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     

DIVIDENDS, DIVIDEND POLICY AND OPTION VALUATION: A NEW PERSPECTIVE

This paper develops a model which explicitly incorporates the impact of the payment of dividends on the underlying stock into the valuation of both American and European calls and puts. Unlike earlier models, what we call the Dividend Adjustment Merton (DAM) model neither assumes arbitrary continuous dividends nor uses ad hoc methods to adjust for discrete dividend payments. Instead, it assumes the existence of a Miller and Modigliani (1961) valuation neutral dividend policy and adjusts Merton's constant proportional dividend model to incorporate any known schedule of discrete cash dividends of this type. The DAM model produces results which are equal to or superior to those of the separate models now used to value American calls (the Roll-Geske-Whaley model) and American puts (the Geske-Johnson model) on dividend paying stocks. It has the virtue of being internally consistent in that the same model can be used to value both calls and puts. In developing the DAM model, the paper clarifies the role of dividends and dividend policy in determining option values. It also produces significantly tightened boundary conditions for option values.     

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.     

Dividend Forecast Biases in Index Option Valuation

Since the early days of option pricing theory,the assumption that the dividends on the underlying stock or index over the life of the contract are known has not been challenged. We examine the sensitivity of index option prices to the assumption of dividend uncertainty. We consider a number of issues related to the forecasting of dividends and build a dividend forecasting model that passes several rigorous tests for unbiasedness. We then generate option prices using contemporary market levels and interest rates. We find that prices generated with the actual dividends are unbiased with respect to those generated using the forecasted dividends. The magnitudes of the forecast errors, however, are sufficiently large to suggest a concern, but the percentage errors are consistently small, typically amounting to less than two percent of the option price. We conclude that the convenient assumption that the stream of future dividendsis known is probably innocuous. This revised version was published online in November 2006 with corrections to the Cover Date.     

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.     

American option valuation under stochastic interest rates

By applying Ho, Stapleton and Subrahmanyam's (1997, hereafter HSS) generalised Geske–Johnson (1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of American options in a stochastic interest rate economy. The proposed method simplifies HSS's three-dimensional solution to a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate than the HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary between the stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness of the option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlation between them. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Note on Credit Risk of Vertical <Emphasis Type="Italic">Keiretsu</Emphasis> Firms: Preliminary Evidence from the Japanese Automobile Industry

This paper empirically examines the relationship between the credit risk of Toyota, Nissan and Honda keiretsu-affiliated firms and the credit risk of the respective parent company. As credit spread data for keiretsu-affiliated firms were not available we create a keiretsu default index, as a proxy, using expected default probabilities obtained from the KMV and Leland and Toft (J. Finance 51, 987–1019, 1996) option pricing models. We find parent credit spreads do not Granger cause our keiretsu default index and vice versa in a bivariate vector autoregressive (VAR) framework.JEL classification: G3, L62     

Confined exponential approximations for the valuation of American options

We provide an alternative analytic approximation for the value of an American option using a confined exponential distribution with tight upper bounds. This is an extension of the Geske and Johnson compound option approach and the Ho et al. exponential extrapolation method. Use of a perpetual American put value, and then a European put with high input volatility is suggested in order to provide a tighter upper bound for an American put price than simply the exercise price. Numerical results show that the new method not only overcomes the deficiencies in existing two-point extrapolation methods for long-term options but also further improves pricing accuracy for short-term options, which may substitute adequately for numerical solutions. As an extension, an analytic approximation is presented for a two-factor American call option.     

Biases in option prices : Evidence from the foreign currency option market

This paper examines a European call model of option pricing over a data set which does not suffer from the early exercise problems that have plagued earlier studies of call options on common stocks. We specifically examine a data set of American call prices on spot foreign exchange for which it is plausible to apply an adjusted version of the Garman-Kohlhagen (1983) and Grabbe (1983) European call option model. We make adjustments for interest rate risk and find that the model is nearly unbiased in the valuation of foreign currency options. We conclude that the Geske-Roll (1984) conjecture about dividend uncertainty creating biases in stock option prices holds analogously in the foreign currency option market. Interest rate differential risk (analogous to risky dividends) thus appears to be an important element in the valuation of foreign currency options.     

THE PROPOSED INTRODUCTION OF FUTURES‐STYLE MARGINING IN THE UNITED STATES: AN AUSTRALIAN COMPARISON

We extend the quadratic approximation method to examine American‐style options traded using futures‐style margining and show that an early exercise premium can exist when the cost of carry is negative. Empirical results based on a reduced form of the model using futures‐style call options traded on the Australian All Ordinaries Share Price Index are consistent with previous research: call option early exercise premiums are economically zero. Full option prices are examined by comparing observed futures‐style with theoretical stock‐style values. We find futures‐style values exceed stock‐style values and argue that the increase results from improvements in liquidity. The findings are particularly relevant given the pending decision at the Commodity Futures Trading Commission to introduce a futures‐style system in the United States. JEL classification: G13, C13     

The Distributional Efficiency of Alternative Regulatory Regimes: A Real Option Approach

This paper studies the effects of regulatory constraints on firm's irreversible investment decisions. The RPI–x rule is compared to a profit sharing rule, which increases the x factor in case profits go beyond a given level.When the firm has an option to delay investment, these rules have the same impact on investment choices. As profit sharing has a greater ability to extract rents, however, it is more efficient than the RPI–x rule.     

