the underlying asset. We use simulations and empirical tests of crude oil, jet fuel cross hedges to demonstrate the hedging effectiveness of the model. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:736–756, 2009     

Robust estimation of the optimal hedge ratio

When using derivative instruments such as futures to hedge a portfolio of risky assets, the primary objective is to estimate the optimal hedge ratio (OHR). When agents have mean‐variance utility and the futures price follows a martingale, the OHR is equivalent to the minimum variance hedge ratio,which can be estimated by regressing the spot market return on the futures market return using ordinary least squares. To accommodate time‐varying volatility in asset returns, estimators based on rolling windows, GARCH, or EWMA models are commonly employed. However, all of these approaches are based on the sample variance and covariance estimators of returns, which, while consistent irrespective of the underlying distribution of the data, are not in general efficient. In particular, when the distribution of the data is leptokurtic, as is commonly found for short horizon asset returns, these estimators will attach too much weight to extreme observations. This article proposes an alternative to the standard approach to the estimation of the OHR that is robust to the leptokurtosis of returns. We use the robust OHR to construct a dynamic hedging strategy for daily returns on the FTSE100 index using index futures. We estimate the robust OHR using both the rolling window approach and the EWMA approach, and compare our results to those based on the standard rolling window and EWMA estimators. It is shown that the robust OHR yields a hedged portfolio variance that is marginally lower than that based on the standard estimator. Moreover, the variance of the robust OHR is as much as 70% lower than the variance of the standard OHR, substantially reducing the transaction costs that are associated with dynamic hedging strategies. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:799–816, 2003     

Pricing variance swaps under the Hawkes jump-diffusion process

This paper presents an analytical approach for pricing variance swaps with discrete sampling times when the underlying asset follows a Hawkes jump-diffusion process characterized with both stochastic volatility and clustered jumps. A significantly simplified method, with which there is no need to solve partial differential equations, is used to derive a closed-form pricing formula. A distinguished feature is that many recently published formulas can be shown to be special cases of the one presented here. Some numerical examples are provided with results demonstrating that jump clustering indeed has a significant impact on the price of variance swaps.     

CLOSED FORM PRICING FORMULAS FOR DISCRETELY SAMPLED GENERALIZED VARIANCE SWAPS

Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one‐dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.     

Pricing models of equity swaps

This article provides a generalized formula for pricing equity swaps with constant notional principal when the underlying equity markets and settlement currency can be set arbitrarily. To derive swap values using the risk‐neutral valuation method, the swap payment is replicated at each settlement date by constructing a self‐financing portfolio. To obtain the foreign equity index return denominated in the domestic or in a third currency, equity‐linked foreign exchange options are used to hedge the exchange rate risk. It is found that if the swap involves international equity markets, then the swap value contains an extra term which reflects the currency hedging costs. This methodology can easily be applied to price various types of equity swaps simply by modifying the specifications of the model presented here as required. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:751–772, 2003     

A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed‐form exact solution for the partial differential equation (PDE) system based on the Heston's two‐factor stochastic volatility model embedded in the framework proposed by Little and Pant. In comparison with the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed‐form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous‐sampling‐time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.     

Futures hedging when the structure of the underlying asset changes: The case of the BIFFEX contract

This article is concerned with the hedging effectiveness of futures contracts whose underlying asset is an index, when the structure of this index is changing. The case of the freight futures (BIFFEX) contract is examined here. Investigation of this issue is particularly interesting as the composition of its underlying asset, the Baltic Freight Index (BFI), has been revised on a number of occasions in order to improve the hedging performance of the market; previous empirical evidence on the market indicates substantially lower variance reduction (4–19%), compared to other markets (up to 98%). The BFI is a weighted average dry‐cargo freight rate index, compiled from actual freight rates on 11 shipping routes that are dissimilar in terms of vessel sizes and transported commodities. The hedging effectiveness of the market is investigated using both constant and time‐varying hedge ratios, estimated through bivariate error correction GARCH models. Our results indicate that the effectiveness of the BIFFEX contract as a centre for risk management has strengthened over the recent years as a result of the more homogeneous composition of the index. This by itself indicates that the latest restructuring of the index, in November 1999, which is aimed at increasing its homogeneity even further, is likely to have a beneficial impact on the market. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:775–801, 2000     

