Convertible Securities,Employee Stock Options and the Cost of Equity

We provide a method for calculating the cost of equity and the cost of capital in the presence of convertible securities and employee stock options. We demonstrate how this approach can be applied if a company already has issued convertible claims or if it is considering doing so for the first time. We provide several numerical examples illustrating the significance of errors in estimating the cost of capital that can result when (1) employee stock options are ignored or (2) the observable stock price is used as a proxy for the unobservable underlying asset.     

Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model

This paper develops two novel methodologies for pricing and hedging European-style barrier option contracts under the jump to default extended constant elasticity of variance (JDCEV) model, namely: a stopping time approach based on the first passage time densities of the underlying asset price process through the barrier levels; and a static hedging portfolio approach in which the barrier option is replicated by a portfolio of plain-vanilla and binary options. In doing so, both valuation methodologies are extended to a more general set-up accommodating endogenous bankruptcy, time-dependent barriers and the commonly observed stylized facts of a positive link between default and equity volatility and of a negative link between volatility and stock price. The two proposed numerical methods are shown to be accurate, easy to implement and efficient under both the JDCEV model and the nested constant elasticity of variance model.     

Volatility spreads and earnings announcement returns

Prior research documents that volatility spreads predict stock returns. If the trading activity of informed investors is an important driver of volatility spreads, then the predictability of stock returns should be more pronounced during major information events. This paper investigates whether the predictability of equity returns by volatility spreads is stronger during earnings announcements. Volatility spreads are measured by the implied volatility differences between pairs of strike price and expiration date matched put and call options and capture price pressures in the option market. During a two-day earnings announcement window, the abnormal returns to the quintile that includes stocks with relatively expensive call options is more than 1.5% greater than the abnormal returns to the quintile that includes stocks with relatively expensive put options. This result is robust after measuring volatility spreads in alternative ways and controlling for firm characteristics and lagged equity returns. The degree of announcement return predictability is stronger when volatility spreads are measured using more liquid options, the information environment is more asymmetric, and stock liquidity is low.     

An adjusted binomial model for pricing Asian options

We propose a model for pricing both European and American Asian options based on the arithmetic average of the underlying asset prices. Our approach relies on a binomial tree describing the underlying asset evolution. At each node of the tree we associate a set of representative averages chosen among all the effective averages realized at that node. Then, we use backward recursion and linear interpolation to compute the option price.     

On-market share buybacks, exercisable share options and earnings management

Chan et al. (2006b ) suggest that managers might announce a share buyback to manipulate investors’ perceptions and capitalize on the positive price reaction usually associated with the announcement. The incentive to do so is greater when managers have exercisable options. Prior studies document that managers engage in upwards earnings management for opportunistic reasons related to option holdings (Bergstresser and Philippon, 2006). We examine the association between earnings management and exercisable option holdings for buyback firms to investigate if earnings management in the pre‐buyback period is greater for firms with equity incentives to increase share price. Our results, using 138 buybacks over the period 1996–2003, support our prediction. We find that buyback firms with both exercisable options that are in‐the‐money prior to the buyback announcement as well as options that are exercised in the buyback period have higher discretionary current accruals than buyback firms with no exercisable options, unexercised options or with out‐of‐the‐money options. Overall, our results are consistent with buyback firms with exercisable options using earnings management and buyback announcements to maximize option payoffs, and buyback firms without exercisable options signalling undervaluation.     

Analytical pricing of American options

By using the homotopy analysis method, we derive a new explicit approximate formula for the optimal exercise boundary of American options on an underlying asset with dividend yields. Compared with highly accurate numerical values, the new formula is shown to be valid for up to 2?years of time to maturity, which is ten times longer than existing explicit approximate formulas. The option price errors computed with our formula are within a few cents for American options that have moneyness (the ratio between stock and strike prices) from 0.8 to 1.2, strike prices of 100 dollars and 2?years to maturity.     

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.     

On the Upper Bound of a Call Option

Using only a weak set of assumptions, Merton (1973) shows that the upper bound of a European or American call option on a non-dividend paying stock is the underlying stock price: a result which is often extended to options on dividend paying stocks. In this short technical piece we show that the underlying stock price is in fact not the least upper bound of either a European or an American call option on a stock that pays one or more known dividends prior to maturity. Based on Merton's (1973) original framework, new upper bounds are established which depend on the size(s) of the dividend(s) compared to the size of the strike. JEL Classification: G12, G13     

When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

Expensing stock options: a fair-value approach

Now that companies such as General Electric and Citigroup have accepted the premise that employee stock options are an expense, the debate is shifting from whether to report options on income statements to how to report them. The authors present a new accounting mechanism that maintains the rationale underlying stock option expensing while addressing critics' concerns about measurement error and the lack of reconciliation to actual experience. A procedure they call fair-value expensing adjusts and eventually reconciles cost estimates made at grant date with subsequent changes in the value of the options, and it does so in a way that eliminates forecasting and measurement errors over time. The method captures the chief characteristic of stock option compensation--that employees receive part of their compensation in the form of a contingent claim on the value they are helping to produce. The mechanism involves creating entries on both the asset and equity sides of the balance sheet. On the asset side, companies create a prepaid-compensation account equal to the estimated cost of the options granted; on the owners'-equity side, they create a paid-in capital stock-option account for the same amount. The prepaid-compensation account is then expensed through the income statement, and the stock option account is adjusted on the balance sheet to reflect changes in the estimated fair value of the granted options. The amortization of prepaid compensation is added to the change in the option grant's value to provide the total reported expense of the options grant for the year. At the end of the vesting period, the company uses the fair value of the vested option to make a final adjustment on the income statement to reconcile any difference between that fair value and the total of the amounts already reported.     

