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Appendix D  Chapter 6 Technical Topic:  Stock Price Dynamics

D.1  Stock Price Dynamics:  Certainty Model


ssume that the stock price S grows at the continuously compounded rate r  in a market with no uncertainty.  The equation describing this growth has the following form:



If you differentiate this with respect to t, you get


Thus, the certainty model assumes that the rate of change of the stock price through time is a constant proportion of the current stock price.

This expression describes the dynamic stock price process in a world with certainty.  Multiplying both sides by dt and rearranging,  we can rewrite the equation as:


In this form, the instantaneous rate of return on the stock is r.  Here, r is also called the drift rate of the stock price process.   In the technical topic Ito Processes, the dynamic stock price process presented in this current topic is extended to a world with uncertainty.  In this extension, a constant volatility term is added so that the price process is assumed to follow a specific type of diffusion process.