Production and hedging under state‐dependent preferences

This study examines the behavior of the competitive firm under output price uncertainty and state‐dependent preferences. When there is a futures market for hedging purposes, the firm's optimal production decision is independent of the output price uncertainty and of the state‐dependent preferences. If the futures contracts are unbiased, the firm's optimal futures position is an over‐hedge or an under‐hedge, depending on whether the firm is correlation averse or correlation loving, and on whether the output price is positively or negatively expectation dependent on the state variable. When the firm has access not only to the unbiased futures but also to fairly priced options, sufficient conditions are derived under which the firm's optimal hedge position includes both hedging instruments. This study thus establishes a hedging role of options, which is over and above that of futures, in the case of state‐dependent preferences. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:945–963, 2012     

VALUATION OF CLAIMS ON NONTRADED ASSETS USING UTILITY MAXIMIZATION

A topical problem is how to price and hedge claims on nontraded assets. A natural approach is to use for hedging purposes another similar asset or index which is traded. To model this situation, we introduce a second nontraded log Brownian asset into the well-known Merton investment model with power law and exponential utilities. The investor has an option on units of the nontraded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete markets. Employing utility maximization and duality methods we obtain a series approximation to the optimal hedge and reservation price using the power utility. The problem is simpler for the exponential utility, and in this case we derive an explicit representation for the price. Price and hedging strategy are computed for some example options and the results for the utilities are compared.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

Conditional OLS minimum variance hedge ratios

The paper presents a new methodology to estimate time dependent minimum variance hedge ratios. The so‐called conditional OLS hedge ratio modifies the static OLS approach to incorporate conditioning information. The ability of the conditional OLS hedge ratio to minimize the risk of a hedged portfolio is compared to conventional static and dynamic approaches, such as the naïve hedge, the roll‐over OLS hedge, and the bivariate GARCH(1,1) model. The paper concludes that, both in‐sample and out‐of‐sample, the conditional OLS hedge ratio reduces the basis risk of an equity portfolio better than the alternatives conventionally used in risk management. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:945–964, 2004     

Hedging and value at risk: A semi‐parametric approach

The non‐normality of financial asset returns has important implications for hedging. In particular, in contrast with the unambiguous effect that minimum‐variance hedging has on the standard deviation, it can actually increase the negative skewness and kurtosis of hedge portfolio returns. Thus, the reduction in Value at Risk (VaR) and Conditional Value at Risk (CVaR) that minimum‐variance hedging generates can be significantly lower than the reduction in standard deviation. In this study, we provide a new, semi‐parametric method of estimating minimum‐VaR and minimum‐CVaR hedge ratios based on the Cornish‐Fisher expansion of the quantile of the hedged portfolio return distribution. Using spot and futures returns for the FTSE 100, FTSE 250, and FTSE Small Cap equity indices, the Euro/US Dollar exchange rate, and Brent crude oil, we find that the semiparametric approach is superior to the standard minimum‐variance approach, and to the nonparametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently generates hedge portfolios that in most cases have lower VaR and CVaR. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:780–794, 2010     

The Effect of the Hedge Horizon on Optimal Hedge Size and Effectiveness When Prices are Cointegrated

This study compares two alternative regression specifications for sizing hedge positions and measuring hedge effectiveness: a simple regression on price changes and an error correction model (ECM). We show that, when the prices of the hedged item and the hedging instrument are cointegrated, both specifications yield similar results which depend on the hedge horizon (i.e., the time frame for measuring price changes). In particular, the estimated hedge ratio and regression R² will both be small when price changes are measured over short intervals, but as the hedge horizon is lengthened both measures will converge toward one. These results imply that, when prices are cointegrated, a longer hedge horizon will yield an optimal hedge ratio closer to one, while at the same time enhancing the ability to qualify for hedge accounting. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:837–876, 2012     

