Hedging errors with Leland's option model in the presence of transaction costs

Nonzero transaction costs invalidate the Black–Scholes [1973. Journal of Political Economy 81, 637–654] arbitrage argument based on continuous trading. Leland [1985. Journal of Finance 40, 1283–1301] developed a hedging strategy which modifies the Black–Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Kabanov and Safarian [1997. Finance and Stochastics 1, 239–250] calculated the limiting hedging error of the Leland strategy and pointed out that it is nonzero for the approximate pricing of an European call option, in contradiction to Leland's claim. As a further contribution, we first identify the mathematical flaw in the argument of Leland's claim and then quantify the expected percentage of hedging losses in terms of the hedging frequency and the level of the option strike price.     

Static hedging and pricing American options

This paper utilizes the static hedge portfolio (SHP) approach of Derman et al. [Derman, E., Ergener, D., Kani, I., 1995. Static options replication. Journal of Derivatives 2, 78–95] and Carr et al. [Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190] to price and hedge American options under the Black-Scholes (1973) model and the constant elasticity of variance (CEV) model of Cox [Cox, J., 1975. Notes on option pricing I: Constant elasticity of variance diffusion. Working Paper, Stanford University]. The static hedge portfolio of an American option is formulated by applying the value-matching and smooth-pasting conditions on the early exercise boundary. The results indicate that the numerical efficiency of our static hedge portfolio approach is comparable to some recent advanced numerical methods such as Broadie and Detemple [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211–1250] binomial Black-Scholes method with Richardson extrapolation (BBSR). The accuracy of the SHP method for the calculation of deltas and gammas is especially notable. Moreover, when the stock price changes, the recalculation of the prices and hedge ratios of the American options under the SHP method is quick because there is no need to solve the static hedge portfolio again. Finally, our static hedging approach also provides an intuitive derivation of the early exercise boundary near expiration.     

Cephalon, Inc. Taking risk management theory seriously

Cephalon Inc., a biotech firm, bought call options on its own stock to meet its conditional cash flow needs. We analyze this decision by using the cash flow hedging concepts of Froot et al., (1993. Journal of Finance 5, 1629–1658). We identify the managerial analyses necessary to apply this theory and discuss managerial considerations absent from the theory. We find that managers consider deadweight costs of risk management, which theory tends to ignore. Theory provides little guidance in how to measure these and other deadweight costs. Finally, uncertainty about the availability of external financing and accounting considerations are critical considerations by managers.     

Long term spread option valuation and hedging

This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are in dynamic equilibrium (i.e., cointegrated). The spread exhibits properties different from its two underlying commodity prices and should therefore be modelled directly. This approach offers significant advantages relative to the traditional two price methods since the correlation between two asset returns is notoriously hard to model. In this paper, we propose a two factor model for the spot spread and develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spreads in energy markets – the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results.     

A new computational tool for analysing dynamic hedging under transaction costs

This paper highlights a framework for analysing dynamic hedging strategies under transaction costs. First, self-financing portfolio dynamics under transaction costs are modelled as being portfolio affine. An algorithm for computing the moments of the hedging error on a lattice under portfolio affine dynamics is then presented. In a number of circumstances, this provides an efficient approach to analysing the performance of hedging strategies under transaction costs through moments. As an example, this approach is applied to the hedging of a European call option with a Black–Scholes delta hedge and Leland's adjustment for transaction costs. Results are presented that demonstrate the range of analysis possible within the presented framework.     

Tests of the efficiency of the U.S. rights offering market: An option pricing approach

The efficiency of the U.S. market for stock purchase rights is empirically analyzed in an options framework, in which prices of rights, given the prices of underlying stock, are examined with regard to the possibilities of actually earning above-normal profits, considering the risk taken. Two neutral hedging tests for market efficiency, along with a simple buy-and-exercise trading strategy, are applied to daily traded rights data. Results from ex-post hedging tests suggest that the trading strategy based on the rights valuation model is able to differentiate between overpriced and underpriced rights so as to generate substantial book profits. The positive ex-ante hedge return, found to exist empirically, is completely eliminated once transaction costs are introduced, lending support for the efficient U.S. rights offering market on an after-transaction cost basis.     

