On Valuing American Call Options with the Black-Scholes European Formula

Empirical papers on option pricing have uncovered systematic differences between market prices and values produced by the Black-Scholes European formula. Such “biases” have been found related to the exercise price, the time to maturity, and the variance. We argue here that the American option variant of the Black-Scholes formula has the potential to explain the first two biases and may partly explain the third. It can also be used to understand the empirical finding that the striking price bias reverses itself in different sample periods. The expected form of the striking price bias is explained in detail and is shown to be closely related to past empirical findings.     

Skewness and Stock Option Prices

Abstract

In the classical Black-Scholes model, the logarithm of the stock price has a normal distribution, which excludes skewness. In this paper we consider models that allow for skewness. We propose an option-pricing formula that contains a linear adjustment to the Black-Scholes formula. This approximation is derived in the shifted Poisson model, which is a complete market model in which the exact option price has some undesirable features. The same formula is obtained in some incomplete market models in which it is assumed that the price of an option is defined by the Esscher method. For a European call option, the adjustment for skewness can be positive or negative, depending on the strike price.     

The valuation of compound options

This paper presents a theory for pricing options on options, or compound options. The method can be generalized to value many corporate liabilities. The compound call option formula derived herein considers a call option on stock which is itself an option on the assets of the firm. This perspective incorporates leverage effects into option pricing and consequently the variance of the rate of return on the stock is not constant as Black-Scholes assumed, but is instead a function of the level of the stock price. The Black-Scholes formula is shown to be a special case of the compound option formula. This new model for puts and calls corrects some important biases of the Black-Scholes model.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

Companies' Modest Claims About the Value of CEO Stock Option Awards

This paper analyzes company disclosures of CEO stock option values in compliance with the SEC's regulations for reporting executive compensation data to stockholders. Companies appear to exploit the flexibility of the regulations to reduce the apparent value of managerial compensation. Companies shorten the expected lives of stock options and unilaterally apply discounts to the Black-Scholes formula. Theoretical support for these adjustments is often thin, and companies universally ignore reasons that the Black-Scholes formula might underestimate the value of executive stock options. The findings not only cast light upon how corporations value executive stock options, but also provide a means of forecasting compliance with controversial new FASB requirements for firms to disclose the compensation expense represented by executive stock options.     

SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.     

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.     

Executive Stock Options

Abstract

As a part of the compensation package many companies provide executives with executive stock options, which are call options with additional restrictions. They provide some financial advantages to the executives and help the company retain the service of the executives who improve the company’s earnings and management.

Until recently the values of the executive stock options were not required to be disclosed in the company?s financial reports. But recent statements from the Financial Accounting Standards Board (FASB) have made it necessary to value these executive stock options. The valuation of executive stock options is also required for investors and financial practitioners. This paper considers the award of performance-based executive stock options when the stock price at the time of stock option award exceeds a given preassigned value. It is assumed that the stock price follows a geometric Brownian motion, and that the number of stock options awarded at any time depends on the stock price at that time.

A valuation formula is derived using the method of Esscher transforms for a multiyear award plan. The closed-form formula derived is similar to the Black-Scholes formula for options and utilizes the standard bivariate normal distribution function, which is available in statistical software. In this paper the number of stock options awarded is assumed to be in a specific form, but the theory presented can be modified to suit other forms of award structure. Moreover, by suitable choice of parameters, a valuation formula is also presented for the award of fixed-value executive stock options grants; this formula is also in a closed form and involves cumulative distribution values of the standard normal random variable. Numerical illustrations of the use of the valuation formulas are presented.     

THE DESIGN OF MULTI-YEAR STOCK OPTION PLANS

Despite the explosion in the corporate use of stock options, the incentives created by stock options are not well understood by either the boards who grant them or the executives who are meant to be motivated by them. A major source of confusion stems from the corporate practice of using multi-year stock option plans. Such multi-year grants create subtle, potentially important links between current performance and future grants that can significantly dilute incentives for better performance.
For example, so-called "fixed value" plans provide very weak, even perverse, incentives ex ante since the value of future option grants is completely insulated from current performance. Under such plans, an executive's reward for superior performance is to receive fewer options, and to receive more options for substandard performance. In contrast, the fixed number plan creates an intrinsic link between changes in this year's stock price and changes in the value of future option grants.
The author also reports the findings of new empirical research that shows that stock option plans, taken as a whole, have a pay-to-performance correlation that is eight times stronger than that of salary and bonus. But, consistent with the analysis above, fixed value option plans have pay-to-performance that is only six times that of salary and bonus, as compared to ten times for fixed number plans.     

