Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations

On the basis of the theory of a wedge between the physical and risk‐neutral conditional volatilities in Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010), we develop a modification of the GARCH option pricing model with the filtered historical simulation proposed in Barone‐Adesi, G., Engle, R. F., & Mancini, L. (2008). The one‐day‐ahead conditional volatilities under physical and risk‐neutral measures are the same in the previous model, but should have been allowed to be different. Using extensive data on S&P 500 index options, our approach, which employs one‐day‐ahead risk‐neutral conditional volatility estimated from the cross‐section of the option prices (in contrast to the existing GARCH option pricing models), maintains theoretical consistency under conditional non‐normality, and improves the empirical performances. Remarkably, the risk‐neutral volatility dynamics are stable over time in this model. In addition, the comparison between the VIX index and the risk‐neutral integrated volatility economically validates our approach. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:1–28, 2013     

Multifactor and analytical valuation of treasury bond futures with an embedded quality option

A closed‐form pricing solution is proposed for the quality option embedded in Treasury bond futures contracts, under a multifactor and D. Heath, R. Jarrow, and A. Morton (1992) Gaussian framework. Such an analytical solution can be obtained through a conditioning approximation, in the sense of M. Curran (1994) and L. Rogers and Z. Shi (1995), or via a rank 1 approximation, following A. Brace and M. Musiela (1994). Monte Carlo simulations show that both approximations are extremely accurate and easy to calculate. Application of the proposed pricing model to the EUREX market from January 2000 through May 2004, yields an excellent fit and an insignificant estimate of the quality option magnitude. On average, this delivery option accounts for only of the futures prices. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:275–303, 2007     

A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013     

A modified static hedging method for continuous barrier options

This study modifies the static replication approach of Derman, E., Ergener, D., and Kani, I. (1995, DEK) to hedge continuous barrier options under the Black, F. and Scholes, M. (1973) model. In the DEK method, the value of the static replication portfolio, consisting of standard options with varying maturities, matches the zero value of the barrier option at n evenly spaced time points when the stock price equals the barrier. In contrast, our modified DEK method constructs a portfolio of standard options and binary options with varying maturities to match not only the zero value but also zero theta on the barrier. Our numerical results indicate that the modified DEK approach improves performance of static hedges significantly for an up‐and‐out call option under the BS model even if the bid–ask spreads are considered. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Are credit spreads too low or too high? A hybrid barrier option approach for financial distress

Based on the works of Brockman, P. and Turtle, H. J. (2003) and Giesecke, K. (2004), we propose in this study a hybrid barrier option model to explain observed credit spreads. It is free of problems with the structural model, which underprescribed credit spreads for investment grade corporate bonds and overprescribed the high‐yield issues. Unlike the standard barrier option approach, our hybrid model does not imply, for high‐yield issues with firms under financial stress, a reduction of credit spreads while firm value actually falls. Our empirical analysis supports that when credit spreads are quoted abnormally higher or rising faster than expected, unexpected changes tend to persist. Otherwise a significant and prompt reversion to long‐term equilibrium takes place. This asymmetric pricing phenomenon is validated with a method introduced by Enders, W. and Granger, C. W. J. (1998) and Enders, W. and Siklos, P. L. (2001). The pricing asymmetry could not have been produced by a structural model employing only standard option. But it is consistent with a hybrid barrier option model. Our model characterizes the valuation of debt under financial stress and the asymmetric price pattern better than both the classical structural and the standard barrier option approaches. It can be extended to the study of individual CDS for its better liquidity than individual corporate bonds. This study provides helpful implications especially for the medium and high‐yield issues in pricing as well as portfolio diversification. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:1161–1189, 2009     

Too many options? Theory and evidence on option exchange design

A model of option exchange design is proposed and tested. The model allows investors to choose among several exchange‐traded options based on a trade‐off between standardization costs and liquidity/transaction costs. It employs a spatial economics approach to provide results for the existence of markets for particular option contracts on the exchange, a comparison of exchange design by a social planner and a profit‐maximizing monopolist (corresponding to the idea that most derivatives exchanges centralize the design and creation of option contracts), and comparative statics that can potentially aid decision makers in the design of option exchanges. In the empirical work, open interest is analyzed for Chicago Board Options Exchange (CBOE) options on the stocks in the S&P 100 index. In accordance with the model's predictions, open interest forms a previously undocumented seesaw pattern across strike prices, clustering around certain strike prices, and dropping off for the adjacent strike prices. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:533–570, 2006     

