Stochastic Dividend Research Articles

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Optimal investment and consumption with stochastic dividends

AbstractWe use the statistical model of bandit processes to formulate and solve two kinds of optimal investment and consumption problems. The payoffs from the investment are dividend payments with fixed return rates, but the payment frequency is stochastic following a Poisson distribution. The financial market consists of assets which follow Poisson distributions with known or unknown intensity rates. Two kinds of consumption patterns are defined and the optimality of the myopic strategy, the Gittins index strategy, and the play‐the‐winner strategy are discussed. Copyright © 2009 John Wiley & Sons, Ltd.

A note on calibration of Markov processes

AbstractUsing Malliavin calculus, we derive under the forward probability measure Ito's formula for anticipating processes in the multi‐dimensional case. We use it when the instantaneous volatility, the spot rate and the dividend yield of the stock price are Markov processes to approximate solutions of the Black type to the prices of call options as well as binary options. The stochastic dividend yield makes it possible to consider the pricing of contingent claims not only in the equity market but also in the commodities, fixed income and forex markets. We then derive the Greeks of the model, which will be used for the risk management of the exotic products, and approximate the implied volatility surface induced by the model. Copyright © 2010 Wilmott Magazine Ltd.

A REGULARIZED FOURIER TRANSFORM APPROACH FOR VALUING OPTIONS UNDER STOCHASTIC DIVIDEND YIELDS

This paper studies the option pricing problem in a class of models in which dividend yields follow a time-homogeneous diffusion. Within this framework, we develop a new approach for valuing options based on the use of a regularized Fourier transform. We derive a pricing formula for European options which gives the option price in the form of an inverse Fourier transform and propose two methods for numerically implementing this formula. As an application of this pricing approach, we introduce the Ornstein-Uhlenbeck and the square-root dividend yield models in which we explicitly solve the pricing problem for European options. Finally we highlight the main effects of a stochastic dividend yield on option prices.

Analytical Formulas for Local Volatility Model with Stochastic Rates

This paper presents new approximation formulae of European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1275872 for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data.

Analytical formulas for a local volatility model with stochastic rates

This paper presents new approximation formulae for European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models [Int. J. Theor. Appl. Finance, 2010, 13(4), 603–634] for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data.

A Note on Calibration of Markov Processes

Using Malliavin calculus, we derive under the forward probability measure Ito's formula for anticipating processes in the multi-dimensional case. We use it when the instantaneous volatility, the spot rate and the dividend yield of the stock price are Markov processes to approximate solutions of the Black type to the prices of call options as well as binary options. The stochastic dividend yield makes it possible to consider the pricing of contingent claims not only in the equity market but also in the commodities, fixed income and forex markets. We then derive the Greeks of the model, which will be used for the risk management of the exotic products, and approximate the implied volatility surface induced by the model.

Notes On Stochastic Rates, Dividends, Volatilities in an Equity Framework

Part 1 - Black-Scholes formulas are consistent with a diffusion in the respective forward neutral probability.Part 2 - Mixing Forward Neutral Probabilities for different maturities, we construct a probability more natural for non european payoffs, different from the Risk Neutral measure, that we call the Pricing Probability.Part 3 - Writing the Radon-Nikodym derivative for changing measure to the Pricing Probability, and the modified HJM relationship.Part 4 - We use a simple Ho & Lee model to get a sense of how the path-dependent hybrid terms impact the Stock diffusions. Due to hybrid effects, BS vols term structure is usually increasing.Part 5 - The Pricing Probability seems to be the natural Probability to be used to price Equity-Linked Structures with any kind of Hybrid Risks - ie all non Strictly European Structures. The alternative - working in the Risk Neutral Probability - would necessitate to recalibrate all volatility, correlation parameters from the Black / Black Scholes world (Forward Neutral Probabilities) to the Risk Neutral world.Part 6 - Some examples of Interest Rate models.Part 7 - Stochastic Dividends diffusions. Ho & Lee example.Part 8 - Cross-Currency framework. Ho & Lee example.Part 9 - Risky Curve Diffusions in the Survival Neutral Probabilities.Analogy with the Currency Quanto World.A quick look at Counterparty Risk and CVA adjustments.Part 10 - Multi-Underlying Framework. Ho & Lee example. Due to hybrid effects, BS correlations term structure is usually increasing.Part 11 - Stochastic Volatility Framework.A statistical bias between vol of realized vols and vol of implied vols.Variance Swaps on Spots, Variances Swaps on Forwards, European Options.Part 12 - Natural parametrizations of volsurfaces coherent with vol dynamics and risks, based on vol, volvol and correl spot/vol breakevens.Historical backtests of implied vs realized breakevens.Generalization with 6 parameters per Maturity.Part 13 - Correlation Skew and Stochastic Volatility.We show that we can calibrate a basket skew independently of individual skews without introducing a spot-dependent correlation: this is typically the case when individual vols move more with the basket spot than with individual spots.

