On the valuation of compositions in Lévy term structure models

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Lévy process.     

Option pricing based on hybrid GARCH-type models with improved ensemble empirical mode decomposition

The exploration of option pricing is of great significance to risk management and investments. One important challenge to existing research is how to describe the underlying asset price process and fluctuation features accurately. Considering the benefits of ensemble empirical mode decomposition (EEMD) in depicting the fluctuation features of financial time series, we construct an option pricing model based on the new hybrid generalized autoregressive conditional heteroskedastic (hybrid GARCH)-type functions with improved EEMD by decomposing the original return series into the high frequency, low frequency and trend terms. Using the locally risk-neutral valuation relationship (LRNVR), we obtain an equivalent martingale measure and option prices with different maturities based on Monte Carlo simulations. The empirical results indicate that this novel model can substantially capture volatility features and it performs much better than the M-GARCH and Black–Scholes models. In particular, the decomposition is consistently helpful in reducing option pricing errors, thereby proving the innovativeness and effectiveness of the hybrid GARCH option pricing model.     

Option Profit and Loss Attribution and Pricing: A New Framework

This paper develops a new top-down valuation framework that links the pricing of an option investment to its daily profit and loss attribution. The framework uses the Black-Merton-Scholes option pricing formula to attribute the short-term option investment risk to variation in the underlying security price and the option's implied volatility. Taking risk-neutral expectation and demanding no dynamic arbitrage result in a pricing relation that links an option's fair implied volatility level to the underlying volatility level with corrections for the implied volatility's own expected direction of movement, its variance, and its covariance with the underlying security return.     

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.      

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.     

American options and callable bonds under stochastic interest rates and endogenous bankruptcy

A new characterization of the American-style option is proposed under a very general multifactor Markovian and diffusion framework. The efficiency of the proposed pricing solutions is shown to depend only on the use of a viable valuation method for the corresponding European-style option and for the transition density of the model’s state variables. Under a Gauss-Markov stochastic interest rates setup, these new American option pricing solutions are shown to offer a much better accuracy-efficiency trade-off than the approximations already available in the literature. This result is also used to price callable corporate bonds under an endogenous bankruptcy structural approach, by decomposing the option to call or default into a European put on the firm value plus two early exercise premium components.     

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.     

Pricing double barrier options under a volatility regime-switching model with psychological barriers

The prices of lots of assets have been proved in literature to exhibit special behaviors around psychological barriers, which is an important fact needed to be considered when pricing derivatives. In this paper, we discuss the valuation problem of double barrier options under a volatility regime-switching model where there exist psychological barriers in the prices of underlying assets. The volatility can shift between two regimes, that is to say, when the asset price rises up or falls down through the psychological barrier, the volatility takes two different values. Using the Laplace transform approach, we obtain the price of the double barrier knock-out call option as well as its delta. We also provide the eigenfunction expansion pricing formula and examine the effect of the psychological barrier on the option price and delta, finding that the gamma of the option is discontinuous at such barriers.     

The valuation of American call options and the expected ex-dividend stock price decline

This study focuses on the ex-dividend stock price decline implicit within the valuation of American call options on dividend-paying stocks. The Roll (1977) American call option pricing formula and the observed structure of CBOE call option transaction prices are used to infer the expected ex-dividend stock price decline as a proportion of the amount of the dividend. The relative decline is shown to be not meaningfully different from one, confirming some recent evidence from studies which examined stock prices in the days surrounding ex-dividend.     

Share Option Rewards and Managerial Performance: an Abnormal Performance Index Model

Managerial share option schemes are widely used as a means of motivating and rewarding corporate performance. Such schemes normally adopt a static exercise price; when additional exercise criteria are employed they are often based on earnings per share. A static exercise price does not adjust for economic changes outside the control of management, and earnings per share hurdles have similar limitations. This paper presents a ‘phantom’ managerial option based on relative performance, together with a pricing model for the valuation of the option. The option is developed and demonstrated using an abnormal performance index. It offers a structure which could be used for different forms of performance measurement, and resolves some important criticisms of the reward and incentive effects of traditional schemes.     

An Empirical Investigation of Alternative Contingent Claims Models for Pricing Residential Mortgages

Researchers have employed option pricing techniques to analyze mortgage financing and valuation. Alternative models (one-, two-, and three-variable models) employing different variables (short- and long-term interest rates and building value) have been designed to price mortgage securities. No prior research has addressed the question of whether the pricing accuracy of these contingent claims models improves as states increase or whether contingent claims models' valuation abilities generate reasonable estimates of primary mortgage market prices. The articles investigates the relative efficiency of each of these alternative mortgage valuation models in predicting primary market mortgage values. Our results show that a two-variable model (short rate and building value) is the most efficient. Valuation results indicate a positive pricing spread between the primary market and the theoretically estimated value.     

SUBORDINATED BINOMIAL OPTION PRICING

We extend the binomial option pricing model to allow for more accurate price dynamics while retaining computational simplicity. The asset price in each binomial period evolves according to two independent and successive Bernoulli trials on trade occurrence/nonoccurrence and up/down price movement. Subordination leads to a trinomial tree with stochastic volatility in calendar time. We derive utility‐dependent valuation results incorporating the leverage effect and test the model empirically.     

Option pricing using a binomial model with random time steps (A formal model of gamma hedging)

This paper provides a new option pricing model which justifies the standard industry implementation of the Black-Scholes model. The standard industry implementation of the Black-Scholes model uses an implicit volatility, and it hedges both delta and gamma risk. This industry implementation is inconsistent with the theory underlying the derivation of the Black-Scholes model. We justify this implementation by showing that these adhoc adjustments to the Black-Scholes model provide a reasonable approximation to valuation and delta hedging in our new option pricing model.     

Good-Deal Option Price Bounds for a Non-Traded Event with Stochastic Return: A Note

Cochrane and Sa'a-Requejo (2000, Journal of Political Economy) proposed the good-deal price bounds for the European call option on an event that is not a traded asset, but is correlated with a traded asset that can be used as an approximate hedge. One remarkable feature of their model is that the return on an event process explicitly appears in the option price bounds formula, which offered a contrast with the standard option pricing model. We show that the good-deal option price bounds on a non-traded event are obtained as a closed-form formula, when the return on an event is governed by a mean reverting process.     

Employee Stock Options, Equity Valuation, and the Valuation of Option Grants Using a Warrant-Pricing Model

We investigate the use of a warrant‐pricing approach to incorporate employee stock options (ESOs) into equity valuation and to account for the dilutive effect of ESOs in the valuation of option grants for financial reporting purposes. Our valuation approach accounts for the jointly determined nature of ESO and shareholder values. The empirical results show that our stock price estimate exhibits lower prediction errors and higher explanatory powers for actual share price than does the traditional stock price estimate. We use our valuation approach to assess the implications of dilution on the fair‐value estimates of ESO grants. We find that the fair value is overstated by 6% if we ignore the dilutive feature of ESOs. Furthermore, this bias is larger for firms that are heavy users of ESOs, small, and R&D intensive, and for firms that have a broad‐based ESO compensation plan.     

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.     

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.     

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.