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In some contexts, this illiquidity of executive stock options is referred to as non-transferability. In others, the problem is cast in terms of the highly concentrated portfolios that managers hold, an implication of which is that managers could not trade the options to diversify. The notion of option liquidity usually conjures up images of trading pits at the Chicago Board Options Exchange or other exchanges. The existence of an active trading pit gives a powerful visual image of liquidity, but, as evidenced by the success of electronic options exchanges such as New York's International Securities Exchange and Frankfurt's EUREX, a trading pit is hardly a requirement for liquidity. The existence of a guaranteed market for standardized options as implied by options exchanges (whether pit-based or electronic) further gives a misleading appearance of high liquidity. There is also a very large market for customized over-the-counter options. It is a misconception to think that these options are not liquid when they are simply not standardized. If an investor can create a highly customized long position in an option, that investor should be able to create a highly customized short position in the same option at a later date before expiration. If both options are created through the same dealer, they will usually be treated as an offset, as they would if they were standardized options clearing through a clearinghouse. If the two transactions are not with the same dealer, they would both remain alive, but the market risks would offset. Only the credit risk, a factor we ignore in this paper, would remain. Hence, these seemingly illiquid options are, for all practical purposes, liquid.²

The purpose of this paper is to present an arbitrarily accurate approximation for the value of European options written on a Black‐Scholes stock paying a discrete dividend.

The proposed method is a computational method for the analytical solution of the problem.

It was found that the proposed method is computationally faster than any other exact computational available method, including Monte‐Carlo simulations.

The method is applied for a single dividend payment, but can be extended for several payments. The exact amount of the dividend must be known ex‐ante, as well as the precise date of payment.

The paper provides the most efficient way to compute with absolute precision the value of European options on dividend‐paying assets, under the Black‐Scholes assumption.

The computing time in the approach is several orders of magnitude faster than with traditional Monte Carlo methods, for the same desired accuracy.

The purpose of this paper is to explain the Black–Scholes model with minimal technical requirements and to illustrate its impact from a business perspective.

The paper employs simple accounting concepts and an argument part based on business need.

The Black–Scholes partial differential equation can be derived in many ways, some easy to understand, some hard, some useful and others not. The two methods in this paper are extremely insightful.

The paper takes a big-picture view of derivatives valuation. As such, it is a simple accompaniment to more complex methods and aims to keep modelling grounded in reality.

The development of standardized measures of institution‐wide volatility exposures has so far lagged that for measures of asset price and interest‐rate exposure—largely because it is difficult to reconcile the various mathematical models used to value options. Recent mathematical results, however, can be used to construct standardized measures of volatility exposure. We consider here techniques for reconciling “vegas” for financial options valued using stochastic models that may be mathematically inconsistent with each other.

This exercise exposes students to the accounting for stock option modifications and option service and performance conditions, requiring research in the Financial Accounting Standards Board (FASB) Accounting Standards Codification and the use of the Black-Scholes option pricing model.

Students identify and apply accounting standards to account for stock option plans, stock option modifications, acquired stock option plans, and service and performance conditions that relate to stock option plans. Indirect student feedback suggests that students view the exercise as valuable. Comments include that the exercise reinforces and expands their knowledge of real-world stock compensation plans. Direct assessment data using grading rubrics finds that most students meet instructor expectations.

The exercise enhances critical thinking skills, increases professional research practice, and improves written skills. It introduces students to common real-world events and reinforces their learning related to stock compensation.

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.

Using volatility cones as the estimate of actual volatility instead of GARCH models, the purpose of this paper is to explore whether volatility arbitrage strategy can provide positive profits and how the transaction costs existed in the real market affect the effectiveness of volatility arbitrage strategy.

A number of hedging approaches proposed to improve the hedging results and final returns of Black-Scholes model are analyzed and compared.

The general finding is that volatility arbitrage strategy can provide satisfactory returns based on the samples in Chinese market. Regarding transaction costs, the variable bandwidth delta and delta tolerance approach showed better results. Besides, choosing futures together with ETFs as hedging underlying can increase the VaR for better risk management.

This paper offers a new method for volatility arbitrage in Chinese financial market.

This paper researches the profitability of the volatility arbitrage strategy on ETF 50 options using volatility cones method for the first time. This method has advantage over the point-wise estimation such as GARCH model and stochastic volatility model.

Reviews previous research on the timing of employee stock option exercise decisions and share price performance before and after insider trading. Analyses the 1992‐1993 exercise behaviour of top executives at 65 large US firms using the Black‐Scholes (1973) value (less anticipated dividends) as a benchmark to compare with the intrinsic value (market price minus exercise price) at the date of exercise. Finds options are exercised when the two values are roughly equal, i.e. that executives’ decisions are not risk‐averse or biased by private information. Also shows a tendency for the subsequent change in share prices to be lower when the intrinsic value is less than the Black‐Scholes value at the time of exercise. Considers consistency with other research and the implications of the findings.

The purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model.

In this paper, the authors employ the block-pulse operational matrix algorithm to approximate the solution of the fractional Black-Scholes equation with the initial condition for a European option pricing problem.

The fractional derivative will be described in the Caputo sense in this paper. The authors show the accuracy and computational efficiency of the proposed algorithm through some numerical examples.

This is the first paper that considers an alternative algorithm for pricing a European option using the fractional Black-Scholes model.