Stock Price Process Research Articles

On Modeling Questions In Security Valuation

After mentioning some deficiencies of the standard Black‐Scholes model for the valuation of call options, we discuss discrete models which allow price changes of the underlying security at discrete time points only. It is shown that, given any distribution with a moment higher than 2, the paths of the Black‐Scholes stock price process can be approximated uniformly as closely as one wishes by discrete paths generated by this distribution. Based on this approximation, discrete‐time trading strategies are defined. Convergence (in measure and almost surely) of the corresponding financial gain processes is obtained. the results show the robustness of the Black‐Scholes model.

Perfect option hedging and the hedge ratio

This paper examines the common practice of combining stock options and the underlying stock to eliminate risk using solely the hedge ratio. It provides necessary and sufficient conditions on the stock price process for this approach to work.

Perfect option hedging and the hedge ratio

This paper examines the common practice of combining stock options and the underlying stock to eliminate risk using solely the hedge ratio. It provides necessary and sufficient conditions on the stock price process for this approach to work.

Ex-Dividend Stock Price Behavior and Arbitrage Opportunities

This paper investigates the relation between ex-dividend stock price behavior and arbitrage oppor tunities. In a continuous trading, frictionless economy, the authors demonstrate that it is possible for the ex-dividend stock price drop to differ from the dividend, and still short-term traders cannot gene rate arbitrage profits. The argument is independent of transactions c osts. The relevance of this insight to estimating the marginal tax br acket based on ex-dividend stock price drops is explored. Furthermore , this insight is also applied to the area of option pricing in which the special class of escrowed dividend stock price processes is stud ied. The authors show that most elements from this class of stock pri ce processes generate invalid option-pricing formulas. Copyright 1988 by the University of Chicago.

The American Put Option Valued Analytically

ABSTRACTAn analytic solution to the American put problem is derived herein. The hedge ratio and other derivatives of the solution are presented. The formula derived implies an exact duplicating portfolio for the American put consisting of discount bonds and stock sold short. The formula is extended to consider put options on stocks paying cash dividends. A polynomial expression is developed for evaluating these formulae. Values and hedge ratios for puts on both dividend and nondividend paying stocks are calculated, tabulated, and compared with values derived by numerical integration and binomial approximation. As with European options, evaluating an analytic formula is more efficient than approximating the stock price process or the partial differential equation by binomial or finite difference methods. Finally, applications of this American put solution are discussed.

Stochastic Portfolio Theory and Stock Market Equilibrium

HARRY MARKOWITZ [1952, 1956, 1959] developed a theory of portfolio selection based on the optimization of a quadratic function subject to linear constraints. His work led to the development of a single period equilibrium model, the SharpeLintner capital asset pricing model (CAPM) (see Sharpe [1964], Lintner [1965a, 1965b]). R. C. Merton [1973] extended the CAPM to a continuous time model using lognormal diffusion processes to represent stock price series and showed that the original conclusions continued to hold virtually without change. In this continuous time model it became possible to observe the dynamic interaction between investors' behavior and the behavior of the stocks. In fact, Rosenberg and Ohlson [1976] showed that his interaction led to internal inconsistencies in the continuous time CAPM. In this paper we analyze long term portfolio performance compatible with equilibrium constraints on global portfolio structure. We present a portfolio theory which is based on Markowitz's theory, but which emphasizes the long term performance of the portfolios in continuous time. The concept of is introduced, a quantity which measures the relative performance of a portfolio compared to that of its component stocks. The equilibrium model we present does not consider questions dealing with optimal strategies for investors, and specifically avoids all normative issues. Rather, it provides constraints on global portfolio structure based on the principle that excess growth is conserved, that the total excess growth in the market at any instant is zero. These constraints generate a distribution of portfolios in the market similar to the energy distribution in thermodynamic equilibrium. This approach provides an alternative to classical supply-demand equilibrium. We adopt some standard conventions here, but most of the results in the paper will carry over to more general settings. The assumptions we make are: (1) Each stock price follows a lognormal diffusion process with constant drift and variance parameters. The covariance parameters between stocks are constant. Stock portfolios can be represented as Ito integrals in the various stock price processes. (2) There are no transaction costs, taxes, or problems with the indivisibility of assets. (3) The number of shares of each corporation remains constant.

Hedging options in the incomplete market with stochastic volatility

We show that it is possible to avoid the discrepancies of continuous path models for stock prices and still be able to hedge options if one models the stock price process as a birth and death process. One needs the stock and another market traded derivative to hedge an option in this setting. However, unlike in continuous models, the number of extra traded derivatives required for hedging does not increase when the intensity process is stochastic. We obtain parameter estimates using Generalized Method of Moments and describe the Monte Carlo algorithm to obtain option prices. We show that one needs to use filtering equations for inference in the stochastic intensity setting. We present real data applications to study the performance of our modeling and estimation techniques.