Prices in futures and options markets reflect expectations about future price movements in spot markets, but these prices can also be influenced by risk premia. Futures and forward prices are sometimes interpreted as market expectations for future spot prices, and option prices are used to calculate the market’s expectations for future volatility of spot prices. Do these prices accurately reflect market expectations? The information that is reflected in futures prices and option prices is examined in this paper through a review of both the relevant analytical models and the empirical evidence. [JEL G13, G12, G14]
Derivative security markets have experienced phenomenal growth in recent years. A wide variety of options and futures contracts are traded on stock indices, bonds, interest rates, foreign currencies, gold, oil, and numerous commodities. An important issue for financial economists and market analysts is the information content of these prices. Prices on traditional assets, like stocks and bonds, are determined by the discounting of expected future cash flows, and the prices reflect expectations about future events that may affect the underlying cash flows. What kind of information is reflected in the prices of derivative securities?1 Futures and forward prices are prices for future delivery of some specified asset. Do these prices reflect expectations of future prices on the asset? Option prices depend on future prices and the potential variability of those prices. Do the option prices reflect expectations of the price and its potential volatility? In this paper I examine the information content of prices in derivative security markets and the manner in which arbitrage, expectations, and risk premia influence market prices. The theoretical analysis is balanced with a review of the relevant empirical research and a presentation of some new empirical results. Because volatility of the underlying asset price plays an important role in the pricing of options, I examine the behavior of implied volatilities, which traders and market analysts compute from option prices.
In Section I, which focuses on futures and forward prices, I argue that the arbitrage relationship is so strong that these prices are determined primarily as functions of spot prices and interest rates. From this perspective, the expectations reflected in futures or forward prices are the same as those reflected in the spot asset prices. One cannot infer expected future spot prices from futures and forward prices without measuring the relevant risk premia. In Section II, which focuses on option prices, I show that the linkage between implied volatilities from the options market and expectations of future volatility is weakened by the presence of risk premia associated with volatility. There is some empirical evidence that implied volatilities are useful for forecasting future volatility. By contrast, futures and forward prices, particularly forward foreign exchange rates, do not seem to be useful as predictors of future spot prices.
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Louis O. Scott is Associate Professor of Finance at the University of Georgia, and he holds degrees from Duke University, Tulane University, and the University of Virginia. This paper was written while the author was a Visiting Scholar in the Research Department. He would like to thank Kellett Hannah for assistance with the data and computations.
These futures and options are sometimes called derivative instruments because their payoffs are derived from asset prices or economic variables.
The notable exceptions are the longer-term Eurodollar futures and swaps.
This same convention will be followed in the discussion of foreign currency options.
Examples include banks, investment houses, and futures trading firms.
No distinction is made here between real prices and nominal prices. Cox, Ingersoll, and Ross (1985b), at the end of their term structure paper, showed that the important results of their valuation model also worked if one used nominal cash flows and nominal interest rates to determine nominal asset prices. The results for futures and forward prices follow from propositions 1 and 2 in Cox, Ingersoll, and Ross (1981).
The dividends are reported in the CBOT Financial Update. To calculate the arbitrage model for the Standard and Poor (S&P) 500 futures contracts, one must collect the dividends on the 500 stocks in the index.
One basis point is equal to 0.01 percent.
I used a spectral estimator for the variance of the parameter estimates. Let X be the T × 2 matrix of observations on the two right-hand-side variables, and let x, be the vector of observations at time t. The variance matrix for the parameter estimates is T(X’X)-1f(X’X)-1 where f is 2∏ times the spectral density matrix of (xtet) evaluated at the zero frequency. To estimate the spectral density, I prewhitened the series first, and then used a smoothed periodogram estimator with a flat window. The last step is to recolor the estimate by the appropriate filter. For a description of this estimator, see Nerlove, Grether, and Carvalho (1979). Generalized least squares was not used because it is not consistent in this application.
European put options can be valued by using the following relationship known as put-call parity: call – put = S – e-r(T-t)K.
For American call options, the effects of early exercise should also be considered.
A similar model can be derived if one allows interest rates to vary, but assumes that the interest rate differential remains fixed.
The models work by allowing for variability in the bond futures price or the futures interest rate. Models for prices on bond and interest rate futures options that incorporate random interest rates can be found in Chen and Scott (1992).
In the next section, I present some regressions for volatility in foreign exchange rates, and the correlation across exchange rates is substantial.
The implied volatilities have been taken from joint research with Marc Chesney. Implied volatilities from a model that incorporates an analytic approximation for the American premium have also been used, and the results are virtually the same.