# The impact of stochastic volatility on short-term call option prices

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## Abstract

The Black-Scholes option pricing model is widely used to calculate fundamental values of options, but has several demonstrated biases. This dissertation describes an alternative option pricing model that relaxes the non-stochastic volatility assumption for the underlying asset, then compares the pricing accuracy of the models using sixteen Dow-Jones Industrial Average firms.

The first problem occurs with estimating weekly volatilities, since stock returns are serially correlated. French, Schwert, and Stambaugh estimated serially correlated volatilities by adding a sample covariance term to the normal sample variance formula, allowing each observation period to define a covariance structure. Schwert used a two-step regression approach to estimate volatilities from one observation per period. Both of these approaches displayed problems in this study. To avoid these problems, another approach was developed that uses a correlation structure estimated over longer periods.

Once weekly volatilities were estimated, the distribution of these volatilities was considered. Weekly volatility distributions appear approximately log-normal or Gamma distributed. The latter distribution was used for these option pricing model tests.

Given stochastic volatility, the Black-Scholes model must be modified since it assumes known volatility over the life of the option being priced. The model developed in this study approximates the continuous distribution as a series of discrete non-stochastic volatilities. Assuming a single expected return yields a mixture of normal distributions defining the possible results. The appropriate option price for this mixture-of-normals model is a weighted average of the Black- Scholes option prices for each of these possible normal distributions. Thus the Black-Scholes option pricing model is a special case of the mixtures model.

The mixtures model improved implied volatilities and model prices compared to those obtained using the Black-Scholes model, with most of the improvement coming from reduced variability. Considering that options with different strike-to-stock price ratios have different implied volatilities may result in even greater improvements with less computational effort, so the mixtures model does not dominate the Black- Scholes model. However the results do support the hypothesis that stochastic volatility contributes to the short-term Black-Scholes option pricing model bias.