An option pricing approach to corporate dividends and the capital investment financing decision

In light of a growing trend toward viewing dividends as an investable asset class, this article opens up a new perspective on their valuation. We show that dividends can be viewed as options on the cash flow of the firm. That is, a firm either pays zero dividends, in which case the option expires out‐of‐the‐money, or it pays a positive dividend, the value of which corresponds to the option's moneyness. The exercise price is determined by the capital budget, the flexibility of the company to use external financing, and whether it has minimum and maximum dividends. The model is also capable of accommodating a stochastic capital budget, which allows for uncertain growth opportunities and their correlation with the firm's cash flows. We also present an application of the model using actual data for a large multinational company.     

Complete markets,informed trading and equity option introductions

This paper proposes and tests a new hypothesis concerning the price impact of option introductions on the underlying asset. We argue that the leverage properties of options induce a higher level of informed trading in the aggregate market (underlying plus derivative), resulting in excess listing-day price movements in the newly optioned equity. Using an alternative dataset, our results suggest that this may be an explanation for the observed positive than negative excess listing-day returns of US optioned stocks over the past thirty years.     

Option Pricing with Heterogeneous Expectations

This paper re-derives the finite mixture option pricing model of Ritchey (1990), based on the assumption that the option investors hold heterogeneous expectations about the parameters of the lognormal process of the underlying asset price. By proving that the model admits no riskless arbitrage, this paper justifies that the entire family of finite mixture of lognormal distributions is a desirable candidate set for recovering the risk-neutral probability distributions from contemporaneous options quotes. The parametric method derived from the model is significantly simpler than the nonparametric method of Rubinstein (1994) for recovering the risk-neutral probability distributions from contemporaneous option prices.     

On the Distribution of CBOE Option Trade Prices Occurring Between Consecutive Stock Trades

This paper examines the volume distribution of option trade prices that occurs when the underlying stock price remains constant. The width of these option trade price bands provides direct evidence on the law of one price and the redundancy of options assumed in many option models. We find that index option bands are narrower than equity option bands. Furthermore, for both equity and index options, puts have narrower bandwidths than calls. In general, option price bandwidth is narrow and can be explained by the minimum price movement allowed by the Chicago Board Options Exchanges (CBOE). This supports the single price law and the redundancy assumption. The existence of bid/ask quotes on the option does not materially affect the above results although it does alter the frequency of multiple option trade prices for a given underlying stock price. We note that over 53% of option trading volume occurs without bid/ask quotes on the CBOE compared to less than 15% a decade ago. Our results suggest that the effective bid/ask spread on options is probably no larger than the minimum price movements allowed by the CBOE. Furthermore, the need for the liquidity services of market makers may be declining if the decline in quoting activity stems from cross trading (i.e. trades not involving market makers).     

Two-dimensional risk-neutral valuation relationships for the pricing of options

The Black–Scholes model is based on a one-parameter pricing kernel with constant elasticity. Theoretical and empirical results suggest declining elasticity and, hence, a pricing kernel with at least two parameters. We price European-style options on assets whose probability distributions have two unknown parameters. We assume a pricing kernel which also has two unknown parameters. When certain conditions are met, a two-dimensional risk-neutral valuation relationship exists for the pricing of these options: i.e. the relationship between the price of the option and the prices of the underlying asset and one other option on the asset is the same as it would be under risk neutrality. In this class of models, the price of the underlying asset and that of one other option take the place of the unknown parameters.      

High-order accurate implicit finite difference method for evaluating American options

A numerical method is presented for valuing vanilla American options on a single asset that is up to fourth-order accurate in the log of the asset price, and second-order accurate in time. The method overcomes the standard difficulty encountered in developing high-order accurate finite difference schemes for valuing American options; that is, the lack of smoothness in the option price at the critical boundary. This is done by making special corrections to the right-hand side of the differnce equations near the boundary, so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr and Eaguet. Results of numerical experiments are presented comparing the method with more standard finite difference methods.     

The Value of Wildcard Options

Wildcard options are embedded in many derivative contracts. They arise when the settlement price of the contract is established before the time at which the wildcard option holder must declare his intention to make or accept delivery and the exercise of the wildcard option closes out the underlying asset position. This paper provides a simple method for valuing wildcard options and illustrates the technique by valuing the sequence of wildcard options embedded in the S&P 100 index (OEX) option contract. The results show that wildcard options can account for an economically significant fraction of OEX option value.     

Partial differential equations for Asian option prices

In this article, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this paper, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and as they perform numerically well or better than the others, must be considered the preferred choice.