The performance of alternative futures buy‐write strategies

This study compares the performance of a conventional buy‐write (or covered call writing) and a dynamic buy‐write strategy. The conventional strategy generally enhances portfolio returns in low volatility conditions but underperforms the underlying cash asset in sharply rising markets. The dynamic strategy adjusts the moneyness of the option according to market conditions. The study extends Hill, J. M., Balasubramanian, V., Gregory, K., and Tierens, I. ( 2006 ) and tests how and to what extent market volatility and market direction affect the performance of these two strategies. The study finds that both strategies offer significant positive α, higher returns and lower standard deviations than the market. Consistent with prior research, the abnormal returns of the buy‐write strategies can be attributed to a volatility premium embedded in the options prices. The buy‐write returns from the Hong Kong market appear to be lower than those found in the U.S. and U.K. markets. The conventional buy‐write outperforms the dynamic strategy in both high and low volatility environments, and in sharply falling markets. However, by targeting exercise probability, the dynamic strategy provides a greater upside in sharply rising markets. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Relationships and networks in marketing: International perspectives

Gemünden, H.G., Ritter, T., and Walter, A. (eds.): (1997) Relationships and Networks in International Markets, NY: Pergamon, 460 pages, ISBN 0‐08‐ 043963‐5 (hardcover). Naudé, P. and Turnbull, P. (eds.): (1998) Network Dynamics in International Markets, NY: Pergamon, 321 pages, ISBN 0‐08‐043358‐8 (hardcover).     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

On the Rate of Convergence of Discrete‐Time Contingent Claims

This paper characterizes the rate of convergence of discrete‐time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all‐or‐nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete‐time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous‐time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.     

PRICING DERIVATIVES ON MULTISCALE DIFFUSIONS: AN EIGENFUNCTION EXPANSION APPROACH

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class of derivative‐assets. The payoff of the derivative‐assets may be path‐dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast‐varying and one slow‐varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative‐assets: a vanilla option on a defaultable stock, a path‐dependent option on a nondefaultable stock, and a bond in a short‐rate model.     

Multifactor implied volatility functions for HJM models

This study evaluates two one‐factor, two two‐factor, and two three‐factor implied volatility functions in the HJM class, with the use of eurodollar futures options across both strike prices and maturities. The primary contributions of this article are (a) to propose and test three implied volatility multifactor functions not considered by K. I. Amin and A. J. Morton (1994), (b) to evaluate models using the AIC criteria as well as other standard criteria neglected by S. Y. M. Zeto (2002), and (c) to .nd that multifactor models incorporating the exponential decaying implied volatility functions generally outperform other models in .tting and prediction, in sharp contrast to K. I. Amin and A. J. Morton, who find the constantvolatility model superior. Correctly specified and calibrated simple constant and square‐root factor models may be superior to inappropriate multifactor models in option trading and hedging strategies. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:809–833, 2006     

Examination of long‐term bond iShare option selling strategies

This article examines volatility trades in Lehman Brothers 20+ Year US Treasury Index iShare (TLT) options from July 2003 through May 2007. Unconditionally selling front contract strangles and straddles and holding for one month is highly profitable after transactions costs. Short‐term option selling strategies are enhanced when implied volatility is high relative to time series volatility forecasts. Risk management strategies such as stop loss orders detract from profitability, while take profit orders have only modest favorable effects on profitability. Overall, the results demonstrate that TLT option selling strategies offered attractive risk‐return tradeoffs over the sample period. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:465–489, 2010     

On a Mean—Generalized Semivariance Approach to Determining the Hedge Ratio

A new mean‐risk hedge ratio based on the concept of generalized semivariance (GSV) is proposed. The proposed mean‐GSV (M‐GSV) hedge ratio is consistent with the GSV‐based risk–return model developed by Fishburn (1977), Bawa (1975, 1978), and Harlow and Rao (1989). The M‐GSV hedge ratio can also be considered an extension of the GSV‐minimizing hedge ratio considered by De Jong, De Roon, and Veld (1997) and Lien and Tse (1998, 2000). The M‐GSV hedge ratio is estimated for Standard & Poor's (S&P) 500 futures and compared to six other widely used hedge ratios. Because all the hedge ratios considered are known to converge to the minimum‐variance (Johnson) hedge ratio under joint normality and martingale conditions, tests for normality and martingale conditions are carried out. The empirical results indicate that the joint normality and martingale hypotheses do not hold for the S&P 500 futures. The M‐GSV hedge ratio varies less than the GSV hedge ratio for low and relevant levels of risk aversion. Furthermore, the M‐GSV hedge ratio converges to a value different from the values of the other hedge ratios for higher values of risk aversion. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 581–598, 2001     

Valuing real options using implied binomial trees and commodity futures options

A real option on a commodity is valued using an implied binomial tree (IBT) calibrated using commodity futures options prices. Estimating an IBT in the absence of spot options (the norm for commodities) allows real option models to be calibrated for the first time to market‐implied probability distributions for commodity prices. In addition, the existence of long‐dated futures options means that good volatility estimates may now be incorporated into capital budgeting evaluations of real options projects with long planning horizons. An example is given using gold futures options and a real option to extract gold from a mine. A detailed out‐of‐sample test is included that shows how IBT option pricing errors evolve on subtrees emanating from future levels of the underlying asset. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:203–226, 2007     

Is there information in the volatility skew?

