Testing Greeks and price changes in the S&P 500 options and futures contract: A regression analysis

a r t i c l e i n f o Keywords: Price-change implied volatility Implied volatility S&P 500 options Futures contracts We use a regression model to test observed price changes with Greeks as regressors. Greeks are computed using implied volatility, price-change implied volatility and historical volatility. We find sufficient evidence to reject model Greeks as unbiased responses to underlying price as well as sufficient evidence that the American version of binomial model results in biased estimates of price changes. We use options on the S&P 500 futures contracts and their underlying. We also evaluate the frequency of " wrong signs. " Call prices and their underlying move in the opposite direction almost 10 percent of the time. In recent years, capital markets have experienced an increasing use of derivative instruments. The important factors allowing for this growth in derivative markets have been both the hedging requirements of investors as well as advances in the development of valuation models. The arbitrage-free pricing model was developed by Black and Scholes, 1973. Since then, new models that allow for additional variables , such as stochastic volatility have been developed. The empirical research testing the ability of these models to correctly price derivatives is extensive. Most of the papers in the empirical literature test models using the absolute error or absolute percentage error in prices as the loss function. In this paper we present an empirical test of price changes instead of price levels. The advantage of this approach is its direct implications for delta hedging. The local price change of the option in response to a change in the underlying is captured by the first partial (" delta ") of the option with respect to the underlying and a drift component that is due to convexity (" gamma ") and time (" theta "). By allowing time change to be small but finite, we are able to separate the effects of gamma and theta. We examine price changes in S&P 500 futures options using the American version of the binomial model (ABM). The performance of the price change model depends in large measure on how the Greeks are calculated. In addition to traditional volatility measures such as implied volatility and historical volatility, we compute Greeks using a volatility implied by price changes (price-change implied volatility). To be more specific, the price-change implied volatility is the volatility that equates the observed price …