How Duration Between Trades of Underlying Securities Affects Option Prices

Cartea, Álvaro and Meyer-Brandis, Thilo (2009): How Duration Between Trades of Underlying Securities Affects Option Prices. Forthcoming in: Review of Finance

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Duration between trades, waiting-times, stochastic volatility, operational clock, transaction time, high frequency data.
• Subjects: G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

A¨ıt-Sahalia, Yacine, and Per A. Mykland, 2003, The Effects of random and discrete sampling when estimating continuous-time diffusions, Econometrica 71, 483–549.

Andersen, Torben G., 1996, Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility, The Journal of Finance 51, 169–204.

Barndorff-Nielsen, Ole E., and Neil Shephard, 2001, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, Journal of the Royal Statistical Society Series B, 167–241.

Bossaerts, Peter, Eric Ghysels, and Christian Gouri´etoux, 1997, Arbitrage-Based Pricing When Volatility is Stochastic, Social Science Working Paper 977, Caltech.

Cameron, A. Colin, and Pravin K. Trivedi, 1996, Count data models for financial data, in G.S. Maddala, and Calyampudi Rao, eds.: Handbook of Statistics (Elsevier, North-Holland, Amsterdam, ).

Carr, Peter, H´elyette Geman, Dilip Madan, and Marc Yor, 2002, The fine Structure of asset returns: an empirical investigation, Journal of Business 75, 305–332.

Carr, Peter, H´elyette Geman, Dilip B. Madan, and Marc Yor, 2003, Stochastic Volatility for L´evy Processes, Mathematical Finance 13, 345–382.

Carr, Peter, and Dilip Madan, 1999, Option Valuation Using the Fast Fourier Transform, Journal of Computational Finance 2, 61–73.

Carr, Peter, and Liuren Wu, 2003, The Finite Moment Logstable Process and Option Pricing, The Journal of Finance LVIII, 753–777.

Carr, Peter, and Liuren Wu, 2004, Time-Changes L´evy Processes and Option Pricing, Journal of Financial Economics 71, 113–141.

Clark, Peter K., 1973, A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices, Econometrica 41, 135–155.

Cox, John C., Stephen A. Ross, and Mark Rubinstein, 1979, Option pricing: A simplified approach, Journal of Financial Economics 7, 229–263.

Diamond, Douglas W., and Robert E. Verrechia, 1987, Constraints on short-selling and asset price adjustment to private information, Journal of Financial Economics 18, 277–311.

Dufour, Alfonso, and Robert F. Engle, 2000, Time and the Price Impact of a Trade, The Journal of Finance LV, 2467–2498.

Easley, David, and Maureen O’Hara, 1992, Time and the Process of Security Price Adjustment, The Journal of Finance XLVII, 577–605.

Engle, Robert F., 2000, The Econometrics of Ultra-High-Frequency Data, Econometrica 68, 1–22.

Engle, Robert F., and Jeffrey R. Russell, 1998, Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica 66, 1127–1162.

Epps, Thomas W., and Mary Lee Epps, 1976, The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis, Econometrica 44, 305–21.

Gallant, A. Ronald, Peter E. Rossi, and George Tauchen, 1992, Stock Prices and Volume, Review of Financial Studies 5, 199–242.

Jacobsen, Martin, 2006, Point Process Theory and Applications. (Birkh¨auser).

Jacod, Jean, and Albert N. Shiryaev, 2002, Limit Theorems for Stochastic Processes vol. 228 of A Series of Comprehensive Studies in Mathematics. (Springer).

Jones, Charles M., Gautam Kaul, and Marc L. Lipson, 1994, Transactions, volume and volatility, The Review of Financial Studies 7, 631–651.

Karpoff, Jonathan M., 1987, The Relation between price changes and trading volume: a survey, Journal of Financial and Quantitative Analysis 22, 109–126.

Mainardi, Francesco, Marco Raberto, Rudolf Gorenflo, and Enrico Scalas, 2000, Fractional Calculus and Continuous-time Finance II: the waiting-time distribution, Physica A 287, 469–481.

Osborne, M.F.M., 1959, Brownian Motion in the stock market, Operations Research 7, 145–173.

Podlubny, Igor, 1999, Fractional Differential Equations vol. 198 of Mathematics in Science and Engineering. (Academic Press San Diego, California) 1st edn.

Ross, Stephen A., 1989, Information and Volatility: The No-Arbitrage Martingale Approach to Timing and Resolution Irrelevancy, Journal of Finance 44, 1–17.

Tauchen, George E., and Mark Pitts, 1983, The Price Variability-Volume Relationship on Speculative Markets, Econometrica 51, 485–505.