Since the 1987 crash, option prices have exhibited a strong negative skew, implying higher implied volatility for out‐of‐the‐money puts than at‐ and in‐the‐money puts. This has resulted in incorporating multiple jumps and stochastic volatility within the data generating process to improve the Black–Scholes model in an attempt to capture negative skewness and a highly leptokurtic distribution. The general conclusion is that there is a large jump premium in the short term, which best explains the significant negative skew for short maturity options. Alternative explanations for the negative skew are related to market liquidity driven by demand shocks and supply shortages. Regardless of the explanation for the negative skew, we assess the information content in the shape of the skew to infer if the option market can accurately forecast stock market crashes and/or spikes upward. We demonstrate, using all options on the S&P 100 from 1984–2006, that the shape of the skew can reveal with significant probability when the market will crash or spike. However, we find the magnitude of the spike prediction is not economically significant. Our findings are strongest for the short‐term out‐of‐the money puts, consistent with the notion of investors' aversion to large negative movements. We also find that the power of the crash/spike prediction decreases with an increase in the time to option maturity. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:921–959, 2007     

Efficient static replication of European options under exponential Lévy models

This study proposes a new scheme for the static replication of European options and their portfolios. First, a general approximation formula for efficient static replication as an extension of Carr P. and Chou A. (1997, 2002) and Carr P. and Wu L. (2002) is derived. Second, a concrete procedure for implementing the scheme by applying it to plain vanilla options under exponential Lévy models is presented. Finally, numerical examples in a model developed by Carr, P., Geman, H., Madan, D., and Yor M. (2002) are used to demonstrate that the replication scheme is more efficient and more effective in practice than a standard static replication method. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 29:1–15, 2009     

Richardson extrapolation techniques for the pricing of American‐style options

In this article, the authors reexamine the American‐style option pricing formula of R. Geske and H.E. Johnson (1984), and extend the analysis by deriving a modified formula that can overcome the possibility of nonuniform convergence (which is likely to occur for nonstandard American options whose exercise boundary is discontinuous) encountered in the original Geske–Johnson methodology. Furthermore, they propose a numerical method, the Repeated‐Richardson extrapolation, which allows the estimation of the interval of true option values and the determination of the number of options needed for an approximation to achieve a given desired accuracy. Using simulation results, our modified Geske–Johnson formula is shown to be more accurate than the original Geske–Johnson formula for pricing American options, especially for nonstandard American options. This study also illustrates that the Repeated‐Richardson extrapolation approach can estimate the interval of true American option values extremely well. Finally, the authors investigate the possibility of combining the binomial Black–Scholes method proposed by M. Broadie and J.B. Detemple (1996) with the Repeated‐Richardson extrapolation technique. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:791–817, 2007     

Effects of rollover strategies and information stability on the performance measures in options markets: An examination of the KOSPI 200 index options market

One of the most widely used option valuation models among practitioners is the ad hoc Black–Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider two rollover strategies (nearest‐to‐next strategy and next‐to‐next) used in the AHBS model to investigate their effect on pricing errors. We suggest a new rollover strategy, next‐to‐next strategy, and demonstrate that our rollover strategy produces more consistent estimates between in‐sample market and model option prices. Probably even more important is that our new rollover strategy makes more accurate out‐of‐sample forecasts for 1‐day or 1‐week ahead prices. Prior literature has documented some anomalies associated with the use of AHBS model, for example, an overfitting problem. A secondary contribution is that our new rollover strategy does not suffer from this overfitting critique. Third, this study uses the mean square error for out‐of‐sample pricing and price changes to determine how the options investors are influenced by moneyness. The results indicate that underpricing (or overpricing) by the AHBS model for the near‐the‐money category is more likely to be maintained for the next several trading days but that such a phenomenon is disappeared for the deep out‐of‐the‐money category. Finally, we suggest the ratio of the number of option contracts to differences in strike prices available for trading between the current day and the previous day(s) as a good categorizing factor for options, such as moneyness. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark