Discrete Dividend Research Articles

The valuation of options on discrete dividend-paying stocks

ABSTRACT In this paper, the valuation of European options in which the underlying stock pays a discrete dividend is investigated. A specific value is set in advance, and a dividend is paid when the underlying share price reaches it. The risk-neutral price of the associated European call option is derived. Numerical simulations are presented to illustrate the effects of model parameters on the option prices.

Estimating Option Prices with Discrete Dividend Payment Using Finite Difference Method and Monte Carlo Simulation: A Comparative Study

Estimating Option Prices with Discrete Dividend Payment Using Finite Difference Method and Monte Carlo Simulation: A Comparative Study

Compound Option Pricing and the Roll-Geske-Whaley Formula under the Conjugate-Power Dagum Distribution

We explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.

Discrete Dividend Payments in Continuous Time

We propose a model in which dividend payments occur at regular, deterministic intervals in an otherwise continuous model. This contrasts traditional models where either the payment of continuous dividends is controlled or the dynamics are given by discrete time processes. Moreover, between two dividend payments, the structure allows for other types of control; we consider the possibility of equity issuance at any point in time. The value is characterized as the fixed point of an optimal control problem with periodic initial and terminal conditions. We prove the regularity and uniqueness of the corresponding dynamic programming equation and the convergence of an efficient numerical algorithm that we use to study the problem. The model enables us to find the loss caused by infrequent dividend payments. We show that under realistic parameter values, this loss varies from around 1%–24% depending on the state of the system and that using the optimal policy from the continuous problem further increases the loss.

Comparison of American Binomial Options with Discrete and Continuous Dividend

This study discusses the effect of dividend on option pricing by using a binomial method. It also investigated the initial stock value, number of steps, and strike price effects on the behavior of options pricing. From several simulations conducted, it was found that the values of call options with discrete dividend are greater than the continuous dividend. While on the put option, the values of the put options with a continuous dividend are greater than the discrete dividend.

Range-Curtailing for Options with Discrete Dividend Payments under General Diffusions

Lattice methods are often employed to price contingent claims with discrete dividends under the lognormal diffusion, but they are inclined to suffer from large decreases in execution speed as the number of dividends increases. Heteroskedastic assumptions for the stock price dynamics in between ex-dividend dates exacerbate these difficulties, and the option pricing problem with discrete dividends has thus been limited to the lognormal framework. This article proposes strategies to speed up lattice-based approximations under these general diffusions. A range-curtailing technique that bypasses superfluous computations of numerous subtrees at unrealistic stock prices is considered for European, American, and barrier options. The effect of discrete dividends on the premature exercise of American options is also studied. A benchmark method based on numerical integration is described to validate results obtained in the heteroskedastic framework. TOPICS:Options, statistical methods

Early Exercise Decision in American Options with Dividends, Stochastic Volatility, and Jumps

Using a fast numerical technique, we investigate a large database of investors’ suboptimal nonexercise of short-maturity American call options on dividend-paying stocks listed on the Dow Jones. The correct modeling of the discrete dividend is essential for a correct calculation of the early exercise boundary, as confirmed by theoretical insights. Pricing with stochastic volatility and jumps instead of the Black–Scholes–Merton benchmark cuts the amount lost by investors through suboptimal exercise by one-quarter. The remaining three-quarters are largely unexplained by transaction fees and may be interpreted as an opportunity cost for the investors to monitor optimal exercise.

Portfolio optimization with early announced discrete dividends

In this article we include discrete dividends in the stock price model and solve the corresponding generalized portfolio optimization problem. For this, we develop a new discrete dividend model that allows for the possibility of early announcement and ensures that the drop of the stock price at the ex-dividend date equals the dividend. The resulting portfolio problem can be solved explicitly for both the wealth and the trading strategy. We find that the resulting optimal portfolio process differs from the Merton strategy.