PRICING AND HEDGING SHORT STERLING OPTIONS USING NEURAL NETWORKS

This paper compares the performance of artificial neural networks (ANNs) with that of the modified Black model in both pricing and hedging short sterling options. Using high‐frequency data, standard and hybrid ANNs are trained to generate option prices. The hybrid ANN is significantly superior to both the modified Black model and the standard ANN in pricing call and put options. Hedge ratios for hedging short sterling options positions using short sterling futures are produced using the standard and hybrid ANN pricing models, the modified Black model, and also standard and hybrid ANNs trained directly on the hedge ratios. The performance of hedge ratios from ANNs directly trained on actual hedge ratios is significantly superior to those based on a pricing model, and to the modified Black model. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Exchange rate exposure and foreign exchange derivatives: do ineffective hedgers modify future derivatives use?

Recent studies examining the relationship between stock returns and exchange rate changes have provided evidence that the exchange rate exposure of non-financial companies is reduced by the use of foreign exchange derivatives. Building on such research, this study investigates whether past ineffective derivative hedging contributes to explaining future derivatives use. To the extent that companies monitor the effectiveness of their currency risk management practices, past ineffective hedgers can be expected to modify their future use of foreign exchange derivatives accordingly. In our study of 94 non-financial US multinationals, we provide evidence that the change in derivatives use from 1996–1998 to 1998–2000 can be explained in part by the ineffective hedging of currency risk in 1996–1998, controlling for variables associated with theories of optimal hedging. Additional analyses confirm that such primary results are robust to firm size, the level of foreign operations, and the use of derivatives to partially hedge currency risk. Our results imply that as exchange markets and risk management practices change, the use of derivatives to manage exchange rate risk also changes. Our contribution to this field of study is that we find evidence that past ineffective hedgers tend to increase their future use of FXDs.     

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.     

Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.     

The hedging effect of green bonds on carbon market risk

This paper explores effective hedging instruments for carbon market risk. Examining the relationship between the carbon futures returns and the returns of four major market indices, i.e., the VIX index, the commodity index, the energy index and the green bond index, we find that the connectedness between the carbon futures returns and the green bond index returns is the highest and this connectedness is extremely pronounced during the market's volatile period. Further, we develop and evaluate hedging strategies based on three dynamic hedge ratio models (DCC-APGARCH, DCC-T-GARCH, and DCC-GJR-GARCH models) and the constant hedge ratio model (OLS model). Empirical results show that among the four market indices the green bond index is the best hedge for carbon futures and performs well even in the crisis period. The paper also provides evidence that the dynamic hedge ratio models are superior to the OLS model in the volatile period as more sophisticated models can capture the dynamic correlation and volatility spillover between the carbon futures and market index returns.     

Option pricing in a lognormal securities market with discrete trading

This paper derives a call option valuation equation assuming discrete trading in securities markets where the underlying asset and market returns are bivariate lognormally distributed and investors have increasing, concave utility functions exhibiting skewness preference. Since the valuation does not require the continouus time riskfree hedging of Black and Scholes, nor the discrete time riskfree hedging of Cox, Ross and Rubinstein, market effects are introduced into the option valuation relation. The new option valuation seems to correct for the systematic mispricing of well-in and well-out of the money options by the Black and Scholes option pricing formula.     

A call on art investments

The art market has seen several booms and busts during the last 20 years and, despite its recent downturn, has received more attention from investors given the low interest environment following the financial crisis. However, participation has been reserved for a few investors and the hedging of exposures remains difficult. This paper proposes to overcome these problems by introducing a call option on an art index, derived from one of the most comprehensive data sets of art market transactions. The option allows investors to optimize their exposure to art. For pricing purposes, non-tradability of the art index is acknowledged and option prices are derived in an equilibrium setting as well as by replication arguments. In the former, option prices depend on the attractiveness of gaining exposure to a previously non-traded risk. This setting further overcomes the problem of art market exposures being difficult to hedge. Results in the replication case are primarily driven by the ability to reduce residual hedging risk. Even if this is not entirely possible, the replication approach serves as a pricing benchmark for investors who are significantly exposed to art and try to hedge their art exposure by selling a derivative.     

Multiperiod Strip Hedging of Forward Commitments

This paper empirically compares two multiperiod hedging strategies—a strip hedge and a stack-and-roll hedge—to hedge a forward commitment. The multiperiod strip hedge is found to outperform the stack-and-roll hedge when forward prices are subject to multiple sources of price uncertainty and to perform no better when only one source of uncertainty is present. Moreover, the relative superiority of the strip hedge increases with the presence of multiple sources of uncertainty. Last, the strip hedge is found to be more costly to trade than the stack-and-roll hedge; however, its cost varies directly with its superiority at reducing risk.     

