Employee Reload Options: Pricing, Hedging, and Optimal Exercise

Reload options, call options granting new options on exercise,are popularly used in compensation. Although the compound optionfeature may seem complicated, there is a distribution-free dominantpolicy of exercising reload options whenever they are in themoney. The optimal policy implies general formulas for numericalvaluation. Simpler formulas for valuation and hedging followfrom Black–Scholes assumptions with or without continuousdividends. Time vesting affects the optimal policy, but numericalresults indicate that it is nearly optimal to exercise in themoney whenever feasible. The results suggest that reload optionsproduce similar incentives as employee stock options and sharegrants.     

Valuing employee reload options under the time vesting requirement

Upon the exercise of an employee stock option, the embedded reload provision entitles the holder to receive additional units of new options from the employer. The number of units of new options received is equal to the number of shares tendered as payment of strike and the new strike is set at the prevailing stock price. The reload provision may be subject to a time vesting requirement, that is, after each exercise, the employee is prohibited from exercising the reload until the end of a vesting period. In this paper, we construct an efficient numerical algorithm that computes the market value of the employee reload options under a time vesting requirement. Also, we explore the analytic properties of the price functions and optimal exercise policies of the employee reload options.     

Implicit Options in Life Insurance: Valuation and Risk Management

Participating life insurance contracts typically contain various types of implicit options. These implicit options can be very valuable and can thus represent a significant risk to insurance companies if they practice insufficient risk management. Options become especially risky through interaction with other options included in the contracts, which makes their evaluation even more complex. This article provides a comprehensive overview and classification of implicit options in participating life insurance contracts and discusses the relevant literature. It points out the potential problems particularly associated with the valuation of rights to early exercise due to policyholder exercise behavior. The risk potential of the interaction of implicit options is illustrated with numerical examples by means of a life insurance contract that includes common implicit options, i.e., a guaranteed interest rate, stochastic annual surplus participation, and paid-up and resumption options. Valuation is conducted using risk-neutral valuation, a concept that implicitly assumes the implementation of risk management measures such as hedging strategies.     

Valuing reload options

Over the past quarter century, the use of stock options as pay for performance has grown enormously. Option grants now account for 32% of CEO pay—more than twice that of salaries. In addition options are now being granted to many more employees than before. During this same time period, there have been numerous innovations in the features on compensation options. One of these features is the reload—the grant of new options to replace shares tendered in the payment of the exercise. Within the past year, the long-delayed FASB requirement that options be expensed for financial reporting has finally become a fact. It is incumbent upon financial researchers to provide methods to achieve the goal of valuing options, not only to serve the accounting needs, but also to provide ways of determining their true costs and incentive effects. This paper analyzes the various forms of reload options and provides simple Black-Scholes like formulas for evaluating them. JEL Classification G13     

The Valuation of Multivariate Contingent Claims in Discrete Time Models

There are several examples in the literature of contingent claims whose payoffs depend on the outcomes of two or more stochastic variables. Familiar cases of such claims include options on a portfolio of options, options whose exercise price is stochastic, and options to exchange one asset for another. This paper derives risk neutral valuation relationships (RNVRs) in a discrete time setting that facilitate the pricing of such complex contingent claims in two specific cases: joint lognormally distributed underlying variables and constant proportional risk aversion on the part of investors, and joint normally distributed underlying variables and constant absolute risk aversion preferences, respectively. This methodology is then applied to the valuation of several interesting complex contingent claims such as multiperiod bonds, multicurrency option bonds, and investment options.     

An IFRS 2 and FASB 123 (R) Compatible Model for the Valuation of Employee Stock Options

In this paper, we show how employee stock options can be valued under the new reporting standards IFRS 2 and FASB 123 (revised) for share-based payments. Both standards require companies to expense employee stock options at fair value. We propose a new valuation model, referred to as Enhanced American model, that complies with the new standards and produces fair values often lower than those generated by traditional models such as the Black–Scholes model or the adjusted Black–Scholes model. We also provide a sensitivity analysis of model input parameters and analyze the impact of the parameters on the fair value of the option. The valuation of employee stock options requires an accurate estimation of the exercise behavior. We show how the exercise behavior can be modeled in a binomial tree and demonstrate the relevance of the input parameters in the calibration of the model to an estimated expected life of the option. JEL Classification G13, G30     

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.     

