EARLY EXERCISE OF AMERICAN INDEX OPTIONS

We show that exercise of American call options on stock indexes frequently occurs before expiration and attribute this early exercise to the “wild card” option that results from the cash settlement exercise process. The wild card represents an “implied option” to sell the index option at the fixed settlement price; it is therefore a put option on the index call option. We derive a simple one-period valuation model using compound option pricing. Analysis of observed early exercise demonstrates that the wild card feature is a factor influencing early exercise of index options.     

THE PRICING AND HEDGING OF LIMITED EXERCISE CAPS AND SPREADS

Capped options are barrier option spreads that automatically create simultaneous long and short positions. Exchange-traded capped options were introduced in 1991, though with limited volume. Such options, however, have traded on the over-the-counter markets for several years. Most of these options have the unusual feature that they automatically exercise when the underlying asset closes beyond a critical strike, making them a hybrid of European and American options. In this paper I present their boundary conditions and examine the prices, deltas, gammas, and thetas of caps as well as spreads constructed with European and American options. I also examine the effect of permitting exercise based only on the closing price as opposed to exercise at any time the critical strike is reached. I show that assuming that exercise can occur at any time can lead to serious pricing errors. The results have implications for the pricing of barrier options in general, which nearly always exercise early based only on the closing price.     

Jump Diffusion Option Valuation in Discrete Time

We develop a simple, discrete time model to value options when the underlying process follows a jump diffusion process. Multivariate jumps are superimposed on the binomial model of Cox, Ross, and Rubinstein (1979) to obtain a model with a limiting jump diffusion process. This model incorporates the early exercise feature of American options as well as arbitrary jump distributions. It yields an efficient computational procedure that can be implemented in practice. As an application of the model, we illustrate some characteristics of the early exercise boundary of American options with certain types of jump distributions.     

The early exercise premia of America put options on stocks

Using the put-call parity, this paper finds that early exercise premia of short-lived American put options on stocks account for a significant portion of put prices. This finding holds even for out-of-the-money put options. The magnitude of the early exercise premia of American put options with no dividend is positvely related to the degree of moneyness, time to maturity of the put option, and the volatility. The magnitude of the early exercise premia of American put options with dividend is positvely related to the degree of moneyness and the risk-free interest rates.     

The Valuation of Options on Futures Contracts

Rational restrictions are derived for the values of American options on futures contracts. For these options, the optimal policy, in general, involves premature exercise. A model is developed for valuing options on futures contracts in a constant interest rate setting. Despite the fact that premature exercise may be optimal, the value of this American feature appears to be small and a European formula due to Black serves as a useful approximation. Finally, a model is developed to value these options in a world with stochastic interest rates. It is shown that the pricing errors caused by ignoring the location of the interest rate (relative to its long-run mean) range from ?5% to 7%, when the current rate is ±200 basis points from its long-run value. The role of interest rate expectations is, therefore, crucial to the valuation. Optimal exercise policies are found from numerical methods for both models.     

Early Exercise of Put Options on Stocks

U.S. exchange‐traded stock options are exercisable before expiration. While put options should frequently be exercised early to earn interest, they are not. In this paper, we derive an early exercise decision rule and then examine actual exercise behavior during the period January 1996 through September 2008. We find that more than 3.96 million puts that should have been exercised early remain unexercised, representing over 3.7% of all outstanding puts. We also find that failure to exercise cost put option holders $1.9 billion in forgone interest income and that this interest is systematically captured by market makers and proprietary firms.     

Exercises of Executive Stock Options on the Vesting Date

We investigate the motives for executives to exercise executive stock options on the options’ vesting date versus a later early exercise. We find that executives frequently exercise on the vesting date, executives with a greater need for portfolio diversification and riskier underlying stocks are more likely to exercise their options on the vesting date versus a later early exercise, and private information appears less relevant to vesting date exercises.     

Options on the Spot and Options on Futures

This paper analyzes and compares the valuation of two types of options that relate to the same asset: options on the asset itself and options on the futures on the asset. The early exercise privilege plays a central role in explaining the differences between the values of the two options. It is shown that in the case of a cash instrument that does not make interim payments, such as gold, the value of a call option on the spot is smaller than the call option on the futures contract; the opposite is true for put options. The early exercise boundaries, which characterize when it pays to exercise, are also compared and analyzed.     

A lattice model for option pricing under GARCH-jump processes

This study extends the GARCH pricing tree in Ritchken and Trevor (J Financ 54:366–402, 1999) by incorporating an additional jump process to develop a lattice model to value options. The GARCH-jump model can capture the behavior of asset prices more appropriately given its consistency with abundant empirical findings that discontinuities in the sample path of financial asset prices still being found even allowing for autoregressive conditional heteroskedasticity. With our lattice model, it shows that both the GARCH and jump effects in the GARCH-jump model are negative for near-the-money options, while positive for in-the-money and out-of-the-money options. In addition, even when the GARCH model is considered, the jump process impedes the early exercise and thus reduces the percentage of the early exercise premium of American options, particularly for shorter-term horizons. Moreover, the interaction between the GARCH and jump processes can raise the percentage proportions of the early exercise premiums for shorter-term horizons, whereas this effect weakens when the time to maturity increases.     

