Dividend-paying Assets Research Articles

On asset pricing in a binomial model with fixed and proportional transaction costs, portfolio constraints and dividends

AbstractWe extend the classical binomial model proposed by Cox, Ross, and Rubinstein for derivative security pricing to encompass both fixed and proportional transaction costs, portfolio constraints including margin requirements, and dividend-paying assets. Our focus is on studying option hedging within this enriched framework. Initially, we establish the existence of a hedging strategy in this context. Subsequently, we determine the optimal hedging strategy and its associated initial cost by decomposing the problem into a sequence of hedging problems. To illustrate our approach, we present a numerical example within a 3-period binomial model.

The stochastic balance equation for the American option value function and its gradient

In the paper we consider the problem of valuation and hedging of American options written on dividend-paying assets whose price dynamics follow a multidimensional diffusion model. We derive a stochastic balance equation for the American option value function and its gradient. We prove that the latter pair is the unique solution of the stochastic balance equation as a result of the uniqueness in the related adapted future-supremum problem. The latter problem has an attractive interpretation: the given adapted stochastic process can be adjusted by a martingale in such a manner, that the observer will gain the perfect foresight of the resulting future-supremum process via the Snell envelope of the given stochastic process.

Consuming Dividends

Abstract This paper studies why investors buy dividend-paying assets and how they time consumption accordingly. We combine administrative bank data linking customers’ consumption and income to portfolio data and survey responses on financial behavior. We find that private consumption is excessively sensitive to dividend income. Investors across wealth, income, and age distributions increase spending precisely around days of dividend receipt. Our results are at odds with a number of existing rational and behavioral explanations, such as financial constraints and impulsiveness. Instead, consumption responses reflect “planned” excess sensitivity, driven by investors who select dividend portfolios, anticipate dividend income, and plan consumption accordingly.

Short time asymptotics for American maximum options with a dividend-paying asset

<abstract><p>We investigate the asymptotic behaviors of American maximum options with dividend-paying assets near maturity. Using the exercise conditions of American options, we obtain the asymptotic forms of the two boundaries with respect to time-to-maturity. Furthermore, we derive the matched asymptotic expansion for the rescaled value function of American maximum option. The all results are provided with detailed computations and derivations. Numerical examples show that the asymptotic value function and exercise boundaries can provide an efficient alternative for the true ones, respectively.</p></abstract>

On perpetual American options in a multidimensional Black–Scholes model

We consider the problem of pricing perpetual American options written on dividend-paying assets whose price dynamics follow a multidimensional Black and Scholes model. For convex Lipschitz continuous reward functions, we give a probabilistic characterization of the fair price in terms of a reflected BSDE, and an analytical one in terms of an obstacle problem. We also provide the early exercise premium formula.

Pricing of general European options on discrete dividend-paying assets with jump-diffusion dynamics

• Closed form option prices when underlying asset follows a jump-diffusion process AND pays discrete or continuous dividends. • Distribution of jumps is not specified, so formula includes the cases of lognormal and double exponential distributions. • The pricing formula offers a unified approach to pricing of options in the presence of discrete and continuous dividends. • The formula is written in terms of the Black–Scholes formula, so that the role of the parameters can be inferred. Stocks regularly pay dividends at discrete intervals of time while statistical evidence indicates the existence of small “jumps” in the stock price dynamics. In this paper, we find closed-form solutions for the valuation of European options when the underlying asset is modeled by a jump-diffusion process and pays discrete or continuous dividends. The formula is very general and can be used with any specification on the distribution of the jump. Moreover, the formula is written in terms of the Black–Scholes formula with no jumps or dividends and thus indicates the effect of the jumps and the effect of the inclusion of discrete (or continuous) dividends on the price of the option.

A Radial Point Interpolation Method for Pricing Options on a Dividend Paying Asset

A Radial Point Interpolation Method for Pricing Options on a Dividend Paying Asset

Valuing American floating strike lookback option and Neumann problem for inhomogeneous Black–Scholes equation

This paper presents our study of American floating strike lookback options written on dividend-paying assets. The valuation of these options can be mathematically formulated as a free boundary inhomogeneous Black–Scholes PDE with a Neumann boundary condition, which we, by using a Mellin transform, convert into a relatively simple ordinary differential equation with Dirichlet boundary conditions. We then use these results to derive an integral equation that can be used to calculate the price of American floating strike lookback options. In addition, we also used Mellin transforms to derive the closed-form of the perpetual case.

The Early Exercise Premium Representation for American Options on Multiply Assets

In the paper we consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follow the classical multidimensional Black and Scholes model. We provide a general early exercise premium representation formula for options with payoff functions which are convex or satisfy mild regularity assumptions. Examples include index options, spread options, call on max options, put on min options, multiply strike options and power-product options. In the proof of the formula we exploit close connections between the optimal stopping problems associated with valuation of American options, obstacle problems and reflected backward stochastic differential equations.

