Topic

# Volatility smile

About: Volatility smile is a(n) research topic. Over the lifetime, 8702 publication(s) have been published within this topic receiving 318440 citation(s).

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Abstract: This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

9,393 citations

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Abstract: 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 BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

7,295 citations

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Abstract: We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH-M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance. Using the modified GARCH-M model, we also show that monthly conditional volatility may not be as persistent as was thought. Positive unanticipated returns appear to result in a downward revision of the conditional volatility whereas negative unanticipated returns result in an upward revision of conditional volatility. THE TRADEOFF BETWEEN RISK and return has long been an important topic in asset valuation research. Most of this research has examined the tradeoff between risk and return among different securities within a given time period. The intertemporal relation between risk and return has been examined by several authors-Fama and Schwert (1977), French, Schwert, and Stambaugh (1987), Harvey (1989), Campbell and Hentschel (1992), Nelson (1991), and Chan, Karolyi, and Stulz (1992), to name a few. This paper extends that research.

7,201 citations

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Abstract: One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. ONE OPTION-PRICING PROBLEM that has hitherto remained unsolved is the pricing of a European call on a stock that has a stochastic volatility. From the work of Merton [12], Garman [6], and Cox, Ingersoll, and Ross [3], the differential equation that the option must satisfy is known. The solution of this differential equation is independent of risk preferences if (a) the volatility is a traded asset or (b) the volatility is uncorrelated with aggregate consumption. If either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way. This paper produces a solution in series form for the situation in which the stock price is instantaneously uncorrelated with the volatility. We do not assume that the volatility is a traded asset. Also, a constant correlation between the instantaneous rate of change of the volatility and the rate of change of aggregate consumption can be accommodated. The option price is lower than the BlackScholes (B-S) [1] price when the option is close to being at the money and higher when it is deep in or deep out of the money. The exercise prices for which overpricing by B-S takes place are within about ten percent of the security price. This is the range of exercise prices over which most option trading takes place, so we may, in general, expect the B-S price to overprice options. This effect is exaggerated as the time to maturity increases. One of the most surprising implications of this is that, if the B-S equation is used to determine the implied volatility of a near-the-money option, the longer the time to maturity the lower the implied volatility. Numerical solutions for the case in which the volatility is correlated with the stock price are also examined. The stochastic volatility problem has been examined by Merton [13], Geske [7], Johnson [10], Johnson and Shanno [11], Eisenberg [5], Wiggins [16], and

4,191 citations

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Abstract: A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex-post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, we show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex-post interdaily volatility measurements based on high-frequency intradaily data are also discussed.

2,901 citations