I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Many plaudits have been aptly used to describe Black and Scholes ’ (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original Black-Scholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns I thank Hans Knoch for computational assistance. I am grateful for the suggestions of Hyeng Keun (the referee) and for comments by participants

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.

Journal Article A Closed-Form GARCH Option Valuation Model Get access Steven L. Heston, Steven L. Heston Goldman Sachs & Company Search for other works by this author on: Oxford Academic Google Scholar Saikat Nandi Saikat Nandi Research Department, Federal Reserve Bank of Atlanta Address all correspondence to Saikat Nandi, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, N.W, Atlanta, GA 30303, or e-mail: [email protected]. Search for other works by this author on: Oxford Academic Google Scholar The Review of Financial Studies, Volume 13, Issue 3, July 2000, Pages 585–625, https://doi.org/10.1093/rfs/13.3.585 Published: 15 June 2015

The Black-Scholes-Merton option valuation method involves deriving and solving a partial differential equation (PDE). But this method can generate multiple values for an option. We provide new solutions for the Cox-Ingersoll-Ross (CIR) term structure model, the constant elasticity of variance (CEV) model, and the Heston stochastic volatility model. Multiple solutions reflect asset pricing bubbles, dominated investments, and (possibly infeasible) arbitrages. We provide conditions to rule out bubbles on underlying prices. If they are not satisfied, put-call parity might not hold, American calls have no optimal exercise policy, and lookback calls have infinite value. We clarify a longstanding conjecture of Cox, Ingersoll, and Ross.

This paper studies seasonal predictability in the cross section of international stock returns. Stocks that outperform the domestic market in a particular month continue to outperform the domestic market in that same calendar month for up to 5 years. The pattern appears in Canada, Japan, and 12 European countries. Global trading strategies based on seasonal predictability outperform similar nonseasonal strategies by over 1% per month. Abnormal seasonal returns remain after controlling for size, beta, and value, using global or local risk factors. In addition, the strategies are not highly correlated across countries. This suggests they do not reflect return premiums for systematic global risk.

We examine the influence of industrial structure on the cross-sectional volatility and correlation structure of country index returns for 12 European countries between 1978 and 1992. We find that industrial structure explains very little of the cross-sectional difference in country return volatility, and that the low correlation between country indices is almost completely due to country-specific sources of return variation. Diversification across countries within an industry is a much more effective tool for risk reduction than industry diversification within a country.

We develop a model-free measure of the variance premium by constructing option portfolios whose returns are highly correlated with realized stock variance. This effectively decomposes returns into realized variance minus implied variance. We apply this decomposition to document a novel quarterly cross-sectional continuation pattern in both realized variance and implied variance of individual stocks. Implied variance underanticipates the seasonality of realized variance, so options that performed well at quarterly lags continue to earn high returns in the future. Quarterly periodicity in realized stock variance only occurs on days with analyst earning revisions, suggesting an informational channel for this pattern.

This paper investigates the performance of option investments across different stocks by computing monthly returns on at‐the‐money straddles on individual equities. We find that options with high historical returns continue to significantly outperform options with low historical returns over horizons ranging from 6 to 36 months. This phenomenon is robust to including out‐of‐the‐money options or delta‐hedging the returns. Unlike stock momentum, option return continuation is not followed by long‐run reversal. Significant returns remain after factor risk adjustment and after controlling for implied volatility and other characteristics. Across stocks, trading costs are unrelated to the magnitude of momentum profits.