Options markets, self-fulfilling prophecies, and implied volatilities

This paper answers the following often asked question in option pricing theory: if the underlying asset's price does not satisfy a lognormal distribution, can market prices satisfy the Black-Scholes formula just because market participants believe it should? In complete markets, if the underlying asset's objective distribution is not lognormal, then the answer is no. But, in an incomplete market, if the underlying asset's objective distribution is not lognormal and all traders believe it is, then the answer is yes! The Black-Scholes formula can be a self-fulfilling prophecy. The proof of this second assertion consists of generating an economy where self-confirming beliefs sustain the Black-Scholes formula as an equilibrium. An asymmetric information model is provided, where the underlying asset's price has stochastic volatility and drift. This model is distinct from the existing pricing models in the literature, and it provides new empirical implications concerning Black-Scholes implied volatilities and the bid/ask spread. Similar to stochastic volatility models, this model is consistent with the implied volatility “smile” pattern in strike prices. In addition, it is consistent with implied volatilities being biased predictors of future volatilities.     

The stop-loss start-gain paradox and option valuation: a new decomposition into intrinsic and time value

The downside risk in a leveraged stock position can be eliminatedby using stop-loss orders. The upside potential of such a positioncan be captured using contingent buy orders. The terminal payoffto this stop-loss start-gain strategy is identical to that ofa call option, but the strategy costs less initially. This articleresolves this paradox by showing that the strategy is not self-financingfor continuous stock-price processes of unbounded variation.The resolution of the paradox leads to a new decomposition ofan option's price into its intrinsic and time value. When thestock price follows geometric Brownian motion, this decompositionis proven to be mathematically equivalent to the Black-Scholes(1973) formula.     

OPTION PRICING AND THE ARBITRAGE PRICING THEORY

This paper applies the arbitrage pricing theory to option pricing. Under certain distribution assumptions or the assumption that there is only one common factor, the underlying asset of an option is the sole risky factor that explains its expected return. Based upon this relationship, a new and simple option-pricing formula is derived, and some important existing option-pricing formulae are reproduced. Empirical results show that the new formula performs as well as the Black-Scholes formula.     

Call options and the risk of underlying securities

Merton (1973) in his seminal article ‘Theory of Rational Option Pricing’ showed that the rationally determined price of a call option is a non-decreasing function of the ‘riskness’ of its associated common stock. In deriving his results, Merton made restrictive assumptions about the way the market prices payoff distributions, and used the Rothschild-Stiglitz (1970) measure to compare the riskiness of securities. I show by means of an example that the Merton result will not in general be true. I then derive a sufficient condition for the option on one stock to have higher market value than the option on another stock, when both the stocks have the same price, and explain why the Merton result is valid in the Black-Scholes environment.     

Information and option pricings

How can one relate stock fluctuations and information-based human activities? We present a model of an incomplete market by adjoining the Black-Scholes exponential Brownian motion model for stock fluctuations with a hidden Markov process, which represents the state of information in the investors' community. The drift and volatility parameters take different values depending on the state of this hidden Markov process. Standard option pricing procedure under this model becomes problematic. Yet, with an additional economic assumption, we provide an explicit closed-form formula for the arbitrage-free price of the European call option. Our model can be discretized via a Skorohod embedding technique. We conclude with an example of a simulation of IBM stock, which shows that, not surprisingly, information does affect the market.     

Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978

The tests reported here differ in several ways from those of most other papers testing option pricing models: an extremely large sample of observations of both trades and bid-ask quotes is examined, careful consideration is given to discarding misleading records, nonparametric rather than parametric statistical tests are used, reported results are not sensitive to measurement of stock volatility, special care is taken to incorporate the effects of dividends and early exercise, a simple method is developed to test several option pricing formulas simultaneously, and the statistical significance and consistency across subsamples of the most important reported results are unusually high. The three key results are: (1) short-maturity out-of-the-money calls are priced significantly higher relative to other calls than the Black-Scholes model would predict, (2) striking price biases relative to the Black-Scholes model are also statistically significant but have reversed themselves after long periods of time, and (3) no single option pricing model currently developed seems likely to explain this reversal.     

The Effect of Executive Stock Option Plans on Stockholders and Bondholders

Executive stock option plans have asymmetric payoffs that could induce managers to take on more risk. Evidence from traded call options and stock return data supports this notion. Implicit share price variance, computed from the Black-Scholes option pricing model, and stock return variance increase after the approval of an executive stock option plan. The event is accompanied by a significant positive stock and a negative bond market reaction. This evidence is consistent with the notion that executive stock options may induce a wealth transfer from bondholders to stockholders.     

An analytical approximation to the option formula for the GARCH model

This article derives an analytical approximation to the option formula for a spot asset price whose conditional variance equation follows a nonlinear asymmetric GARCH (NGARCH) process. The approximate option formula, which is just a volatility adjustment in comparison to the Black-Scholes (BS) formula, is very simple and provides the volatility term structure of spot asset prices. Also, the formula shows that the most characteristic feature of an NGARCH model appears in the vega of a European option, which depends on both the spread between the long-run variance and the current one and a parameter reproduced from the stationary property of the conditional variance. This methodology can be easily extended to an option formula for the generalized GARCH process.     

IRC Section 162(m) and the law of unintended consequences

In 1993, Section 162(m) of the U.S. Internal Revenue Code was passed into law with the intent to reign in outsized executive compensation by eliminating the tax-deductibility of executive compensation above $1 million unless the excess compensation was performance-based. An unintended consequence of the legislation was that executives' total compensation actually increased in the post-1993 period, largely due to a dramatic increase in employee stock options. Employee stock options have unintended consequences of their own. The economic value of stock options may be influenced by executive decision-making when the options are valued using the Black-Scholes model or some variant thereof. Our findings suggests an unintended consequence that executives used their discretion to positively impact the performance-based component of their compensation through actions increasing share price volatility and reducing dividend yields, assumptions implicit in option valuation models.     

Valuation of American call options on dividend-paying stocks: Empirical tests

This paper examines the pricing performance of the valuation equation for American call options on stocks with known dividends and compares it with two suggested approximation methods. The approximation obtained by substituting the stock price net of the present value of the escrowed dividends into the Black-Scholes model is shown to induce spurious correlation between prediction error and (1) the standard deviation of stock return, (2) the degree to which the option is in-the-money or out-of-the-money, (3) the probability of early exercise, (4) the time to expiration of the option, and (5) the dividend yield of the stock. A new method of examining option market efficiency is developed and tested.     

A Proposal for Expensing Employee Compensatory Stock Options for Financial Reporting Purposes

This article presents an accounting approach for employee stock options based on the insight that the currentperiod compensation expense should reflect only that part of the option value that is earned independent of the obligation of continued employment. Given that the maturity of vested options is typically shortened to 90 days when an employee resigns or is terminated, this method views the employee as owning a 90-day option (even if the stated maturity of the option is ten years) and earning a 90-day extension to that option each quarter as a result of the employee's continued employment. In the case of vested options, the compensation expense in each quarterly accounting period is thus the value of the 90-day extension of the option's maturity. There is no option expense in the quarter when the option is either exercised or expires.
In the case of unvested options, the expected option value at vesting should be estimated quarterly starting at the time of grant and the corresponding estimated expense should be revised and allocated as a pro rata accrual each quarter over the vesting period. The cumulative expense over the entire vesting period will equal the fair market value of the option at its vesting date.
Besides reflecting the economics of the exchange of value for labor involved in stock option grants, this approach has a number of practical advantages:

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  The 90-day maturity permits the use of publicly traded options to determine fair market value and makes Black-Scholes and other (lattice) pricing models more reliable.