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How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the recent stock market crash and the subsequent prices of U.S. index options. With the development of rich and deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black–Scholes, naïve volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-to-expiration, or times. The latter amounts to examining predictions of future implied volatilities.

Certain naïve predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better-performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an “efficient markets result” makes these forecasts difficult, and improvements to the option-pricing models might then be limited.

The common view of many regulators and practitioners is that the minimum risk maturity gap is equal to zero. However, because of the interest sensitivity of such non‐gap items as the average spread between asset and liability rates, lending activity, fee income and prepayments, the minimum risk gap could be significantly different from zero. We formulate and test a model for a sample of four hundred and twenty six thrift institutions. The results strongly suggest that the minimum risk maturity gap is positive for the average firm in the sample and that there is substantial cross‐sectional variability in the ratio of the minimum risk gap to assets. This suggests that attempts to regulate interest rate risk using a uniform gap as a benchmark are misdirected. Finally, we provide some evidence that there is, in fact, a positive cross‐sectional relationship between measured maturity gap positions and our estimates of the minimum risk maturity gap.

Many investors have been disappointed with the practical results of portfolio insurance programs, which attempt to achieve option‐like results through dynamic hedging. This article takes the simplest form of dynamic hedging, constant proportion portfolio insurance (CPPI), as a model for developing a more optimal approach. The author uses Monte Carlo simulation to model the multi‐period median growth in discretionary wealth. In addition, he constructs leverage policies that mitigate the practical drawbacks to dynamic hedging. The article also shows that self‐imposed ex ante borrowing constraints (not the ex post constraint imposed by a margin call) can, under certain conditions, improve the performance of dynamic hedging with respect to median terminal wealth.

This paper aims to find optimal emission reduction investment strategies for the manufacturer and examine the effects of carbon cap-and-trade policy and uncertain low-carbon preferences on emission reduction investment strategies.

This paper studied a supply chain consisting of one manufacturer and one retailer, in which the manufacturer is responsible for emission reduction investment. The manufacturer has two emission reduction investment strategies: (1) invest in traditional emission reduction technologies only in the production process and (2) increase investment in smart supply chain technologies in the use process. Then, three different Stackelberg game models are developed to explore the benefits of the manufacturer in different cases. Finally, this paper coordinates between the manufacturer and the retailer by developing a revenue-sharing contract.

The manufacturer's optimal emission reduction strategy is dynamic. When consumers' low-carbon preferences are low and the government implements a carbon cap-and-trade policy, the manufacturer can obtain the highest profit by increasing the emission reduction investment in the use process. The carbon cap-and-trade policy can encourage the manufacturer to reduce emissions only when the initial carbon emission is low. The emission reduction, order quantity and the manufacturer's profit increase with the consumers' low-carbon preferences. And the manufacturer can adjust the emission reduction investment according to the emission reduction cost coefficient in two processes.

This paper considers the investment of emission reduction technologies in different processes and provides theoretical guidance for manufacturers to make a low-carbon transformation. Furthermore, the paper provides suggestions for governments to effectively implement carbon cap-and-trade policy.

Option pricing theory enables computation of the price of an option using different variables associated with the underlying security and options contract. The purpose of this study is to assess research trends that emerged in the field of option pricing. This study reviews existing literature of the option pricing domain, both qualitatively and quantitatively, and identifies potential themes for future research.

This study adopts bibliometric analysis method to explore literature published in the option pricing domain. As part of bibliometric analysis, this study considers both descriptive and network analysis to assess publication trends. For descriptive analysis, the “bibliometrix” package proposed by Aria and Cuccurullo (2017) is used and for network analysis, VOS viewer (Van Eck and Waltman, 2017) and Gephi (Bastian et al., 2009) are used.

This study identifies research trends, top researchers, articles, journals and contributions from institutions and countries in the option pricing domain. It identifies four clusters that show different directions and also focuses on past studies on the same subject. It explores research gaps by performing an in-depth analysis of existing literature on option pricing and suggests the way forward for research in this area.

To the best of the authors’ knowledge, no previous studies have attempted to analyze the literature published in the option pricing domain. This study fulfils this research gap by conducting a comprehensive analysis of studies in the option pricing area. This study identifies quality research work published in the domain, research trends, contribution by most relevant researchers, contributions across geographies and institutions and the connections among these aspects. This study also identifies important themes and provides directions for future research.

In this second installment, the author addresses some of the problems associated with empirically validating contingent‐claim models for valuing risky debt. The article uses a simple contingent claims risky debt valuation model to fit term structures of credit spreads derived from data for U.S. corporate bonds. An essential component to fitting this model is the use of expected default frequency; the estimate of the firms' expected default probability over a specific time horizon. The author discusses the statistical and econometric procedures used in fitting the term structure of credit spreads and estimating model parameters. These include iteratively reweighted non‐linear least squares are used to dampen the impact of outliers and ensure convergence in each cross‐sectional estimation from 1992 to 1999.

Three years after the introduction of exchange‐traded options on the American scene, a call options market was made with ten popular common stocks in Singapore in early 1977. Only calls were traded and no puts were introduced. After six months of trading actively, volume dwindled, and the market was withdrawn in early 1980. Three currency options markets introduced in 1987 continues to thrive at the time of this study. The reason for the demise of the call options market is mainly the significant mispricing of the contracts as most contracts were systematically above the theoretical fair prices. Low volatility in the spot market after the calls were introduced, availability of alternative speculative instrument for traders, high transaction costs and the lack of knowledge about the complexity of options trades are suggested as reasons for the failure of the market. As a new options market has been introduced again in March, 1993, it is worthwhile to learn from the past.

This case is a vehicle for discussing the theory and practice of option pricing through the valuation of warrants of Chrysler held by the U.S. government and of the loan guarantee provided by the U.S. government to Chrysler.

Recent US federal court rulings have provided new guidance on the use of economic models of bargaining in estimating reasonable royalty damages in patent cases. After reviewing relevant case law and providing an overview of the bargaining range approach, we describe one analytic method (the Rubinstein Bargaining Model) for developing a quantitative starting point with which to divide a bargaining range and explain how it can be tied, at least in part, to the facts and circumstances of the parties around the time of the Hypothetical Negotiation. We also describe how this approach can be used in conjunction with an analysis of other quantitative and qualitative factors related to the bargaining power of the parties, to help estimate reasonable royalty damages.