How Duration Between Trades of Underlying Securities Affects Option Prices*

Abstract

We propose a model for stock price dynamics that explicitly incorporates random waiting times between trades, also known as duration, and show how option prices can be alculated using this model. We use ultra-high-frequency data for blue-chip companies to motivate a particular choice of waiting-time distribution and then calibrate risk- eutral parameters from options data. We also show that the convexity commonly observed in implied volatilities may be explained by the presence of duration between trades. Furthermore, we find that, ceteris paribus, implied olatility decreases in the presence of longer durations, a result consistent with the findings of Engle (2000) and Dufour and Engle (2000) which demonstrates the relationship between levels of activity and volatility for stock prices. Finally, by directly employing information given by time-stamps of trades, our approach provides a direct link between the literature on stochastic time changes and business time (see Clark (1973)) and, at the same time, highlights the link between number and time of arrival of transactions with implied volatility and stochastic volatility models.