Pricing American Call Options Under the Assumption of Stochastic Dividends - An Application of the Korn-Rogers-Model

In nancial mathematics stock prices are usually modelled directly as a result of supply and demand and under the assumption that dividends are paid continuously. In contrast economic theory gives us the dividend discount model assuming that the stock price equals the present value of its future dividends. These two models need not to contradict each other - in their paper Korn and Rogers (2005) introduce a general dividend model preserving the stock price to follow a stochastic process and to be equal to the sum of all its discounted dividends. In this paper we specify the model of Korn and Rogers in a Black-Scholes framework in order to derive a closed-form solution for the pricing of American Call options under the assumption of a known next dividend followed by several stochastic dividend payments during the option's time to maturity.

International asset market, nonconvergence, and endogenous fluctuations

We develop an overlapping generations model with re-tradeable paper assets and capital accumulation to analyze the interaction between the real economy and an international asset market. The world consists of two homogeneous countries, which differ only in their initial levels of capital. Consumers who live for two periods transfer wealth over time and across countries by holding international mutual funds which pay stochastic dividends. The optimal portfolio decisions of consumers do not necessarily induce convergence of incomes between the two countries. Moreover, interaction through the asset market induces endogenous fluctuation of capital flows between the rich and the poor country.

The Tax-CAPM With Stochastic Dividends (Das Tax-CAPM Mit Stochastischen Dividenden)

The IDW (Institut der Wirtschaftsprufer in Deutschland) proposes to derive the cost of capital, that are required for valuation with the DCF-approach, by using the Tax-CAPM. The original version of the model assumes deterministic dividends. To valuate stochastic cash-flows however, a Tax-CAPM with stochastic dividends is required. The paper deals with integrating stochastic dividends into the Tax-CAPM. In the first part of the paper, the equilibrium equation of the Tax-CAPM with stochastic dividends is derived for different assumptions concerning the tax system. A former version of an equilibrium equation, that does not model the influence of taxation on the beta-factor appropriately, is corrected. In the second part, an interpretation of the equilibrium equation is given. In the third part, single-period valuation equations are derived.

Stochastic dividend yields and derivatives pricing in complete markets

When an underlying yields a stochastic dividend yield, derivatives with linear payoff at their maturities that are written on this underlying have the following properties: (i) they have a unique price only if markets are complete; (ii) the dynamic strategies that replicate these contingent claims contain hedging components against the state variables in the economy; (iii) the prices of these derivatives will depend upon the dynamics of the market prices of risk even when markets are complete. Within an affine framework, we explicitly price forward and futures contracts with stochastic dividends. We also show that the quantitative impact of assuming that dividends are deterministic when they are actually stochastic is significant.

A class of solvable stochastic dividend optimization problems: on the general impact of flexibility on valuation

We consider the determination of an optimal dividend policy in the presence of cash flow uncertainty and transaction costs. We state a set of weak conditions under which the optimal dividend policy can be explicitly characterized for a broad class of diffusions modelling the underlying cash flow dynamics and demonstrate that increased dividend policy flexibility does not only increase the maximal expected cumulative present value of the future dividends, it also increases the rate at which this value grows (i.e. Tobin’s marginal q). We also prove that increased transaction costs result into larger but less frequent dividend payments.

