Tests of Two Models for Valuing Call Options on Stocks with Dividends

Roll has recently formulated an option pricing model which allows dividend payments on the underlying stock. This paper compares the performance of the exact Roll model with a modified, but inexact, Black-Scholes model. The results indicate that the Roll model prices are significantly closer to actual market prices.     

Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978

The tests reported here differ in several ways from those of most other papers testing option pricing models: an extremely large sample of observations of both trades and bid-ask quotes is examined, careful consideration is given to discarding misleading records, nonparametric rather than parametric statistical tests are used, reported results are not sensitive to measurement of stock volatility, special care is taken to incorporate the effects of dividends and early exercise, a simple method is developed to test several option pricing formulas simultaneously, and the statistical significance and consistency across subsamples of the most important reported results are unusually high. The three key results are: (1) short-maturity out-of-the-money calls are priced significantly higher relative to other calls than the Black-Scholes model would predict, (2) striking price biases relative to the Black-Scholes model are also statistically significant but have reversed themselves after long periods of time, and (3) no single option pricing model currently developed seems likely to explain this reversal.     

MACROECONOMIC FACTORS, THE APT AND THE UK STOCKMARKET

This paper examines the macroeconomic sources of risk priced in the UK stockmarket between 1983 and 1990 using monthly data on 840 stocks to form both beta-sorted and market value sorted portfolios using the methodology proposed by Chen, Roll and Ross (1986) and Chan, Chen and Hsieh (1985) for the US. We find that several intuitively plausible macroeconomic variables were priced over this period using the beta sorted portfolios and that once these variables are included there is little role for the return on the market. However, when the market value sorted portfolios were used only inflation and a measure of equity market 'expense' relative to gilts was priced; furthermore with the market value sorted portfolios a role for the market return was found.     

Predicting Volatility in the Foreign Exchange Market

Measures of volatility implied in option prices are widely believed to be the best available volatility forecasts. In this article, we examine the information content and predictive power of implied standard deviations (ISDs) derived from Chicago Mercantile Exchange options on foreign currency futures. The article finds that statistical time-series models, even when given the advantage of “ex post” parameter estimates, are outperformed by ISDs. ISDs, however, also appear to be biased volatility forecasts. Using simulations to investigate the robustness of these results, the article finds that measurement errors and statistical problems can substantially distort inferences. Even accounting for these, however, ISDs appear to be too variable relative to future volatility.     

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.     

Option values under stochastic volatility: Theory and empirical estimates

This paper numerically solves the call option valuation problem given a fairly general continuous stochastic process for return volatility. Statistical estimators for volatility process parameters are derived, and parameter estimates are calculated for several individual stocks and indices. The resulting estimated option values do not differ dramatically from Black-Scholes values in most cases, although there is some evidence that for longer-maturity index options, Black-Scholes overvalues out-of-the-money calls in relation to in-the-money calls.     

On Jumps in Common Stock Prices and Their Impact on Call Option Pricing

The Black-Scholes call option pricing model exhibits systematic empirical biases. The Merton call option pricing model, which explicitly admits jumps in the underlying security return process, may potentially eliminate these biases. We provide statistical evidence consistent with the existence of lognormally distributed jumps in a majority of the daily returns of a sample of NYSE listed common stocks. However, we find no operationally significant differences between the Black-Scholes and Merton model prices of the call options written on the sampled common stocks.     

Randomization and the American put

While American calls on non-dividend-paying stocks may be valuedas European, there is no completely explicit exact solutionfor the values of American puts. We use a technique called randomizationto value American puts and calls on dividend-paying stocks.This technique yields a new semiexplicit approximation for Americanoption values in the Black-Scholes model. Numerical resultsindicate that the approximation is both accurate and computationallyefficient.     

An Asymptotic Expansion Approach to Pricing Financial Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applicable to a wide range of valuation problems including contingent claims associated with stocks, foreign exchange rates, the term structure of interest rates, and even their combinations. We illustrate our method by discussing the Black-Scholes economy when the underlying asset prices follow the continuous diffusion processes, which are not necessarily time-homogeneous. The standard Black-Scholes model on stocks and the Cox-Ingersoll-Ross model on the spot interest rate are simple examples. Then we shall give a series of examples on the valuation formulae including plain vanilla options, average options, and other contingent claims. We shall also give some numerical evidence of the accuracy of the approximations we have obtained for practical purposes. Our approach can be rigorously justified by an infinite dimensional mathematics, the Malliavin-Watanabe-Yoshida theory recently developed in stochastic analysis.     