American-style Indexed Executive Stock Options

This paper develops a new pricing model for American-style indexed executive stock options. We rely on a basic model framework and an indexation scheme first proposed by Johnson and Tian (2000a) in their analysis of European-style indexed options. Our derivation of the valuation formula represents an instructive example of the usefulness of the change-of-numeraire technique. In the paper's numerical section we implement the valuation formula and demonstrate that not only may the early exercise premium be significant but also that the delta of the American-style option is typically much larger than the delta of the otherwise identical (value-matched) European-style option. Vega is higher for indexed options than for conventional options but largely independent of whether the options are European- or American-style. This has important implications for the design of executive compensation contracts. We finally extend the analysis to cover the case where the option contracts are subject to delayed vesting. We show that for realistic parameter values, delayed vesting leads only to a moderate reduction in the value of the American-style indexed executive stock option.     

Optimal exercise of executive stock options

Valuing executive stock options is a challenging problem, because the standard risk-neutral valuation of those options is not appropriate; the executive is not allowed to trade the stock of the firm, so is not operating in a complete market. As this paper shows, an executive holding many American-style call options on his firm’s stock will optimally exercise the options bit by bit, whereas a risk-neutral valuation of the options would assume that all are exercised at the same time. Comparative statics of the optimal exercise policy show many surprising features.      

Commodity Spread Option with Cointegration

We derive the valuation formula of a European call option on the spread of two cointegrated commodity futures prices, based on the Gibson–Schwartz with cointegration (GSC) model. We also analyze the American commodity spread option including the early exercise premium representation and an analytical approximation valuation formulae with cointegration. In the numerical analysis, we compare the spread option values calculated by the GSC model and the Gibson–Schwartz (GS) model that ignores cointegration. Consistent with the intuition that the cointegration prevents the prices from diverging, the GSC model prices the commodity spread option lower than the GS model which have longer maturity of more than 6 years. In other words, the GS model may overprice the commodity spread options for those with longer maturity without taking account of cointegration. Thus, incorporating cointegration is important for valuation and hedging of long-term commodity spread options such as large scale oil refining plant developments.     

Realizing the Potential of Real Options: Does Theory Meet Practice?

The idea of viewing corporate investment opportunities as “real options” has been around for over 25 years. Real options concepts and techniques now routinely appear in academic research in finance and economics, and have begun to influence scholarly work in virtually every business discipline, including strategy, organizations, management science, operations management, information systems, accounting, and marketing. Real options concepts have also made considerable headway in practice. Corporate managers are more likely to recognize options in their strategic planning process, and have become more proactive in designing flexibility into projects and contracts, frequently using real options vocabulary in their discussions. Thanks in part to the spread of real options thinking, today's strategic planners are more likely than their predecessors to recognize the “option” value of actions like the following: ? dividing up large projects into a number of stages; ? investing in the acquisition or production of information; ? introducing “modularity” in manufacturing and design; ? developing competing prototypes for new products; and ? investing in overseas markets. But if real options has clearly succeeded as a way of thinking, the application of real options valuation methods has been limited to companies in relatively few industries and has thus failed to live up to expectations created in the mid‐ to late‐1990s. Increased corporate acceptance and implementations of real options valuation techniques will require several changes coming together. On the theory side, we need more realistic models that better reflect differences between financial and real options, simple heuristic methods that can be more easily implemented (but that have been carefully benchmarked against more precise models), and better guidance on implementation issues such as the estimation of discount rates for the “optionless” underlying projects. On the practitioner side, we need user‐friendly real options software, more senior‐level buy‐in, more deliberate diffusion of real options knowledge throughout organizations, better alignment of managerial incentives with long‐term shareholder value, and better‐designed contracts to correct the misalignment of incentives across the value chain. If these challenges can be met, there will continue to be a steady if gradual diffusion of real options analysis throughout organizations over the next few decades, with real options eventually becoming not only a standard part of corporate strategic planning, but also the primary valuation tool for assessing the expected shareholder effect of large capital investment projects.     

Comments on the valuation of derivative assets

This paper presents an alternative approach to derive the Breeden-Litzenberger valuation formula, which expresses the price of an arbitrary derivative security in terms of call options' prices. This valuation formula follows from the observation that a continuous derivative security can be replicated by a portfolio including a bond and call options with all possible exercise prices. Discrete terms are added to the original Breeden-Litzenberger formula to reflect possible discontinuities of the call option price's derivative with respect to the exercise price. These discontinuities are subsequently shown to correspond to mass points of the probability distribution of the stock price. Several applications of the Breeden-Litzenberger valuation formula are demonstrated.     

