Stopping Times Related to Trading Strategies

by Abramov, Vilen

We use CUSUM [1] procedure to analyze trading the line strategy [2]. Closed form expressions concerning probabilistic characteristics of the CUSUM stopping time and stopped process were obtained in discrete time setting for a wide class of processes. This class of discrete processes was recently defined by K. M. Khan and R. A. Khan [3]. In continuous time, the CUSUM procedure applied to the processes driven by a particular stochastic differential equation was studied [6]. As a result the joint Laplace transform of the maximum process and CUSUM stopping time was derived. Finally, the trading the line strategy was studied for the process driven by the fractional Brownian motion. As in regular Brownian motion case, the Laplace transform was linked to the partial differential equation. Although the lack of optional sampling theorem in this case prevents us from getting a closed form expression, the structure of the Laplace transform is derived. By using these results we point some of the subtle features of the trading the line strategy [7]. References [1] Page, E. S., Continuous Inspection Schemes, Biometrika, 41, (1954), pp. 100-115 [2] Katz, J. O., and McCormick, D. L. The Encyclopedia of Trading Strategies, McGraw-Hill, 2000 [3] Khan, R. A. and Khan, K. M., On the Use of the SPRT in Determining the Properties of Some CUSUM Procedures, Sequentional Analysis, 23, 3 (2004), pp. 1-24. [4] Glynn, P. W., Iglehart, D. L., Trading Securities Using Trailing Stops, Management Science, 41, 6 (1995), pp. 1096-1106 [5] Hadjiliadis, O., Ve?ce?r, J., Drawdowns Preceding Rallies in the Brownian Motion Model, Quantitative Finance, 6, 5 (2006), pp. 403-409 [6] Lehoczky, J. P., Formulas for Stopped Diffusion Processes with Stopping Times Based on the Maximum, The Annals of Probability, Vol. 5, No. 4, (1977), pp. 601-607 [7] Abramov, V., Khan, M. K., Khan, R. A., A Probabilistic Analysis of Trading the Line Strategy, to appear in the Journal of Quantitative Finance