A contingent-claims valuation of convertible securities

This paper examines the pricing of convertible bonds and preferred stocks. The optimal policies for call and conversion of these securities are determined via the criterion of dominance. The techniques underlying the Black-Scholes Option Model are used to price convertible securities as contingent claims on the firm as a whole.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

VOLUME AND R2: A FIRST LOOK

Roll (1988) reports that when days on which public announcements occur are excluded from a regression of stock returns on market returns, the R²s are largely unaffected. To explain his findings, Roll suggests that much of the firm-specific movements in common stocks may be a result of private information or occasional trading frenzy. As a test of Roll's conjecture, volume is used in this study as a proxy to capture the impact of firm-specific information and irrational trading. If Roll's conjecture is correct, the R² should rise when high-volume days are excluded from a regression of stock returns on market returns. The results presented here are consistent with that prediction, but they are not strong.     

Speed of convergence to market efficiency: The role of ECNs

Chordia, Roll and Subrahmanyam (2005, CRS) estimate the speed of convergence to market efficiency based on short-horizon return predictability of the 150 largest NYSE firms. We extend CRS to a broad panel of NYSE stocks and are the first to examine the relation between electronic communication networks (ECNs) and the corresponding informational efficiency of prices. Overall, we confirm CRS's result that price adjustments to new information occur on average within 5–15 min for large NYSE stocks. We further show that it takes about 20 min longer for smaller firms to incorporate information into prices. Most importantly, we demonstrate that the speed of convergence to market efficiency is significantly related to the type of trading platform where orders are executed, even after controlling for relative order flows, trading costs, volatility, informational effects, trading conditions, market quality, institutional trading activity, and other firm-specific characteristics. Our findings provide direct answers and insights to issues raised in a recent SEC concept release document.     

A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks

We propose a nonparametric method for estimating the pricing formula of a derivative asset using learning networks. Although not a substitute for the more traditional arbitrage-based pricing formulas, network-pricing formulas may be more accurate and computationally more efficient alternatives when the underlying asset's price dynamics are unknown, or when the pricing equation associated with the no-arbitrage condition cannot be solved analytically. To assess the potential value of network pricing formulas, we simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis function networks, multilayer perceptron networks, and projection pursuit. To illustrate the practical relevance of our network pricing approach, we apply it to the pricing and delta-hedging of S&P 500 futures options from 1987 to 1991.     

Over-the-Counter Option Market Dividend Protection and “Biases” in the Black-Scholes Model: A Note

Most options are traded over-the-counter (OTC) and are dividend “protected;” the exercise price decreases on the ex date by an amount equal to the dividend. This protection completely inhibits the early exercise of American call options. Nevertheless, OTC-protected options have market values which differ systematically from Black-Scholes values for European options on non-dividend paying stocks. The pricing difference is related to both the variance of the underlying stock return and to time until expiration of the option, but it is quite small in dollar amount.     

Asset Pricing and Expected Inflation

This paper provides an equilibrium model in which expected real returns on common stocks are negatively related to expected inflation and money growth. It is shown that the fall in real wealth associated with an increase in expected inflation decreases the real rate of interest and the expected real rate of return of the market portfolio. The expected real rate of return of the market portfolio falls less, for a given increase in expected inflation, when the increase in expected inflation is caused by an increase in money growth rather than by a worsening of the investment opportunity set. The model has empirical implications for the effect of a change in expected inflation on the cross-sectional distribution of asset returns and can help to understand why assets whose return covaries positively with expected inflation may have lower expected returns. The model also agrees with explanations advanced by Fama [5] and Geske and Roll [10] for the negative relation between stock returns and inflation.     

Arbitrage,Continuous Trading,and Margin Requirements

This paper studies the impact that margin requirements have on both the existence of arbitrage opportunities and the valuation of call options. In the context of the Black-Scholes economy, margin restrictions are shown to exclude continuous-trading arbitrage opportunities and, with two additional hypotheses, still to allow the Black-Scholes call model to apply. The Black-Scholes economy consists of a continuously traded stock with a price process that follows a geometric Brownian motion and a continuously traded bond with a price process that is deterministic.     

Volatility Implied by Option Prices: The Case of Takeover Bids

We model the effect of an impending share price jump on the implied standard deviation (ISD) of a company's options, testing the model by investigating its predictive ability for ISDs of companies subject to a takeover bid. Our model fits the observed ISDs well for all but certain deep in-the-money options. However, the model demonstrates that a discontinuity in the relationship between moneyness and the ISD both explains the combination of high and zero ISDs exhibited by these options, and impairs the predictive power of the model at these levels of moneyness.     

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.