The early exercise of American puts

This paper shows that American puts on dividend paying stocks are most likely to be exercised either just after an ex-dividend date or just prior to expiration. At any other time the option to exercise an American put early may have less value. Thus, put writers and converters can predict when protection against premature exercise will be most valuable. The probability of early exercise is shown to be sensitive to managerial policy regarding the suspension of dividend payments, transaction costs, and interest rates. However, dividend payments are demonstrated to be the primary deterrent to early exercise.     

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.     

Option-Based Compensation,Corporate Investment Policy,and Stock Repurchases

Abstract:

Taking account of the business life cycle, this paper investigates the impact of the proceeds associated with stock option exercises on investment expenditures and stock repurchases. The results reveal that the proceeds associated with option exercises could add internal funds to firms and contribute to investment in research and development and capital expenditures, especially in the growth stage of a firm’s life cycle. This paper also shows the positive relationship between option proceeds and stock repurchases in the stagnant stage of that cycle. The empirical results further suggest that stock repurchases may substitute for dividends. In summary, the paper empirically demonstrates that stock options not only encourage employees to work harder, but also create more funds for the firm.     

A linearly implicit predictor–corrector scheme for pricing American options using a penalty method approach

Pricing of an American option is complicated since at each time we have to determine not only the option value but also whether or not it should be exercised (early exercise constraint). This makes the valuation of an American option a free boundary problem. Typically at each time there is a particular value of the asset, which marks the boundary between two regions: to one side one should hold the option and to other side one should exercise it. Assuming that investors act optimally, the value of an American option cannot fall below the value that would be obtained if it were exercised early. Effectively, this means that the American option early exercise feature transforms the original linear pricing partial differential equation into a nonlinear one. We consider a penalty method approach in which the free and moving boundary is removed by adding a small and continuous penalty term to the Black–Scholes equation; consequently,the problem can be solved on a fixed domain. Analytical solutions of the Black–Scholes model of American option problems are seldom available and hence such derivatives must be priced by stable and efficient numerical techniques. Standard numerical methods involve the need to solve a system of nonlinear equations, evolving from the finite difference discretization of the nonlinear Black–Scholes model, at each time step by a Newton-type iterative procedure. We implement a novel linearly implicit scheme by treating the nonlinear penalty term explicitly, while maintaining superior accuracy and stability properties compared to the well-known θ-methods.     

Measuring portfolio performance and the empirical content of asset pricing models

Recent work by Richard Roll has challenged the worth of portfolio performance measures based on the capital asset pricing model. This paper demonstrates that Roll's conclusions are due to his focusing on a ‘truly’ ex-ante efficient index. Using a choice and information theoretic framework, we show that an appropriate index is efficient relative to the probabilities assessed by the ‘market’. Residual analyses and portfolio performance tests, using such an index, yield meaningful results for a wide class of information structures. Roll's primary criticisms, however, relate to tests of the asset pricing model itself. We argue that these criticisms are vastly overstated.     

Scrip Dividends: the Management's View

This paper examines the motivation underlying the payment of scrip dividends through a questionnaire survey conducted among a sample of companies listed on the London Stock Exchange that offered their shareholders this option, and a control sample of firms that paid only cash dividends. The results show that the overwhelming majority of the respondents from both groups feel that the scrip dividend option is driven by the current and/or potential high irrecoverable advanced corporation tax but the proposition that this option is a substitute for external finance or a cut in cash dividends is strongly rejected. Managers feel that, although the scrip dividend option allows small shareholders to increase their holdings without incurring transaction costs, it is not intended to convey information that will lead to a rise in the share price. Furthermore, the results reveal that the decision between offering this option or paying only cash dividends is substantially affected by shareholders' pressure, suggesting that firms in the UK are subject to direct shareholder monitoring.     

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.     

Pricing American options written on two underlying assets

This paper extends the integral transform approach of McKean [Ind. Manage. Rev., 1965, 6, 32–39] and Chiarella and Ziogas [J. Econ. Dyn. Control, 2005, 29, 229–263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [J. Polit. Econ., 1973, 81, 637–659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches.     

Traffic light options

This paper introduces, prices, and analyzes traffic light options. The traffic light option is an innovative structured OTC derivative developed independently by several London-based investment banks to suit the needs of Danish life and pension (L&P) companies, which must comply with the traffic light solvency stress test system introduced by the Danish Financial Supervisory Authority (DFSA) in June 2001. This monitoring system requires L&P companies to submit regular reports documenting the sensitivity of the companies’ base capital to certain pre-defined market shocks – the red and yellow light scenarios. These stress scenarios entail drops in interest rates as well as in stock prices, and traffic light options are thus designed to pay off and preserve sufficient capital when interest rates and stock prices fall simultaneously. Sweden’s FSA implemented a traffic light system in January 2006, and supervisory authorities in many other European countries have implemented similar regulation. Traffic light options are therefore likely to attract the attention of a wider audience of pension fund managers in the future. Focusing on the valuation of the traffic light option we set up a Black–Scholes/Hull–White model to describe stock market and interest rate dynamics, and analyze the traffic light option in this framework.