A note on the derivation of Black‐Scholes hedge ratios

An option hedge ratio is the sensitivity of an option price with respect to price changes in the underlying stock. It measures the number of shares of stocks to hedge an option position. This article presents a simple derivation of the hedge ratios under the Black‐Scholes option‐pricing framework. The proof is succinct and easy to follow. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1119–1122, 2003     

Minimum‐variance futures hedging under alternative return specifications

It is widely believed that the conventional futures hedge ratio, is variance‐minimizing when it is computed using percentage returns or log returns. It is shown that the conventional hedge ratio is variance‐minimizing when computed from returns measured in dollar terms but not from returns measured in percentage or log terms. Formulas for the minimum‐variance hedge ratio under percentage and log returns are derived. The difference between the conventional hedge ratio computed from percentage and log returns and the minimum‐variance hedge ratio is found to be relatively small when directly hedging, especially when using near‐maturity futures. However, the minimum‐variance hedge ratio can vary significantly from the conventional hedge ratio computed from percentage or log returns when used in cross‐hedging situations. Simulation analysis shows that the incorrect application of the conventional hedge ratio in crosshedging situations can substantially reduce hedging performance. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:537–552, 2005     

Analytic approximation formulae for pricing forward‐starting Asian options

In this article we first identify a missing term in the Bouaziz, Briys, and Crouhy ( 1994 ) pricing formula for forward‐starting Asian options and derive the correct one. First, illustrate in certain cases that the missing term in their pricing formula could induce large pricing errors or unreasonable option prices. Second, we derive new analytic approximation formulae for valuing forward‐starting Asian options by adding the second‐order term in the Taylor series. We show that our formulae can accurately value forward‐starting Asian options with a large underlying asset's volatility or a longer time window for the average of the underlying asset prices, whereas the pricing errors for these options with the previously mentioned formula could be large. Third, we derive the hedge ratios for these options and compare their properties with those of plain vanilla options. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:487–516, 2003     

Asymmetric information and credit quality: Evidence from synthetic fixed‐rate financing

In this article the usage of synthetic fixed‐rate financing (SFRF) with interest rate swaps (i.e., borrowing short‐term and using swaps to hedge interest rate risk, instead of selecting conventional fixed‐rate financing) by Fortune 500 and S&P 500 nonfinancial firms is examined over the period 1991 through 1995. Credit ratings, debt issuance, and debt maturities of these firms are monitored through 1999. Strong evidence is found supporting the asymmetric information theory of swap usage as described by S. Titman (1992), even after controlling for industry, credit quality, size effects, and the simultaneity of the capital structure and the interest rate swap usage decision. Consistent with theoretical predictions, SFRF firms are more likely to undergo credit quality upgrades. When limiting the sample to firms where asymmetric information costs are potentially the greatest, the results are even stronger. These findings are important because they document that swaps serve a highly valuable service for firms subject to information asymmetries. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:595–626, 2006     

MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS

We consider a class of asset pricing models, where the risk‐neutral joint process of log‐price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long‐term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff‐Nielsen–Shephard model.     

The Asymmetric Commodity Inventory Effect on the Optimal Hedge Ratio

Hedging strategies for commodity prices largely rely on dynamic models to compute optimal hedge ratios. This study illustrates the importance of considering the commodity inventory effect (effect by which the commodity price volatility increases more after a positive shock than after a negative shock of the same magnitude) in modeling the variance–covariance dynamics. We show by in‐sample and out‐of‐sample forecasts that a commodity price index portfolio optimized by an asymmetric BEKK–GARCH model outperforms the symmetric BEKK, static (OLS), or naïve models. Robustness checks on a set of commodities and by an alternative mean‐variance optimization framework confirm the relevance of taking into account the inventory effect in commodity hedging strategies.