Efficiency Tests of the Foreign Currency Options Market

Based on a new options transactions data base from the Philadelphia Stock Exchange Foreign Currency Options Market, this paper examines the importance of the effect of nonsynchronous prices and transaction costs on the usual option market efficiency tests. The tests conducted are based on the transaction cost adjusted early exercise and put-call parity pricing boundaries applicable to the American foreign currency options market. The test results show that the put-call parity boundary tests are sensitive to both nonsynchronous prices and transaction costs. The early exercise boundary tests are sensitive to transaction costs but are not very sensitive to simultaneity of the option price and the underlying spot price. Under the no-transaction costs scenario, a large number of early exercise boundary violations is found even when simultaneous spot and option prices are used. These violations disappear when actual transaction costs are taken into account.     

A GUIDE TO VALUING EMPLOYEE STOCK OPTIONS WITH A RELOAD FEATURE

At first glance, executive stock options with reload provisions appear to be more complicated than conventional options, and thus the valuation of such options would appear to be more difficult. But, as the authors demonstrate in this article, such reload options provide the employee with a dominant exercise strategy—namely, to exercise the option whenever it is "in-the-money." And the fact that reload options will always be exercised simplifies their valuation by eliminating a major problem—that associated with employee's risk references and uncertain early exercise—in valuing conventional options.     

An empirical analysis of common stock call exercise: A note

This study tests the hypothesis that common stock call options are exercised rationally and in accordance with the commonly used frictionless markets boundary conditions. Using two years of historical early exercise data for common stock call options, the results show that contrary to the frictionless markets boundary conditions, approximately 20 percent of the early call exercise occurs at times other than ex-dividend dates. While most of the non-dividend related early exercise may be explained by transactions costs, a significant number of contracts appear to be exercised irrationally. These results suggest that failure to incorporate market frictions in option pricing models is likely to lead to specification error.     

Over-the-Counter Option Market Dividend Protection and “Biases” in the Black-Scholes Model: A Note

Most options are traded over-the-counter (OTC) and are dividend “protected;” the exercise price decreases on the ex date by an amount equal to the dividend. This protection completely inhibits the early exercise of American call options. Nevertheless, OTC-protected options have market values which differ systematically from Black-Scholes values for European options on non-dividend paying stocks. The pricing difference is related to both the variance of the underlying stock return and to time until expiration of the option, but it is quite small in dollar amount.     

Private Information and the Exercise of Executive Stock Options

This paper finds strong evidence that executives use private information when exercising their stock options. The most informed executives tend to exercise early, do not exercise on the vest date, do not exercise to capture dividends, exercise a high percentage of their options, and exercise when the option is the least in‐the‐money. We also find that exercises around resignation and retirement are followed by significant negative abnormal returns. Furthermore, the operating performance of firms following exercises motivated by private information is significantly worse than that of firms in which the exercises are not motivated by private information.     

The Pricing of Options with Default Risk

This paper considers the pricing of options with default risk. The comparative statics of such options can differ from those of ordinary options, and early exercise of such American call options can be optimal. Several examples of options with default risk are considered.     

Valuation of American partial barrier options

This paper concerns barrier options of American type where the underlying asset price is monitored for barrier hits during a part of the option’s lifetime. Analytic valuation formulas of the American partial barrier options are provided as the finite sum of bivariate normal distribution functions. This approximation method is based on barrier options along with constant early exercise policies. In addition, numerical results are given to show the accuracy of the approximating price. Our explicit formulas provide a very tight lower bound for the option values, and moreover, this method is superior in speed and its simplicity.     

A note on the efficiency of the binomial option pricing model

We discuss the efficiency of the binomial option pricing model for single and multivariate American style options. We demonstrate how the efficiency of lattice techniques such as the binomial model can be analysed in terms of their computational cost. For the case of a single underlying asset the most efficient implementation is the extrapolated jump-back method: that is, to value a series of options with nested discrete sets of early exercise opportunities by jumping across the lattice between the early exercise times and then extrapolating from these values to the limit of a continuous exercise opportunity set. For the multivariate case, the most efficient method depends on the computational cost of the early exercise test. However, for typical problems, the most efficient method is the standard step-back method: that is, performing the early exercise test at each time step.     

The Valuation of Executive Stock Options in an Intensity-Based Framework

This paper presents a general intensity-based framework to value executive stock options (ESOs). It builds upon the recent advances in the credit risk modeling arena. The early exercise or forfeiture due to voluntary or involuntary employment termination and the early exercise due to the executive's desire for liquidity or diversification are modeled as an exogenous point process with random intensity dependent on the stock price.Two analytically tractable specifications are given where the ESO value, expected time of exercise or forfeiture, and the expected stock price at the time of exercise or forfeiture are calculated in closed-form.     

A closed-form solution to American options under general diffusion processes

This paper investigates American option pricing under general diffusion processes. Specifically, the underlying asset price is assumed to follow a diffusion process in which both the dividend yield and volatility are functions of time and the underlying asset price. Using the generalized homotopy analysis method, the determination of the early exercise boundary is separated from the valuation procedure of American options. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. In addition, the corresponding optimal early exercise boundary and the Greeks are obtained in closed-form solutions. A nonlinear sequence transformation, the Padé technique, is used to effectively accelerate the convergence of the partial sums of the infinite series. As the homotopy constructed in this paper is based on a generalized deformation with a shape parameter and kernel function, the error of the homotopic approximation could be reduced further for a fixed order. Numerical examples demonstrate the validity, effectiveness, and flexibility of the proposed approach.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.