Black-Scholes-Merton revisited under stochastic dividend yields

European options are priced in a framework à la Black-Scholes-Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean–reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well-known properties of the Black-Scholes-Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black-Scholes-Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703–732, 2006

Quadratic hedging for asset derivatives with discrete stochastic dividends

In this paper we analyze the effect of discrete stochastic dividends on the pricing and hedging of contingent claims, formulating the No Arbitrage condition without requiring the continuity of the implied gain process. We allow the stock jump dependence on an additional random source, still preserving the positivity of the stock value at the ex-dividend date. We characterize all the equivalent martingale measures and analyze the quadratic hedging approaches, as local risk minimizing and mean variance hedging.

Stochastic Dividend Yields and Derivatives Pricing in Complete Markets

Stochastic Dividend Yields and Derivatives Pricing in Complete Markets

Black and Scholes Revisited Under Stochastic Dividend Yield

Black and Scholes Revisited Under Stochastic Dividend Yield

On Trees and Logs

In this paper we critically examine the main workhorse model in asset pricing theory, the Lucas (1978) tree model (LT-Model), extended to include heterogeneous agents and multiple goods, and contrast it to the benchmark model in financial equilibrium theory, the real assets model (RA-Model). Households in the LT-Model trade goods together with claims to Lucas trees (exogenous stochastic dividend streams specified in terms of a particular good) and long-lived, zero-net-supply real bonds, and are endowed with share portfolios. The RA-Model is quite similar to the LT-Model except that the only claims traded there are zero-net-supply assets paying out in terms of commodity bundles (real assets) and households' endowments are in terms of commodity bundles as well. At the outset, one would expect the two models to deliver similar implications since the LT-Model can be transformed into a special case of the RA-Model. We demonstrate that this is simply not correct: results obtained in the context of the LT-Model can be strikingly different from those in the RA-Model. Indeed, specializing households' preferences to be additively separable (over time) as well as log-linear, we show that for a large set of initial portfolios the LT-Model -- even with potentially complete financial markets -- admits a peculiar financial equilibrium (PFE) in which there is no trade on the bond market after the initial period, while the stock market is completely degenerate, in the sense that all stocks offer exactly the same investment opportunity -- and yet, allocation is Pareto optimal. We then thoroughly investigate why the LT-Model is so much at variance with the RA-Model, and also completely characterize the properties of the set of PFE, which turn out to be the only kind of equilibria occurring in this model. We also find that when a PFE exists, either (i) it is unique, or (ii) there is a continuum of equilibria: in fact, every Pareto optimal allocation is supported as a PFE. Finally, we show that most of our results continue to hold true in the presence of various types of restrictions on transactions in financial markets. Portfolio constraints however may give rise other types of equilibria, in addition to PFE. While our analysis is carried out in the framework of the traditional two-period Arrow-Debreu-McKenzie pure exchange model with uncertainty (encompassing, in particular, many types of contingent commodities), we show that most of our results hold for the analogous continuous-time martingale model of asset pricing. Financial Equilibrium, Nonuniqueness of Equilibria, Portfolio Constraints

Quadratic Hedging for Asset Derivatives with Discrete Stochastic Dividends

In this paper we analyze the effect of discrete stochastic dividends on the pricing and hedging of contingent claims, formulating the No Arbitrage condition without requiring the continuity of the implied gain process. We allow the stock jump dependence on an additional random source, still preserving the positivity of the stock value at the ex dividend date. We characterize all the equivalent martingale measures and analyze the quadratic hedging approaches, as local risk minimizing and mean variance hedging.

European Option Pricing with Discrete Stochastic Dividends

The original Black-Scholes model prices options on a non-dividend paying stock, but of course, most actual stocks pay dividends. The two standard ways of adjusting for dividends are either to assume future dividends are known with certainty and subtract their present value from the current stock price, or to assume the dividend is paid continuously as a constant proportion of the stock price (making the dividend amount stochastic). This article shows that even when dividends are discrete and stochastic, the first type of dividend correction can still work. The key is that the current market values of the stochastic future dividends must be subtracted from the initial stock price. Chance, Kumar, and Rich then discuss ways in which the necessary market values might be obtained.