Arithmetic Brownian motion and real options

We treat real option value when the underlying process is arithmetic Brownian motion (ABM). In contrast to the more common assumption of geometric Brownian motion (GBM) and multiplicative diffusion, with ABM the underlying project value is expressed as an additive process. Its variance remains constant over time rather than rising or falling along with the project’s value, even admitting the possibility of negative values. This is a more compelling paradigm for projects that are managed as a component of overall firm value. After outlining the case for ABM, we derive analytical formulas for European calls and puts on dividend-paying assets as well as a numerical algorithm for American-style and other more complex options based on ABM. We also provide examples of their use.

Almost sure Nash equilibrium strategies in evolutionary models of asset markets

We consider a stochastic model of a financial market with long-lived dividend-paying assets and endogenous asset prices. The model was initially developed and analyzed in the context of evolutionary finance, with the main focus on questions of “survival and extinction” of investment strategies. In this paper we view the model from a different, game-theoretic, perspective and examine Nash equilibrium strategies, satisfying equilibrium conditions with probability one.

American Fractional Lookback Options: Valuation and Premium Decomposition

This paper deals with valuation and premium decomposition of American fractional lookback options written on dividend-paying assets, for which exact formulas are unknown except for the perpetual case. Via a PDE approach, we derive Laplace transforms of the values of lookback call and put options, which can be decomposed into the associated European values plus the early exercise premiums. Using Abelian theorems of Laplace transforms, we characterize asymptotic behaviors of the early exercise boundaries at a time to close to expiration and at infinite time to expiration. Based on the Gaver–Stehfest inversion method combined with the Newton method, we develop a fast and accurate algorithm for computing both the option value and the early exercise boundary. Numerical analysis reveals some notable features of the American fractional lookback options.

MINIMUM GUARANTEED PAYMENTS AND COSTLY CANCELLATION RIGHTS: A STOPPING GAME PERSPECTIVE

We consider the valuation and optimal exercise policy of a δ-penalty minimum guaranteed payment option in the case where the value of the underlying dividend-paying asset follows a linear diffusion. We characterize both the value and optimal exercise policy of the considered game option explicitly and demonstrate that increased volatility increases the value of the option and postpones exercise by expanding the continuation region where exercising is suboptimal. An interesting and natural implication of this finding is that the value of the embedded cancellation rights of the issuer increase as volatility increases.

Far-from-expiry behavior of the American put option on a dividend-paying asset

We provide a rigorous proof of sharp estimates for the long time behavior of the early exercise boundary and the price for an American put option on a dividend-paying asset that follows a geometric Brownian motion.

Exercise Boundary Near Maturity for an American Option on Several Assets

We consider an American put option on a linear function of d dividend-paying assets. The value function of this option is given as the solution of a free boundary problem. When d = 1, the behavior of the free boundary near the maturity of the option is well known. In this article, we extend to the case d > 1 the study of the free boundary near maturity. A parameterization of the stopping region at time t is given. That enables us to define and give a convergence rate for this region when t goes to the maturity.

ALTERNATIVE RANDOMIZATION FOR VALUING AMERICAN OPTIONS

This paper deals with randomization methods for valuing American options written on dividend-paying assets, which are based on the idea of treating the maturity date as a random variable. In the randomization method introduced by Carr in 1998, he used the Erlangian distributed random variable to develop a recursive algorithm starting from the so-called Canadian option with an exponentially distributed random maturity. The purposes of this paper are (i) to provide much simpler pricing formulas for the Canadian option; (ii) to interpret the Gaver–Stehfest method developed for inverting Laplace transforms as an alternative randomization method in the context of valuing American options; and (iii) to evaluate the performance of the Gaver–Stehfest method in details with theoretical and numerical views. Numerical experiments indicate that the Gaver–Stehfest method works well to generate accurate approximations for the early exercise boundary as well as the option value.

Valuing continuous-installment options

Abstract Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.

A simple approach to valuing Asian rainbow options

Asian rainbow options, also called options on the minimum or the maximum of two or several average prices, are found widely applicable in both risk management and the design of incentive contracts for executive compensation. However, their valuation is very complicated. This paper proposes a new and simple approach to valuing such options on dividend-paying assets. Starting from an observation that the price of an Asian option in some cases can become that of a vanilla European option and then using the method of Johnson (1987), we can easily obtain a closed-form valuing formula for the option with two geometric averaging. Moreover, this method can be directly extended to obtain a closed-form solution for the case of several geometric averaging. We also derive a closed-form approximation pricing formula for an Asian rainbow option with two arithmetic averaging based on the obtained price formula of its counterpart geometric averaging Asian rainbow option and the approach of Zhang (1998). Numerical comparison shows the efficiency and accuracy of our approximation price and the different effects of the interest rate, volatility, dividend and observation frequency on the price of Asian rainbow options.

American Continuous-Installment Options: Valuation and Premium Decomposition

Installment options are weakly path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuous-installment options written on dividend-paying assets. The setup is a standard Black–Scholes–Merton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain the Laplace transform of the initial premium in an explicit form, which is decomposed into the value of the associated European vanilla option with the same payoff p...

INFORMATION-BASED ASSET PRICING

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.