A functional approach to pricing complex barrier options

In this article, a new method for pricing contingent claims, which is particularly well suited for options with complex barrier and volatility structures, is introduced. The approach is based on a high-precision approximation of the Feynman–Kac equation with distributed approximating functionals. The method is particularly well suited for long maturity valuation problems, and it is shown to be faster and more accurate than conventional solution schemes.     

A real-world way to manage real options

Each corporate growth project is an option, in the sense that managers face choices--push ahead or pull back--along the way. Yet many companies hesitate to apply options theory to initiatives such as R&D and geographic expansion, partly because these "real" options are highly complex. In this article, the authors make the case that the complexity of real options can be eased through the use of a binomial valuation model. Many of the problems with real-options analysis stem from the use of the Black-Scholes-Merton model, which isn't suited to real options. Binomial models, by contrast, are simpler mathematically, and you can tinker with a binomial model until it closely reflects the project you wish to value. Suppose your company is considering investing in a new plant. To use the binomial model, you must create an "event tree" to figure out the full range of possible values for the plant during the project's lifetime--next year, at the end of the design phase, upon completion. Then you work backward from the value at completion, factoring in the various investments, to determine the value of the project today. These calculations provide you with numbers for all the possible future values of the option at the various points where a decision needs to be made on whether to continue with the project. The authors also address another criticism of real options: that gaps often arise between theoretical and realized values of options of all types. Such gaps may be largely the result of managers exercising options at the wrong time. To improve the way it manages its real options, a company can look out for the decision trigger points that correspond to the nodes on a binomial decision tree. The trigger points should not only tell managers when they need to decide on exercise but also specify rules governing the exercise decisions.     

MARGIN TRADING: A RISK-RETURN ANALYSIS

This paper attempts to present an integrated valuation analysis of investment options involving margin trading. The analysis is based on valuation theories such as Modigliani and Miller's capital structure model, the capital asset pricing model and the option pricing model. It is shown (i) that in margin trading, the return on equity is given by the return on investment plus a risk premium which increases proportionally with the margin-trading rate; (ii) that both the total risk (variance) and systematic risk (beta) of the return on equity increases proportionally with those associated with the return on investment; and (iii) that, when the option pricing model is applied to the case of margin trading, a more precise valuation formula can be employed.     

Warrant valuation and exercise strategy

This paper demonstrates that warrant valuation and exercise strategy differ fundamentally from call option valuation. Simultaneous exercise of warrants is shown to be suboptimal and a monopolist owning all warrants can achieve a higher value. Unless warrants are perfectly divisible, no satisfactory equilibrium exists for the valuation and exercise of widely held warrants. The problems encountered appear to be quite general and stem from necessary assumptions about future corporate dividend policy and capital structure. Such assumptions are necessary for any model of corporate security valuation.     

The analytic valuation of American options

No analytic solution exists for the valuation of American optionswritten on futures contracts and foreign currencies for whichearly exercise may be optimal. This article formulates the Americanoption valuation problem in economically and mathematicallymeaningful ways. This enables us to derive valuation formulasfor American options. The properties associated with the optimalexercise boundary are examined, and a numerical technique toimplement the valuation formulas is presented.     

Real options give insights into real value

Developments in real options theory are resulting in valuation tools more suited to our increasingly dynamic economy. These tools can help where conventional business valuation techniques fail to capture the full opportunity value of new business strategies. Steve Leppard and Peter Morawitz report.     

THE SURVIVAL ZONE FOR A BOND WITH BOTH CALL AND PUT OPTIONS EMBEDDED

We present a numerical method to price bonds that have multiple embedded options with an emphasis on the case with both long call and short put options. The valuation framework is a one-factor model for the term structure of interest rates, where the instantaneous interest rate is allowed to follow a fairly general stochastic process. The equilibrium interest rates that define the free boundaries for the embedded call and put options are given. We demonstrate the survival zone within which a bond with both long call and short put options remains afloat. We show that even moderate levels of transaction costs can have a significant effect on exercise of options.     

An alternative approach to the valuation of American options and applications

In this paper we examine the structure of American option valuation problems and derive the analytic valuation formulas under general underlying security price processes by an alternative but intuitive method. For alternative diffusion processes, we derive closed-form analytic valuation formulas and analyze the implications of asset price dynamics on the early exercise premiums of American options. In this regard, we introduce useful and interesting diffusion processes into American option-pricing literature, thus providing a wide range of choices of pricing models for various American-type derivative assets. This work offers a useful analytic framework for empirical testing and practical applications such as the valuation of corporate securities and examining the impact of options trading on market micro-structure.