Portfolio Optimization with Early Announced Discrete Dividends

In this article we include discrete dividends in the stock price model and solve a generalized portfolio optimization problem. For this, we develop a new discrete dividend model that allows for the possibility of early announcement and ensures that the drop of the stock price at the ex-dividend date equals the dividend. The resulting portfolio problem can be solved explicitly for both the wealth and the trading strategy. We fi nd that the resulting optimal portfolio process di ffers from the Merton strategy.

Fast quadrature methods for options with discrete dividends

Discontinuities in the stock price at ex-dividend dates make it hard to derive mathematically elegant solutions for European-style options with discrete dividends under the piecewise lognormal model. Numerical schemes such as non-recombining trees are either computationally intensive, lack accuracy when computations are fast, or are unable of handling discrete barriers. Quadrature techniques circumvent the latter two problems but due to the occurrence of time-consuming multifold integrals for multi-dividend options, their implementations have been limited to three-dividend European options only. This research proposes a faster computational method where the integrand is approximated by a Chebyshev polynomial and the fast Fejér quadrature is used to evaluate integral representations of European, American, Bermudan, continuous and discrete barrier options. The method has the capability of accurately handling an arbitrary number of discrete dividend payments in an efficient manner. The computational superiority and higher accuracy of the proposed approach is demonstrated via an extensive set of numerical results. • A fast method for pricing options with discrete dividend payments is developed. • The method can price options with different exercise and barrier features. • The method can efficiently handle more discrete dividends than existing methods. • A wide set of numerical results is given to illustrate the merits of the method.

Early Exercise Decision in American Options with Dividends, Stochastic Volatility and Jumps

Using a fast numerical technique, we investigate a large database of investor suboptimal nonexercise of short maturity American call options on dividend-paying stocks listed on the Dow Jones. The correct modelling of the discrete dividend is essential for a correct calculation of the early exercise boundary as confirmed by theoretical insights. Pricing with stochastic volatility and jumps instead of the Black-Scholes-Merton benchmark cuts by a quarter the amount lost by investors through suboptimal exercise. The remaining three quarters are largely unexplained by transaction fees and may be interpreted as an opportunity cost for the investors to monitor optimal exercise.

Dealing a Discrete Dividend with Basket Option Pricing Models

Dealing a Discrete Dividend with Basket Option Pricing Models

Forecasting Discrete Dividends by No-Arbitrage

We develop and showcase a simple no-arbitrage methodology for the prediction of discrete dividend payments, based exclusively on market prices of options via the put-call parity. Our approach integrates all available option market data and simultaneously calibrates the market-implied discount curve, thus ensuring consistency across spot and derivative markets. We illustrate our method using stocks from the German blue-chip index DAX.

Discrete time optimal dividend problem with constant premium and exponentially distributed claims

Discrete time optimal dividend problem with constant premium and exponentially distributed claims

Pricing barrier stock options with discrete dividends by approximating analytical formulae

Deriving accurate analytical formulas for pricing stock options with discrete dividend payouts is a hard problem even for the simplest vanilla options. This is because the falls in the stock price process due to discrete dividend payouts will significantly increase the mathematical difficulty in pricing the option. On the other hand, much literature uses other dividend settings to simplify the difficulty, but these settings may produce inconsistent pricing results. This paper derives accurate approximating formulae for pricing a popular path-dependent option, the barrier stock option, with discrete dividend payouts. The fall in stock price due to dividend payout at an exdividend date is approximated by an accumulated price decrement due to a continuous dividend yield up to time . Thus, the stock price process prior to time and after time can be separately modelled by two different lognormal-diffusive stock processes which help us to easily derive analytical pricing formulae. Numerical experiments suggest that our formulae provide more accurate and coherent pricing results than other approximation formulae. Our formulae are also robust under extreme cases, like the high volatility (of the stock price) case. Besides, our formulae also extend the applicability of the first-passage model (a type of structural credit risk model) to measure how the firm’s payout influences its financial status and the credit qualities of other outstanding debts.