Convertible Preferred Stock Valuation: Tests of Alternative Models

We undertake a comprehensive test of several contingent claim valuation models adapted to callable, convertible preferred stocks employing a sample of 24 issues and over 27,000 daily price observations. To our knowledge, no large-scale tests of these models have been published. The most complete model tested is an extension of the 1970s developments of Ingersoll and of Brennan and Schwartz, allowing for realistic contract features including delayed callability and nonconstant call prices. The mean and the mean absolute pricing errors are approximately –0.18 percent and 5.4 percent, respectively, and this model fits the data substantially better than the simpler alternatives that ignore such features. Thus, the added computational complexity required for the most complete model examined is evidently merited. Moreover, to the extent that the most complete model accurately mirrors reality, the evidence suggests that investors rationally account for many of the complex features imbedded in typical contracts.     

Behavioral currency hedging for international portfolios

The purpose of this paper is to incorporate behavioral issues as it relates to the active currency hedging of international portfolios in the context of traditional expected utility maximization approach. The uniqueness of the approach is that separate risk aversion parameters are introduced for asset and currency markets. The paper is similar in spirit to Black (Black, F. 1989, Universal hedging, Financial Analysts Journal (July-August), 16-22.), who argued that a portion of foreign equity investments should be permanently unhedged, which is basically postulating that one should take a buy-and-hold position in currency with a fraction of the capital. The behavioral twist included in the traditional expected utility maximization approach results in lower hedge ratios, ceteris paribus, partly due to the asymmetric nature of the compensation structure of currency managers.Since the asymmetric nature of incentive schemes of asset and currency managers dictates how one optimizes the investment portfolio of a pension or endowment fund, the unusual behavior of a given institutional fund manager should not be called “irrational,” only because the optimal currency hedging level deviates from the one derived under rational expectations. This only justifies the use of different hedging strategies by various institutional investors. We describe in detail how the level of hedging should be revised downwards because of behavioral factors. Conclusions are in the context of what people would predict to see in the market, if certain investors behave in an “irrational” way.     

CEO Age,Risk Incentives,and Hedging Strategy

We test whether managerial preferences explain how firms hedge, using hand‐collected data on derivative portfolios in the oil and gas industry. How firms hedge involves choosing between linear contracts and put options, and deciding whether to finance these hedging positions with cash on hand or by selling call options. The likelihood of being a hedger increases with chief executive officer (CEO) age, and near‐retirement CEOs prefer linear hedging instruments. The predictions of the managerial risk incentives theory of hedging strategy, according to which managers with convex compensation schemes avoid hedging strategies that cap upside potential, find no support in the data.     

Determination of Optimal Hedging Strategy for Index Futures: Evidence from Turkey

This paper aims to determine optimal hedge strategy for the Istanbul Stock Exchange (ISE)-30 stock index futures in Turkey by comparing hedging performance of constant and time-varying hedge ratios under mean-variance utility criteria. We employ standard regression and bivariate GARCH frameworks to estimate constant and time-varying hedge ratios respectively. The Turkish case is particularly challenging since Turkey has one of the most volatile stock markets among emerging economies and the turnover ratio as a measure of liquidity is very high for the market. These facts can be considered to highlight the great risk and, therefore, the extra need for hedging in the Istanbul Stock Exchange (ISE). The empirical results from the study reveal that the dynamic hedge strategy outperforms the static and the traditional strategies.     

Combining fair pricing and capital requirements for non-life insurance companies

The aim of this article is to identify fair equity-premium combinations for non-life insurers that satisfy solvency capital requirements imposed by regulatory authorities. In particular, we compare target capital derived using the value at risk concept as planned for Solvency II in the European Union with the tail value at risk concept as required by the Swiss Solvency Test. The model framework uses Merton’s jump-diffusion process for the market value of liabilities and a geometric Brownian motion for the asset process; fair valuation is conducted using option pricing theory. We show that even if regulatory requirements are satisfied under different risk measures and parameterizations, the associated costs of insolvency – measured with the insurer’s default put option value – can differ substantially.