The Journal of Finance

Estimation Bias Induced by Discrete Security Prices

Published: 09/01/1988   |   DOI: 10.1111/j.1540-6261.1988.tb02608.x

CLIFFORD A. BALL

Commonly, equilibrium security prices are modeled by continuous‐state stochastic processes, while observed prices are rounded into discrete units. This paper models the rounding mechanism and examines the probabilistic structure of the resultant rounded process. We provide accurate and simple estimates of the inflation in estimated variance and kurtosis induced by ignoring rounding. In particular, the maximum‐likelihood estimate of security price volatility using rounded prices is developed, and a simulation analysis is performed to examine the small‐sample properties of this estimator. For many practical applications, a simple correction for rounding becomes available.

True Spreads and Equilibrium Prices

Published: 12/17/2002   |   DOI: 10.1111/0022-1082.00390

Clifford A. Ball, Tarun Chordia

Stocks and other financial assets are traded at prices that lie on a fixed grid determined by the minimum tick size. Observed prices and quoted spreads do not correspond to the equilibrium prices and true spreads that would exist in a market with no minimum tick size. Using Monte Carlo Markov Chain methods, this paper estimates the equilibrium prices and true spreads. For large stocks, most of the quoted spread is attributable to the rounding of prices and the adverse selection component is small. The true spread and the adverse selection component are greater for mid‐sized stocks.

The Stochastic Volatility of Short‐Term Interest Rates: Some International Evidence

Published: 12/17/2002   |   DOI: 10.1111/0022-1082.00191

Clifford A. Ball, Walter N. Torous

This paper estimates a stochastic volatility model of short‐term riskless interest rate dynamics. Estimated interest rate dynamics are broadly similar across a number of countries and reliable evidence of stochastic volatility is found throughout. In contrast to stock returns, interest rate volatility exhibits faster mean‐reverting behavior and innovations in interest rate volatility are negligibly correlated with innovations in interest rates. The less persistent behavior of interest rate volatility reflects the fact that interest rate dynamics are impacted by transient economic shocks such as central bank announcements and other macroeconomic news.

Futures Options and the Volatility of Futures Prices

Published: 09/01/1986   |   DOI: 10.1111/j.1540-6261.1986.tb04553.x

CLIFFORD A. BALL, WALTER N. TOROUS

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at‐the‐money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.

On Jumps in Common Stock Prices and Their Impact on Call Option Pricing

Published: 03/01/1985   |   DOI: 10.1111/j.1540-6261.1985.tb04942.x

CLIFFORD A. BALL, WALTER N. TOROUS

The Black‐Scholes call option pricing model exhibits systematic empirical biases. The Merton call option pricing model, which explicitly admits jumps in the underlying security return process, may potentially eliminate these biases. We provide statistical evidence consistent with the existence of lognormally distributed jumps in a majority of the daily returns of a sample of NYSE listed common stocks. However, we find no operationally significant differences between the Black‐Scholes and Merton model prices of the call options written on the sampled common stocks.