Fast Trees for Options with Discrete Dividends

The binomial model is a workhorse for valuing options with early exercise possibilities, including options with American or Bermudan exercise and barrier options. The procedure is straightforward when the underlying asset does not pay dividends, and it can be readily adapted to a payout at a fixed proportional rate. But discrete dividends cause trouble because they make the tree “splinter” rather than recombining at each time step. If the tree recombines, an up move followed by a down move produces the same asset price as a down move followed by an up move. But if the intermediate date corresponds to an ex-dividend date and the same fixed amount is subtracted from the up and down nodes, the tree no longer recombines at the subsequent step. The number of nodes in a non-recombining tree increases at a geometric rate, quickly leading to unreasonably long execution times. A number of alternative procedures have been developed to deal with this problem. In this article, Areal and Rodrigues adopt a new approach, accepting the splintering of the binomial lattice but then applying several techniques to accelerate the calculations. They show that their procedures are faster and more accurate than the current methods that force the tree to recombine despite discrete dividend payouts. <b>TOPICS:</b>Options, quantitative methods

Regularity of the American Put option in the Black–Scholes model with general discrete dividends

We analyze the regularity of the value function and of the optimal exercise boundary of the American Put option when the underlying asset pays a discrete dividend at known times during the lifetime of the option. The ex-dividend asset price process is assumed to follow the Black–Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. This function is assumed to be non-negative, non-decreasing and with growth rate not greater than 1. We prove that the exercise boundary is continuous and that the smooth contact property holds for the value function at any time but the dividend dates. We thus extend and generalize the results obtained in Jourdain and Vellekoop (2011) [10] when the dividend function is also positive and concave. Lastly, we give conditions on the dividend function ensuring that the exercise boundary is locally monotonic in a neighborhood of the corresponding dividend date.

Stochastic Expansion for the Pricing of Call Options with Discrete Dividends

In the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black–Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are, respectively, first-, second- and third- order approximations w.r.t. the fixed part of the dividends. Using Cameron–Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black–Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that this approach yields very accurate results, in particular compared with known approximations of Bos, Gairat and Shepeleva (2003, Dealing with discrete dividends, Risk Magazine, 16, pp. 109–112) and Veiga and Wystup (2009, Closed formula for option with discrete dividends and its derivatives, Applied Mathematical Finance, 16(6), pp. 517–531), and quicker than the iterated integration procedure of Haug, Haug and Lewis (2003, Back to basics: a new approach to the discrete dividend problem, Wilmott Magazine, pp. 37–47) or than the binomial tree method of Vellekoop and Nieuwenhuis (2006, Efficient pricing of derivatives on assets with discrete dividends, Applied Mathematical Finance, 13(3), pp. 265–284).

Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics, and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterized in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is nonincreasing in a left-hand neighborhood of $t_d$ and tends to 0 as time tends to $t_d^-$ with a speed that we can characterize. When the dividend function is linear in a neighborhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighborhood of $t_d$. When it is globally linear, then the right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.

Accurate approximation formulas for stock options with discrete dividends

Pricing options on a stock that pays discrete dividends has not been satisfactorily settled in the literature. Frishling (2002) shows that there are three different models to model stock price with discrete dividends, but only one of these models is close to reality and generates consistent option prices. We follow Frishling (2002) by calling this model Model 3. Unfortunately, there is no analytical option pricing formula for Model 3, and many popular numerical methods such as trees are inefficient when used to implement Model 3. A new stock price model is proposed in this article. To guarantee that the option prices generated by this new model are close to those generated by Model 3, the distributions of the new model at exdividend dates and maturity approximate the distributions of Model 3 at those dates. To achieve this, a discrete dividend in Model 3 is replaced by a continuous dividend yield that can be represented as a function of discrete dividends and stock returns in the new model. Thus, the new model follows a lognormal diffusion process and the analytical option pricing formulas can be easily derived. Numerical experiments show that our analytical pricing formulas provide accurate pricing results.