A discrete-state continuous-time model of financial transactions prices and times: The autoregressive conditional multinomial-autoregressive conditional duration model

Financial transaction prices typically lie on a discrete grid of values and arrive at random times. This paper proposes an econometric model with this structure. The distribution of each price change is a multinomial, conditional on past information and the time interval between the transactions. The proposed autoregressive conditional multinomial (ACM) model is not restricted to be Markov or symmetric in response to shocks; however, such restrictions can be imposed. The duration between trades is modeled as an autoregressive conditional duration (ACD) model following Engle and Russell (1998). Maximum likelihood estimation and testing procedures are developed. The model is estimated with 12 months of tick data on a moderately frequently traded NYSE stock, Airgas. The preferred model is estimated, with three lags for the ACM model and two lags for the ACD model. Both price returns and squared returns influence future durations and present and past durations affect price movements. The model exhibits reversals in transaction prices in the short run due to bid-ask bounce and clustering of large moves of either sign in the longer run. Evidence of symmetry in the dynamics of prices is seen, but the response to durations is clearly nonsymmetric. It is found that the volatility per second of trades is highest for short-duration trades and that expected returns are lower for longer-duration trades.